ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E20-3

Public Investment, National Debt, and Economic
Growth: The Role of Debt Finance under Dynamic
Inefficiency
byToshiki Tamai
Akira Kamiguchi

September 2020

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Public Investment, National Debt, and Economic Growth:
The Role of Debt Finance under Dynamic Ine¢ ciency
Toshiki Tamaiy and Akira Kamiguchiz
September 7, 2020

Abstract
Public investment is one of central issues in the dynamic analyses on …scal policy and economic
growth. Debt …nancing for public investment and its e¤ects has been recently focused because theinterest rates has been low and almost remain below economic growth rates. This paper examines
the impacts of debt-…nanced public investment subject to a simple …scal rule in an overlapping
generations model with public capital. This topic includes capital budgeting and the debt/de…cit
criterion of Maastricht treaty. We show that debt …nancing for public investment enhances economic growth if the economy is dynamically ine¢ cient and public capital has su¢ ciently large
productivity e¤ect while it reduces economic growth rates in the dynamically e¢ cient economy.
Debt and growth could have a monotonic or non-monotonic relationship, depending on steadystate interest rate, growth rate, and productivity e¤ect of public investment. The debt-growth
relations match with controversial empirical evidences. Furthermore, existing generations choose
perfect debt …nance, so-called golden rule of public …nance, if dynamic ine¢ ciency exists. In contrast, balanced budget is preferred in a dynamically e¢ cient economy with low productivity e¤ectof public capital. However, an economy with high productivity e¤ect of public capital may choose
debt …nancing. This paper contributes to the understanding of the currently focused issues ofpublic investment.
Keywords: National Debt; Public capital; Economic growth
JEL classi…cation: H54; H60; O40

We thank Real Arai, Amer Tabakovic, and participants in the 2019 International Conference on Public Economic
Theory (PET 2019) and the seminars at Kansai University and Otaru University of Commerce for their helpful comments.
This work was supported by JSPS KAKENHI Grant Number 16H03625, 18H00865, 19K13731, the Kanpo Foundation,
the Nitto Foundation, and the Zengin Foundation for Studies on Economics and Finance.
y Corresponding author: Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi,
464-8601, Japan. Tel: +81-52-789-4928. E-mail: tamai@soec.nagoya-u.ac.jp.
z Faculty of Economics, Kindai University, 3-4-1 Kowakae, Higashi-Osaka, Osaka, 577-8501, Japan. Tel: +81-6-67212332. E-mail: a-kamiguchi@eco.kindai.ac.jp1

1

Introduction

Public investment and its impact on economic performance has been long analyzed as one of core
issues in the literatures on …scal policy and economic growth. Arrow and Kurz (1970) contributed
to construct a dynamic framework of analyzing public investment and provided its comprehensive
analysis. Futagami et al. (1993) extended Arrow and Kurz’s sophisticated model by developing an
endogenous growth model with public capital after Aschauer (1989) found an empirical evidence ofthe productivity e¤ect of public capital. Empirical studies have supported the positive growth e¤ect ofpublic investment.1 However, succeeding theoretical studies have shown that debt …nancing for public
investment does not enhance in many cases.
IMF (2014, Ch.3) has recently found that public investment has short- and long-term positive
growth e¤ect and debt-…nanced public investment e¤ectively increases output growth. This …nding
suggests that “the time is right for an infrastructure push”. Blanchard (2019) argued that public
debt may have no …scal costs because safe interest rates are expected to remain below growth rates,and suggested that welfare costs through reducing private capital accumulation may be smaller thantypically assumed. The …ndings above is expressed as low …scal costs due to low interest rates lessthan growth rates. Therefore, the controversial outcomes of growth e¤ects of public investment bydebt …nancing mainly arise from the evaluation of debt …nancing costs, such as interest payments.2
Public debt costs essentially concern dynamic e¢ ciency issues. Diamond (1965) showed that private
capital can be overaccumulated and public debt can improve e¢ ciency, because debt issuance decreases
oversaving. As the counterpart of Diamond’s model, Ihori (1978) and Tirole (1985) demonstrated thatpublic debt or bubbles (non-productive assets) can improve e¢ ciency in a dynamically ine¢ cienteconomy. The dynamic ine¢ ciency may occur in the overlapping generations models. Public debt is
one of important policy instruments if the government has no other ways of solving dynamic ine¢ ciency.
Dynamic ine¢ ciency is characterized that interest rates are lower growth rates. Mishkin (1984)
presented the evidence of this phenomenon using the data of seven OECD countries from 1967 to
1979. Von Weizsäcker (2014) recently suggested that real interest rate has become negative in theOECD countries and China. Abel et al. (1989) developed a criterion for dynamic e¢ ciency in lieu
of comparing interest rates and growth rates, and found that major OECD countries are dynamicallye¢ cient. Moreover, Geerolf (2018) revisited their results, updating data and adding countries. In
contrast to Abel et al. (1989), the study clari…es that su¢ cient conditions for dynamic e¢ ciency are
not veri…ed in advanced economies and a criterion for dynamic ine¢ ciency is veri…ed in Japan andSouth Korea. In sum, dynamic ine¢ ciency is possible in practice.
Debt …nancing for public investment is related to capital budgeting.3 Particularly, the budgetary
regime is known as the “golden rule”of public …nance (GRPF). Many countries legally adopted GRPF
and its variations.4 Musgrave (1939) pioneeringly developed the analysis of the budget rule. Debt…nanced public investment in‡uences intergenerational welfare through di¤erent long-term bene…ts
and costs. Hence, debt …nancing may have an important role of improving social welfare. Indeed, Bassetto and Sargent (2006) showed that the GRPF can improve e¢ ciency in an overlapping generations
economy. Yet there is a necessity to consider a debt …nancing rule including the GRPF for identifying
1 See Bom and Ligthart (2014) for an excellent survey of the empirical analysis on the productivity e¤ect of public
capital and Agénor (2013) for a comprehensive analysis and survey of theoretical and empirical literature. Agénor andNeanidis (2015) estimated the output elasticity of public capital to be between 0.2 and 0.4 and concluded that theprevious studies may have underestimated its value.
2 Onori (2018) investigated the role of external debt based on Barro’s (1990) model. The study has shown that…nancial openness is bene…cial in high productivity and low interest rate scenario.
3 Poterba (1995) provided comparative analyses on the levels and composition of government spending in U.S. states
by focusing on budgetary schemes, such as separate budgets for capital and operating expenditures and uni…ed budgets.
The empirical …ndings revealed that states with separate capital budgets spend more on public capital projects thanstates with uni…ed budgets.
4 For example, Brazil, Costa Rica, Germany, Japan, Luxembourg, Malaysia, and the United Kingdom. The GRPF
incorporates the possibility of borrowing to …nance productive public investment that can pay for itself over the long-termwith a balanced current budget (IMF, 2014, Ch. 3).

2

growth e¤ects of debt-…nanced public investment.5
In infrastructure-led growth models without overlapping generations structure, such as Futagami etal. (1993), dynamic ine¢ ciency does not occur and the presence of debt negatively impacts economic
performance regardless of the productivity e¤ect of public capital. To address the possibility of dynamic
ine¢ ciency and debt …nancing for public investment, the overlapping generations model of endogenous
growth with public capital must be developed. In the overlapping generations models, Yakita (2008)
demonstrated the existence of an initial debt threshold for the sustainability of government budget
de…cits under the GRPF. In addition, Teles and Mussolini (2014) showed that the level of the debt-toGDP ratio negatively impacts growth e¤ect of …scal policy. They focus on debt sustainability or the
relationship between debt and economic growth.
The goal of this paper is to elucidate why debt …nancing for public investment is chosen. Thus,
we develop an overlapping generations model of endogenous growth with public capital under thegeneralized version of the GRPF to examine growth e¤ects of debt-…nanced public investment. Income
tax and public bonds are used for …nancing public investment and interest payment of public bonds.
The borrowing rule requires that issuing bonds is allowed for public investment only. Total amount ofdebt must not exceed public capital stock.
First, we show that public investment under debt …nance positively impacts economic growth in two
ways. Public investment …nanced by increased tax enhances economic growth through the productivitye¤ect of public capital. In addition, public investment positively associates with public bond issuanceunder the borrowing rule. Then, public investment …nanced by tax increases debt and raises economic
growth rate. Public investment …nanced by bonds may positively a¤ect economic growth if the interest
rate is below the growth rate. Increasing debt hampers private capital accumulation and negatively
impacts economic growth. However, if the productivity e¤ect of public capital is su¢ ciently large,
an increase in the debt dependency ratio of public investment raises economic growth rate. In either
case, excessive public investment diminishes productivity e¤ects of public capital. Therefore, thegrowth-maximizing level of public investment is veri…ed.
Second, we demonstrate that debt …nancing for public investment could be politically preferred byexisting generations whether the economy is dynamically e¢ cient or ine¢ cient. The existing generations are unwilling to increase public investment for economic growth because far future bene…ts byeconomic growth cannot improve their welfare. Thus, they aim to reduce their current burdens. People
depend on debt …nancing to maximize their lifetime income rather than economic growth rate and canignore crowding-out e¤ect on private capital accumulation. If the economy is dynamically ine¢ cient,then existing generations choose the GRPF due to absence of debt costs. Even if the economy is
dynamically e¢ cient, debt …nancing for public investment is chosen with high productivity e¤ect ofpublic capital because the welfare cost is low. However, people prefer a balanced budget scheme when
the productivity e¤ect is not large.
Finally, numerical analyses provide further insights into public investment and debt …nancing.
Under debt …nancing, public investment generates persistent economic ‡uctuations. In particular, theeconomic dynamics exhibits chaos with low total factor productivity. The equilibrium dynamics forhigh degree of debt dependency is complicated, regardless of total factor productivity. Computed
results also show that the equilibrium tax rate and degree of debt …nancing are close to those in thereal economy. Furthermore, an increase in debt-…nanced public investment enhances short- and longterm economic growth in a dynamically ine¢ cient economy. These results are consistent with certain
empirical evidence.
The remainder of this paper is organized as follows. Section 2 discusses the related literature.
Section 3 describes the basic setup of our model and characterizes the stationary equilibrium and its
transitional dynamics. Section 4 examines the growth e¤ects of debt-…nanced public investment andconsiders an endogenous determination of income tax rate and borrowing ratio. Section 5 develops
numerical simulations for additional analyses in Section 4. Finally, Section 6 concludes this paper by
5 Dombi and Dedák (2019) showed that an increase in debt-to-GDP ratio from 0 to 90 percent decreases the output by
2–3 per cent only on a steady state. They used Blanchard’s (1985) model to show this relationship when public capital
is not considered and government expenditure does not a¤ect …rm production.

3

noting possible extensions and issues for further research.

2

Related literature

Numerous studies have examined the relationship between public investment, public debt, and economic growth. The literature is divided into two categories based on research tasks. First categoryof the literature develops a dynamic analysis on the growth e¤ects of public investment …nanced bybonds. Second category of the literature focuses on debt sustainability and …scal rule. Both categories
include theoretical and empirical approaches.
Theoretical analyses in the …rst category have predicted complicated results. Without any productive government expenditure, economic growth negatively associates with public debt (e.g., Saint-Paul
1992). In contrast, Greiner and Semmler (2000) incorporated public capital into their model andshowed that increasing de…cit-…nanced public investment results in a positive growth e¤ect when thegovernment adopts less strict budgetary regimes. They also demonstrated that de…cit …nancing for
public investment tends to produce a negative growth e¤ect if interest payments are high. Ghosh andMourmouras (2004) showed that under the GRPF, less strict budgetary stance lowers optimal level of
public investment, interest rate, and growth rate.6 Using numerical analysis, Minea and Villieu (2009)
argued that the GRPF negatively impacts long-run economic growth compared with balanced-budget
rules, even though it could improve intertemporal welfare.7
Empirical evidence in the …rst category is controversial to the theoretical predictions. Some empirical studies veri…ed a non-monotonic relationship between economic growth and public debt. For
instance, Checherita-Westphal and Rother (2012) found an inverted-U shaped relation between growthand debt-to-GDP ratio from panel data over approximately 40 years in 12 euro area countries. Eberhardt and Presbitero (2015) investigated the long-term relationship between debt and growth using alarge panel data of 118 countries and showed that each country has di¤erent debt-growth interactions,
including non-monotonic relation. They also demonstrated that …nding a negative nonlinear debtgrowth relationship is extremely di¢ cult and sensitive to modeling choices and data coverage. Debt
overhang e¤ect does not arise from common speci…c debt thresholds.8
The second category of related literature includes studies on debt sustainability under various…scal rules. For instance, the No Ponzi-game condition for government budget has been examinedtheoretically and empirically (e.g., Hamilton and Flavin 1986; Blanchard et al. 1990; Bohn 1998).9
Maximum level of government debt has also been clari…ed in the overlapping generations model basedon Diamond (1965). Chalk (2000) showed that current debt stock must not be extremely large for sustainable de…cits and debt even if steady state interest rate is less than growth rate. Bräuninger (2005)
theoretically derived a maximum sustainable debt level in the overlapping generations model withAK production function. These studies considered debt sustainability for non-productive government
expenditure under speci…c …scal rules, such as a de…cit and expenditure rules.
Certain studies in the second category treat productive government expenditure, including public investment. Yakita (2008) considered debt sustainability under three …scal rules, namely, public
investment rule, de…cit rule, and allocation rule of tax revenue.10 Given that the three …scal rules
6 Several studies examined the growth e¤ects of debt-…nanced public investment and optimal public invesment policyin stochastic growth models (e.g., Tamai 2014, 2016). Growth e¤ects depend on the degree of relative risk averse (Tamai2014). The GRPF replicates optimal equilibrium in a decentralized economy (Tamai 2016).
7 Greiner (2010) showed that economic growth rate under the balanced-budget rule coincides with that of under the
GRPF if the growth rate of public debt is less than that of physical capital. Groneck (2011) demonstrated that the
GRPF positively impacts long-run growth and welfare, and mentioned that positive growth e¤ects were observed only
if public consumption expenditures were lowered in the long run.
8 Greiner (2013) and Ueshina (2018) theoretically showed the inverted U-shaped debt-growth relationship under
GRPF. However, a non-monotonic relation between debt and growth has not yet been demonstrated under more generalclass of …scal rules.
9 Numerous studies have empirically investigated this issue based on Bohn’s (1998) approach (e.g., Greiner andKauermann 2007; Greiner et al. 2007; Doi et al. 2011).
1 0 Arai (2011) examines the same issues of Yakita (2008) using the overlapping generations model with productive4

simultaneously bind, tax rate depends on the debt-to-capital ratio as a predetermined variable. It generates multidimensional economic dynamics with private capital, public capital, and public debt. The
existence of a stable steady state indicates a sustainable debt level. The e¤ects of debt-…nanced public
investment are also examined in the paper. Growth e¤ect is ambiguous though the model includes
the GRPF and productive e¤ect of public investment. In principle, the GRPF is a sustainable …scal
scheme.11 Hence, other …scal rules may interfere the stabilizing function and growth e¤ect of public
capital.
Teles and Mussolini (2014) extended Bräuninger’s model by incorporating productive government
expenditure as spending ‡ow rather than public capital accumulation. Public expenditure in theirmodel has three di¤erent growth e¤ects. An increase in public expenditure positively impacts economic
growth through its productivity e¤ect. However, it also raises interest rate and increases tax rate
determined endogenously to balance government budget under …scal rules. Hence, these two indirecte¤ects generate negative growth e¤ect. Teles and Mussolini (2014) found that higher debt-to-GDP
ratio decreases positive growth e¤ect of productive public expenditure by empirical estimation basedon their theoretical model. Their study implies that the impacts of debt costs on di¤erent generations
are important to identify the growth e¤ects of …scal policy.
In our study, we consider the e¤ects of debt-…nanced public investment on economic growth andequilibrium dynamics in the overlapping generations model. If government issues public debt for
public capital investment, then the bene…t accrues not only to current individuals but also to future
ones. Furthermore, dynamic ine¢ ciency has not been assumed in most studies on this literature. As
mentioned in the previous section, the dynamic ine¢ ciency would be the issue we really face. The
overlapping generations model with the GRPF and its variations should be used to clearly capture themechanism of debt …nanced public investment.
Recent studies in the literatures also examined a relationship between debt …nancing of public
investment and economic growth under the GRPF in the overlapping generations model. Bokan et al.
(2016) analyzed the e¤ects of demographic change on economic growth and stability under the GRPF
in Diamond’s overlapping generations model with fertility choice and death uncertainty. They focusedon pure the GRPF as the government borrows for public capital investment only. Hence, their study
do not cover variations of GRPF as part of capital budgeting and the reason why they are chosen.
Kamiguchi and Tamai (2019) examined the growth and welfare e¤ect of …scal policy under the GRPF
in Blanchard’s overlapping generations model. They showed that debt-to-GDP ratio and economic
growth rate have an inverted-U shaped relationship and that di¤erent generations prefer di¤erent tax
rates depending on their ages.
Heterogeneity of preferred policy is crucial to determining an equilibrium policy through a democratic process. With dynamic ine¢ ciency, debt-…nanced public investment may have positive growthe¤ect and be politically chosen as a preferable policy. However, the issue of dynamic ine¢ ciency andhow …scal variables relate to …scal rule determination are not covered in these studies. Our study
takes a di¤erent tack to consider the growth e¤ect of debt-…nanced pubic investment and …scal rule
determination, following the existing literature.

3

The model

Consider the overlapping generations model in which each individual lives for two periods. Time isdiscrete and time period is indexed by the subscript t. The individuals bene…t from their …rst- andsecond-period consumptions. The lifetime utility function for an individual born at period t is
Ut = log cyt + log cot+1(1)where cyt is the consumption for the young in period-t generation, cot+1 is the consumption for the oldin period-t generation, and is the subjective discount rate (0 <1).
government expenditure.
1 1 See Musgrave (1939). Kamiguchi and Tamai (2019) proved that the GRPF sats…es the No-Ponzi game condition.

5

In the …rst period, the individuals are young and supply one unit of labor inelastically. They earnfrom work and spend their earnings on private consumptions and investment (saving). In the second
period, they are old, retiring during the period, expend their savings with interests. The budget
equations arecyt = wt st ;cot+1 = (1 + rt+1 ) st ;

(2a)
(2b)

where wt is the post-tax wage rate, rt+1 is the post-tax interest rate and st is the saving.
Each individual chooses consumption levels to maximize the utility function (1) subject to (2a) and(2b) for given wt and rt+1 . Solving the optimization problem, we obtain the following saving function:st = wt ;

(3)

where
:
1+
Each …rm has an identical production technology. The production function isyt = Akt gt1

lt1

;

(4)

where yt is the output, A is the productivity parameter, is the intensity parameter (0 < < 1), kt
is the private capital, gt is the public capital, and lt is the labor. We assume that public capital islabor-augmenting and taken as given for each …rm. Pro…t maximization conditions lead to
rt = (1wt = (1

) Akt

1 1
gt

lt1

) (1

) Akt gt1

;

(5a)

lt ;

(5b)

where denotes capital depreciation rate (01).
The government issues public bonds and taxes the output to …nance government expenditures forinterest payments and public investment. The government’s budget equation becomes
bt+1 = (1 + rt ) bt + [gt+1(1) gt ]yt ;(6)where bt is the government’s debt outstanding. Note that the second term in the right-hand side ofEquation (6) stands for the investment in public capital.
This economy has two types of assets: private capital and public bonds. The clearing condition ofasset market isst = bt+1 + kt+1 :(7)
Using Equations (2a), (2b), (5a), (5b), (6), and (7), Walras’law holds.12
Suppose that the amount of bond issuance is less than public investment. This borrowing rule has
been adopted in numerous countries and regions (e.g., Germany, Japan, the United Kingdom, andsome states in the United States). This condition can be formalized as
bt+1 = gt+1 ;(8)

1 2 Walras’law can be veri…ed as

yt = cyt + cot + [kt+1= (1) w t lt(1

bt+1

= wt lt + (rt + ) kt

) kt ] + [gt+1(1) gt ]

kt+1 + (bt + kt ) (1 + rt ) + kt+1
yt :

6

(1

) kt + bt+1

(1 + rt ) bt + yt

where 2 [0; 1]. ( ; b0 ) = (0; 0) is referred as a balanced-budget scheme, whereas
the GRPF. Using Equations (4) and (8), the debt rule can be rewritten as
bt
xt
=
;
yt
A
bt+1 bt
bt+1 bt=
yt
bt= 1 is known as

(9a)
bt
;
yt

(9b)

where xt
gt =kt . For given private and public capital stocks, Equations (9a) and (9b) lead to theupper bound of the debt-to-GDP ratio and de…cit-to-GDP ratios, respectively. Hence, the debt rule
corresponds to the Maastricht criteria for the debt outstanding and …scal de…cit. Note that the criterion
levels are endogenously determined. Therefore, the variation occurs unless the ratio of public to private
capital is constant over time.
For 0< 1, Equations (3), (6), (7), and (8) lead to
gt+1 =[

kt+1 = wt(1)

xt ] yt + (11
bt+1 = (1
) (1

) (1

) gt

;

(10a)

) yt

gt+1 ;(10b)

For = 1, we have (rt + ) bt = (rt + ) gt = yt . Then, public-to-private capital ratio under GRPF
becomesxG for = 1:
xt =
(1
)
Equations (10a) and (10b) yield
xt+1 =

(1

[
) (1

) (1

(1
) xt ] A + (1
) (1
) xt
) A
f[
(1
) xt ] A + (1) (1) xt g(xt ) ;

(11)

where
(0) =

(1

) (1

) (1

0;

)

(x) = 0:The dynamic system of Equation (11) is derived from Equations (2a)–(8). Therefore, the time sequenceof xt given by Equation (11) characterizes economic dynamics with all equilibrium conditions.
The equilibrium growth path is de…ned as follows:1

De…nition 1. Equilibrium growth path is a sequence of xt , fxt gt=0 , which satis…es (11).
There may exist multiple equilibrium growth paths with di¤erent initial values and parameters. In
the long-run, the equilibrium growth paths may converge to certain stationary paths with a stationary
value of xt . Formally, we de…ne the stationary equilibrium as follows:
De…nition 2. Stationary equilibrium is a set of xt , which satis…es (11) and xt+1 = xt for t 2 N0 .
We now examine the dynamic properties of Equation (11). Di¤erentiation of Equation (11) withrespect to xt provides
0(xt ) =

(1

(1
) (1
) xt 1
) (1
) (1
) A
f[

, xt Q x
b

(1
) (1
)
(1
) A

(1
(1

1

)
)

A [1 +(xt )]
xt ] A + (1) (1) xt g

R0

1:

The properties of function shows that the graph of in the x- plane is continuous for [0; x]. Figure
1 illustrates
curve. It displays that there exists at least one stationary equilibrium. Dependingon the location of the crossing point between curve and a 45-degree line, the economy traces thedi¤erent equilibrium growth paths.
7

Figure 1.

curve and stationary equilibrium

If the stationary equilibrium is located in the left-hand side of xb, thencurve has a positiveand less than unity slope at the stationary equilibrium. Hence, the economy monotonically converges
to the unique stationary equilibrium. For instance, the balanced budget ( = 0) leads to such asituation. However, de…cit …nancing may cause di¤erent scenarios. If the stationary value of public-toprivate capital ratio, x, is larger than xb, then curve has a negative slope, resulting in an oscillation.
Oscillating convergence and periodic cycles are conceivable. Figure 1 illustrates curve with period-3
cycle, which is the source of Li-Yorke chaos. Then, the economy will not converge to stationary point
E and persistently ‡uctuates.
Under a balanced budget scheme, the economy monotonically converges to the unique stationary
equilibrium. In contrast, the economy with de…cit …nancing has economic ‡uctuations. The dynamics may exhibit oscillatory convergence or complicated periodic cycles. Thus, we have the following
proposition (See Appendix A for the proof of Proposition 1):
Proposition 1. There exists a unique stationary equilibrium. If is su¢ ciently small, then equilibrium
growth paths converges to the unique stationary equilibrium. In contrast, when is insu¢ ciently small
to be j 0 (x)j < 1, equilibrium growth paths may not reach the unique stationary equilibrium, resulting
in economic ‡uctuations. In particular, equilibrium growth paths exhibit a chaotic behavior if
<

(1

(1
) (1

) (1
)
) + (1

)

+

and

(1
) (1
)
(1
) A

1
1

>

(1

) (1

) (1

)

:

If the economy converges to the unique stationary equilibrium, then Equations (10a) and (10b)with the stationary value of xt lead to the equilibrium growth factor as follows:
gt+1
kt+1
(1
=
=
gt
kt

) (1
) Ax1
1+ x

;

(12)

where x stands for the value that x = (x) holds. Growth rate is de…ned as
1. Focusing
on a positive growth rate, we impose
1. Under a balanced budget scheme, B is de…ned as
the equilibrium growth rate with xB
fxjx = (x) ; = 0g. For the GRPF, G is de…ned as theGequilibrium growth rate with x . Similarly, all other variables with superscripts B and G are used forthe economic variables under the balanced budget and GRPF, respectively.
8

4

Fiscal policy and economic growthThis section examines the growth e¤ects of de…cit-…nanced …scal policy and the long-term relationshipbetween economic growth and government debt. We focus on a unique stable stationary equilibrium
to analyze the long-run e¤ect. The government has two policy instruments: output tax rate anddebt-to-public capital ratio (degree of debt …nancing) . We will show that these …scal parametersa¤ect economic growth and debt-to-GDP ratio through di¤erent channels.

4.1

Growth e¤ects of …scal policy

We investigate the growth e¤ects of changes in output tax rate and degree of debt …nancing. Ourmodel has two benchmarks to evaluate the growth e¤ects of the …scal policy: balanced budget ( = 0)
and the GRPF ( = 1). The e¤ects of …scal policy on the accumulation of the two capitals should
be considered to analyze growth e¤ects. Using x = (x) and Equation (11), we obtain the following
equations (See Appendix B for the derivation and properties of Equations (13a) and (13b)):
(1 +
@x
=
@
[(1
@x
=
@

x) A + [(1) (1) A + (1 +x) A] x0 (x)]) (1 + ) x[1[1(1 + x) (r0 (x)]

)] x

R 0;

> 0;

(13a)
(13b)

where
(1
) (1
) (1
) A
f[
(1
) x] A + (1) (1) x g.
Equation (13a) shows that a rise in tax rate increases public-to-private capital ratio. Increased tax
promotes public capital investment by increased tax revenue and decreased interest payment, whereas
it hinders private capital accumulation through decreased saving. Therefore, high tax rate leads tohigh public-to-private capital ratio. Equation (13b) indicates that debt …nancing has an ambiguouse¤ect on public-to-private capital ratio. For given stock levels of the two capitals, a rise in increases
interest payment (negatively in‡uences x) and public investment (positively a¤ects x). The increase
in interest payment is the indirect (negative) e¤ect of a rise in on x, and the increase in public
investment is the direct (positive) e¤ect of a rise in on x. Depending on the two opposite e¤ects,
debt …nancing enhances or declines public capital accumulation. On the other hand, debt …nancing
negatively a¤ects private capital accumulation through the crowding out e¤ect. This e¤ect of debt
…nancing works as another indirect (negative) e¤ect of a rise in on x. Thus, debt …nancing has an
ambiguous e¤ect on x.
The positive direct e¤ect of on x may dominate over the negative indirect e¤ects depending on
the di¤erence between interest and growth rates. If interest rate is less than economic growth rate(r < , i.e., the economy is dynamically ine¢ cient), then the government obtains …nancial resources toincrease net public investment per public capital by debt …nancing because the increase in government
revenue passes increase in interest payment. The direct e¤ect of a rise in on x overweighs the indirecte¤ects. Therefore, additional debt …nancing increases public-to-private capital ratio. However, thispositive e¤ect scenario may not be realized in a dynamically e¢ cient economy (r > ). At least, a risein negatively impacts x for large .
We consider the growth e¤ects of a change in for given . Equations (12) and (13a) yield
@
@

=

1

+ (1

)

@x
x@

x
@x
:
1+ xx@(14)

The …rst and second terms in Equation (14) denote a growth e¤ect of a rise in tax rate through achange in savings. The last term in the equation represents for a negative growth e¤ect of increased
tax through the crowding out e¤ect. The crowding out e¤ect occurs because an increase in public
investment by increased tax revenue positively associates with public debt under a debt …nancing
scheme of Equation (8). A change in tax rate has direct negative and indirect positive e¤ect on
private capital accumulation through a decrease in disposable income and an increase in public capital9

accumulation, respectively. Private and public capital accumulation are sources of economic growth.
Therefore, the growth-maximizing tax rate exists in the range (0; 1).
The analysis on Equation (14) provides the following proposition (See Appendix C for the proof of
Proposition 2):
Proposition 2. Suppose thatis …xed. A growth-maximizing tax rate,?

2 (0; 1), exists.

Proposition 2 ensures the existence of a growth-maximizing tax rate. However, it does not display
its uniqueness and exact value. Nevertheless, we can verify the values in certain special cases. For
instance, ? = 1
holds when = 0. Thus, the growth-maximizing tax rate under a balanced
budget is equal to the output elasticity of public capital (e.g., Futagami et al. 1993).13 In contrast,
the growth-maximizing tax rate may di¤er from the output elasticity of public capital for > 0. Thereare certain channels to be ? < 1.14 Issuing bonds ease the needs for tax …nancing, which has
negative growth e¤ect, through raising government revenue. On the other hand, debt …nancing brings
about the crowding out e¤ect. The mixed e¤ects reduce the growth-maximizing tax rate. For example,?
<1
if = 1, which reveals that the growth-maximizing tax rate is less than the output elasticityof public capital when capital is completely depreciated in one period.
We move to the analysis of the growth e¤ect of debt …nancing. Di¤erentiating Equation (12) withrespect to derives
@
@x
@x
x
= (1
)
1+
:
(15)
@
x@
x@
1+ x
The …rst and second term in Equation (15) correspond to the growth e¤ect of debt …nancing throughsaving and crowding out e¤ect, respectively. The e¤ect of a change in on x, (13b), is a key determinantof the growth e¤ect of a change in .
In a dynamically e¢ cient economy (r > ), a rise in may decrease public-to-private capital ratioor have a weakly positive e¤ect on x. Small x leads to low wage rate, indicating small saving. Hence,a rise in engenders totally negative growth e¤ects through saving and crowding out e¤ect. On theother hand, in a dynamically ine¢ cient economy (r < ) with su¢ ciently large productivity e¤ect ofpublic capital, the result supports debt …nancing to enhance economic growth. The economic intuition
behind the result is explained as follows. Public-to-private capital ratio increases with a rise in . The
growth e¤ect through saving positively associates with public-to-private capital ratio. Hence, for small, the positive growth e¤ect of increasing saving dominates the negative growth e¤ect of crowding out.
An analysis of Equation (15) provides the following result (See Appendix D for the proof of Proposition
3):
Proposition 3. Suppose that is …xed. (i) In a dynamically ine¢ cient equilibrium, there exists a
growth-maximizing ratio of debt …nance ? 2 (0; 1) if
xB <

(1

)

B

( B+ )

rB

and

(1
G+

)
(1

G

rG

)( G

rG )

< xG :(ii) If a dynamically e¢ cient equilibrium, further debt …nancing for public investment decreases theequilibrium growth rate.
Minea and Villieu (2009) and Kamiguchi and Tamai (2019) implied that the equilibrium growthrate under the GRPF is below to the growth rate under a balanced budget. We generalize the resultof these previous studies. In the stationary equilibrium, Equation (6) becomes
(

r) =

y
g

=

R0,

R r;

1 3 Barro (1990) demonstrated that the growth-maximizing tax rate is the output elasticity of productive government

expenditure as public input.
1 4 In Blanchard’s overlapping generations model, Kamiguchi and Tamai (2019) showed that the growth-maximizing
tax rate is less than the output elasticity of public capital because of the presence of generation replacement e¤ect.

10

where is the tax revenue minus capital depreciation rate (Ax). Note that is the net
public investment and growth rate under the balanced budget for = 0 because = B holds. The
models in the previous studies have no dynamic ine¢ ciency. Similarly, we have < . The equilibrium
growth rate under debt …nancing is less than the growth rate under a balanced budget (Proposition 3(i)). In contrast, > holds. The equilibrium growth rate under debt-…nancing may be greater than
the growth rate under a balanced budget (Proposition 3 (ii)).
IMF (2014, Ch.3) found evidence from advanced and developing economies suggesting that debt…nanced public investment would have short- and long-term positive growth e¤ects. Furthermore, somestudies have argued in favor of public debt or dynamic ine¢ ciency because many countries have saving
gluts. Von Weizsäcker (2014) implied that the real rate of interest has become negative in the OECD
countries and China. Geerolf (2018) found that the criterion for dynamic e¢ ciency is unveri…ed forany advanced countries. These …ndings support the second result in Proposition 3. However, strong
productivity e¤ect is necessary to ensure a positive growth e¤ect of debt-…nanced public investment
even if the growth rate exceeds the interest rate.
Checherita-Westphal and Rother (2012) empirically found an inverted-U shaped relationship between growth and debt-to-GDP ratio. It is worthy to investigate whether such relation exists or not.
Under the debt …nancing rule (8), the debt-to-GDP ratio positively associates with public-to-private
capital ratio. Given that Equation (13a) holds, debt-to-GDP ratio increases with output tax rate in
the stationary equilibrium. By Equation (9a) and Proposition 2, if is …xed, then an inverted-U
shaped relationship exists between economic growth and debt-to-GDP ratio, in case of increasing theoutput tax rate. Several studies also showed the inverted-U shaped curve (e.g., Greiner 2010; Ueshina
2018; Kamiguchi and Tamai 2019). However, their results are limited to the GRPF. It corresponds to(8) with = 1. Our result ensures the robustness of a non-monotonic relationship between economic
growth and debt-to-GDP ratio through increasing the output tax rate. We can directly analyze therelationship between economic growth and debt …nancing.
Using Equation (9a), the response of the debt-to-GDP ratio to a change in is
@
@

log

b
y

=1+

@x
> 0:
x@

(16)

If @x=@ > 0, then Equation (16) has a positive sign. Equations (13b) and (16) and Proposition 3
yield the following result:
Proposition 4. Suppose is …xed. (i) In a dynamically ine¢ cient equilibrium, there exists (may notexist) an inverted-U shaped relationship between economic growth and debt-to-GDP ratio in case ofincreasing the degree of de…cit …nancing if the marginal e¤ ect of debt …nancing on public-to-private
capital ratio is (not) su¢ ciently large. (ii) In a dynamically e¢ cient equilibrium, the economic growthis negatively associated with the debt-to-GDP ratio in accordance with increasing the degree of de…cit…nancing.
Bokan et al. (2016) derived the growth-maximizing debt-to-GDP ratio under the GRPF in Diamond’s overlapping generations model. When = 1, we have a similar result by controlling the tax
rate for maximizing equilibrium growth rate. Furthermore, Proposition 4 implies that the GRPF is not
the only contributing factor to the inverted-U shaped curve between debt and growth. This proposition
generalizes the results derived by the previous studies in the sense that the degree of debt …nancing
is ‡exible. Depending on the magnitude relationship between steady state interest rate and growthrate, the debt-to-GDP ratio is di¤erently associated with growth rate. A non-monotonic relationshipbetween debt and growth will be observed if the economy is dynamically ine¢ cient or when the government adopts the GRPF. Reinhart and Rogo¤ (2010), Checherita-Westphal and Rother (2012), andBaum et al. (2013) found an inverted-U shaped relationship between debt and growth.
On the other hand, certain studies have produced evidence against this relationship. Herndon et al.
(2014) refuted Reinhart and Rogo¤’s (2010) result by emphasizing calculation problems. Égert (2015)
demonstrated that …nding a negative nonlinear relationship between debt-to-GDP ratio and economic
11

growth is extremely di¢ cult and is sensitive to modeling choices and data coverage. Furthermore,
Eberhardt and Presbitero (2015) found that each country has di¤erent debt-growth interactions usinga large panel data of 118 countries. The debt overhang e¤ect does not arise from common speci…cdebt thresholds. Proposition 4 implies that the countries have di¤erent levels of debt overhang point
on debt-to-GDP ratio depending on the tax rate. Whether each country is dynamically e¢ cient orine¢ cient is important to investigate the public debt-growth nexus. Thus, our results theoreticallyshow the di¢ culty of identifying the nonlinear relationship suggested by Égert (2015) and also provide
a theoretical mechanism behind the empirical evidence found by Eberhardt and Presbitero (2015).

4.2

Political determination of …scal policy

The previous subsection have examined the growth e¤ects of …scal policy and shown that de…cit…nancing has a positive growth e¤ect in a dynamically ine¢ cient economy. Future generations, including currently young generation, will bene…t from the positive growth e¤ect of debt …nancing andproductivity e¤ect of public capital. On the other hand, the currently old generation does not gain
through public investment. This subsection considers whether debt …nancing is politically chosen ornot.
For analytical tractability, = 1 is assumed in this subsection. Then, Equations (5a) and (12) leadtor = [(1)] (1) Ax1 R 0 , (1) R for = 1:
Therefore, the economy is dynamically e¢ cient ( < r) if and only if (1) <while it is
dynamically ine¢ cient ( > r) if and only if (1) > .
The government’s objective function is based on existing individuals. Speci…cally, the objectivefunction at period t is given by
Wt = log cot + log cyt + log cot+1' log (1
) + f(1 + ) log (1) + [log (1) + (1) log x]g :(17)

Equation (17) can be interpreted as a probabilistic voting model (Lindbeck and Weibull, 1987; Grossman and Helpman, 1998). The …rst-order derivatives of Equation (17) are+ (1 + 2 ) (1
@Wt
=
+
@
1
@Wt
(1
)
@x
=
:
@
x
@

)
x

@x
;
@

(18a)
(18b)

A rise in tax rate directly reduces Wt through a decrease in disposable income of the old andyoung generation and indirectly raises Wt through an increase in future disposable income of theyoung generation by public investment. Therefore, from Equation (18a), we can …nd the tax rate thatmaximizes the value of the government’s objective function (17) for 2 (0; 1). We have the following
result (See Appendix E for the proof of Proposition 5):
Proposition 5. Suppose that
economy.

is …xed. An equilibrium tax rate exists,

2 (0; 1), in the politico

Proposition 5 shows the existence of the equilibrium tax rate but does not ensure its uniquenessand provides any information about an absolute level. The equilibrium tax rate is not necessarily equal
to the growth-maximizing tax rate.
Debt-…nanced public investment is relevant to the young generation’s future disposable income
only. Hence, the sign of Equation (18b) determines that of Equation (13b). From Equations (13b) and(18b), the following proposition holds (See Appendix F for the proof of Proposition 6):

12

Proposition 6. Suppose that is …xed. (i) In a dynamically ine¢ cient economy, the GRPF, = 1,
is chosen. (ii) In a dynamically e¢ cient economy, a balanced budget scheme is preferable if the income
tax rate is su¢ ciently small. In contrast, debt …nancing is preferable and the equilibrium degree ofdebt-…nancing, , exists in the dynamically e¢ cient economy if the income tax rate is su¢ cientlyhigh.
The presence of debt in a dynamically ine¢ cient economy could improve welfare as shown in
Diamond (1965). In our model, the debt is used for …nancing productive public investment. Naturally,
the GRPF is preferred. Surprisingly, debt …nancing could be chosen in a dynamically e¢ cient economy.
Su¢ ciently large productivity e¤ect of public capital ensures low costs of debt …nancing.
Considering the future generations who do not exist currently will change the outcome concerning
the equilibrium rule. Future generations’bene…t depends on the economic growth rate because it canbe evaluated by the present value of their utility functions. The presence of a negative growth e¤ectof debt …nancing will decrease the equilibrium degree of debt …nancing. Therefore, government tend
to avoid debt …nancing if concerned future generations exist.

5

Numerical simulation

This section illustrates a numerical simulation for three issues. First, we numerically verify the existenceof chaos or economic cycles and a su¢ cient condition for stable stationary equilibrium to support
Proposition 1. Second, the income tax rate and debt …nancing ratio will be calculated from a systemcomposed of …rst-order conditions for maximizing the equilibrium growth rate or the government’s
objective function. Finally, we examine the short-term e¤ects of …scal policy to provide the further
policy insights.

5.1

Baseline parameters

We assume that private capital share is = 0:3 from Mankiw et al. (1992) for numerical simulations.15
The preference parameter is set to = 0:95. Then, the value of approximates 0:483. Based on theparameters we assumed, the saving rate to national income becomes 33.8%. Our calculations from
OECD data indicate the 5-year average rates in G5 countries: 49.1%, 50%, 47%, 36.2%, and 35.6%
in France, Germany, Japan, the United Kingdom, and the United States, respectively.16 (1)
is important to determine whether the economy is dynamically e¢ cient or not. Hence, our baseline
parameter of is set below the real values.
= 0:2 is used from OECD data. For instance, the 5-year average tax revenue excepts for social
security contributions (% of GDP) in OECD average indicates 24.9%. Calculations show 29%, 23.3%,
18.2%, 26.5%, and 19.7% in France, Germany, Japan, the United Kingdom, and the United States,
respectively.17 Furthermore, the di¤erent values of ( = 0:2, = 0:3, and = 0:4) are tested. In
practice, a typical infrastructure has a 50-year lifetime. In the overlapping generations model, one
period corresponds to 25-30 years. The realistic value of should be set to 0:5 or above.
Based on Society at Glance 2016 (OECD), OECD average ratio of the voter turnout rate for age
18-24 to age 25-50 is 0:835. The ratios of the voter turnout rate for age 16-35 to age over 55 are 0:761,
0:787, 0:535, and 0:720 in France, Germany, the United Kingdom, and the United States, respectively.
A survey report on 2017 House of Representatives general election in Japan shows that the ratio ofage 20-55 to age over 55 in Japan is 0:730. Hence, we will use = 0:5, = 0:75, and = 1.
1 5 The output elasticity of public capital has been reported in the range [0:1; 0:4]. Bom and Ligthart (2014) found thatthe average output elasticity of public capital in 68 studies is 0.106 after correcting for publication bias and its long-run
value goes up to 0.193.
1 6 These values are calculated from OECD data: Household Spending (doi: 10.1787/b5f46047-en, Accessed on 24 May2020) and Gross National Income (doi: 10.1787/8a36773a-en, Accessed on 24 May 2020).
1 7 Calculations are made from OECD data of Tax Revenue (doi: 10.1787/d98b8cf5-en, Accessed on 24 May 2020).

13

(a)

= 0:2

(b)

= 0:3

(c)

= 0:4

Figure 2. Bifurcation diagrams and Lyapunov exponent ( = 0)

(a)

= 0:2

(b)

= 0:3

(c)

= 0:4

Figure 3. Bifurcation diagrams and Lyapunov exponent ( = 0:5)

14

5.2

Equilibrium dynamics

Proposition 1 shows that a small value of raises the possibility of chaos. Hence, we test two cases:= 0 (as an extreme case) and = 0:5 (as a realistic case).18 Our computations illustrate a bifurcation
diagram to show whether numerous cycles exist. The values of x after 10000 steps are plotted in the
diagrams for the di¤erent ranges of A with di¤erent widths. Furthermore, using values of x, we
calculate Lyapunov exponent such that
T

1

1 X
log j 0 (xt )j :T !1 T
t=0= lim

If chaos exists, then the Lyapunov exponent is positive. Hence, it is helpful to distinguish between
chaotic and non-chaotic regime.
No depreciation ( = 0). Figure 2 displays bifurcation diagrams with the graphs of the Lyapunov
exponent for = 0:2, = 0:3, and = 0:4. In Figure 2, Panels (a) and (b) show that values of x aredensely distributed in a certain range. Furthermore, the graphs of the Lyapunov exponent in panel(a) and (b) of Figure 2 are positive in most parts of the domains. Therefore, the economic dynamics
exhibits chaos for a small value of A with = 0:2 and = 0:3. In panel (c) of Figure 2, values of x
split into two bands for = 0:4. The corresponding values of the Lyapunov exponent are negative inthe domain. Hence, there exists non-chaotic economic ‡uctuations for A 2 [6:324; 7:4].
The computed results provide examples of chaos for a small value of A. On the other hand, theproductivity parameter A should be necessarily large to ensure a positive growth rate, depending onother parameters and endogenous variable x. For example, A must be at least 5 to exhibit a positive
growth rate for the value of x around unity. When = 0:2 and = 0:3, numerical simulations show
the convergence of the dynamic sequence of x to the stationary value. On the contrary, a period-4
cycle exists for = 0:4 and A 2 [7:4; 10]. This result implies that debt …nancing tends to make cyclicalequilibrium dynamics.
Imperfect depreciation ( = 0:5). The possibility of chaos when = 0:5 is rarer than that when
= 0. Hereafter, we focus on a realistic situation with a large value of A to ensure a positive growthrate. In Panels (a) and (b) of Figure 3 show that bifurcation does not exist and the Lyapunov exponents
have negative values for A 2 [5; 10]. These results show that xt converges to a unique stationary value.
In contrast, persistent economic ‡uctuations occur in panel (c) of Figure 3. The complicated dynamics
is not chaos because the corresponding Lyapunov exponents are negative. Numerical analyses imply
that larger dependency on debt …nancing causes economic cycles even if the economy shows a positive
growth trend.

5.3

Simultaneous determination of …scal policy variables

Propositions 2–6 do not ensure that there exists a unique pair of income tax rate and debt …nancing
ratio as interior solutions to maximizing equilibrium growth rate or the government’s objective function.
We calculate the income tax rate and debt-…nancing ratio to verify the existence or non-existence of
the interior solutions, which are consistent with the …rst-order conditions of Equations (14) and (15)or (18a) and (18b). The parameters are set as (A; ; ) = (10; 0:3; 0:483). The values of vary in therange [0:4; 1], and those of are de…ned over [0:5; 1].
Growth-maximizing tax rate and debt-…nancing ratio. The calculated results for 2 (0:4; 1) showthat the growth-maximizing tax rate is equal to 0:7 and the debt-…nancing ratio is almost 0. For
example, x? = 7:764 and ? = 0:7 are obtained when = 0:5. Therefore, the balanced-budget scheme
is good for economic growth within realistic values of parameters. On the other hand, the calculated
value of x is larger than its realistic value, which is expected to be around unity.
Income tax rate and debt-…nancing ratio determined by existing generations. Table 1 reports thecalculation results of x , , , and the de…cit-to-GDP ratio for = 0:4, = 0:5, and = 0:95 with
1 8 When= 1 (perfect depreiation), (xt ) is monotonically decreasing in xt . Complicated dynamics do not occur
even though a two-period cycle exists or growth path oscillatory diverges from a stationary equilibrium.

15

Figure 4.
three di¤erent values of (i.e., = 0:5, = 0:75, and = 1).
= 0:5 complies the convergencecriterion of the government budget de…cit in EU, unlike = 0:75 and = 1. Figure 4 displays thecurves based on the values ofandfor 2 (0:4; 1) and = 0:75. It shows that the values of x are
between 0:824 and 0:919, those ofare between 0:170 and 0:182, and those ofare between 0:346
and 0:534 for 2 (0:4; 1).19 In reality, public-to-private capital ratio is below unity.20 The calculated
result seems to replicate the actual outcomes.

5.4

Short-run e¤ects of …scal policy

Based on the analysis in the previous subsections, we consider two scenarios. Scenario 1 is the baseline
case where (A; ; ; ; ; ) = (10; 0:3; 0:483; 0:5; 0:2; 0:2). Scenario 2 is the situation where output
elasticity of public capital and saving are above the values of the …rst scenario. We use ( ; ) =(0:2; 0:6). Other parameters are same as those of the baseline case. Suppose that the economy isinitially at the stationary equilibrium. At the initial period, the government permanently increases 1%point in income tax rate or debt …nancing ratio, and the policy changes are unexpected for households.
Scenario 1. At period-0 in the baseline case, public-to-private capital ratio x0 , interest rate
r0 , and equilibrium growth factor 0 are 1.054, 1.990, and 2.338, respectively (i.e., growth rate 0 is
1.338). Hence, r0 > 0 holds and the economy is dynamically e¢ cient. At the period 1, the government
unexpectedly raises one percent point in the income tax rate or debt …nancing ratio. Figure 5 illustrates
the dynamics of economic variables after the policy shock in income tax and debt …nancing ratio. Notethat the variable zt in the …gures is de…ned as the percent of the interest payment to GDP.
1%-point increase in income tax rate: The policy shock induces slowdown of private capital accumulation and a surge in public investment at t = 1. Public-to-private capital ratio sharply increases
at t = 1 through these two e¤ects. Private and public capital accumulations a¤ect economic growthrate di¤erently. The positive productivity e¤ect of public capital overweighs the negative impact on
output through slowing down the private capital accumulation. Given that 0 = 1:338 and 1 = 1:409,
a 1%-point increase in the tax rate raises economic growth rate at t = 1 by 7.1% point. However, anegative feedback on x occurs at t = 2. Public capital accumulation increases bond issuance underEquation (8) and interest payment at the next period after the bond issuance. As a result, public capital accumulation slows down while diminishing the crowding-out e¤ect. Thus, the economic growthrate drops at t = 2. Economic variables alternately increase and decrease, and …nally converge to
new stationary values ( = 1:389). In the long-run, tax-…nanced public investment increases economic
1 9 If= 0:5, we have x 2 [0:660; 0:723],
2 [0:140; 0:155] and2 [0:320; 0:566] for 2 [0:4; 1]. When = 0:5, wehave x 2 [0:940; 1:061],
2 [0:190; 0:200] and2 [0:355; 0:516] for 2 [0:4; 1].
2 0 The 5-year average ratios of public to private capital in G5 countries are 0.329, 0.217, 0.529, 0.248, and 0.365 forFrance, Germany, Japan, the United Kingdom, and United States. These values are calculated based on the data for2013-2017 in the Investment and Capital Stock Database provided by IMF.

16

1%-increase in income tax rate

1%-increase in debt …nancing ratio

Figure 5. Short-run e¤ects of …scal policy for Scenario 117

1%-increase in income tax rate

1%-increase in debt …nancing ratio

Figure 6. Short-run e¤ects of …scal policy for Scenario 2

18

growth rate by 5.1%-point.
1%-point increase in debt …nancing ratio: It leads to investment plunge of private and public
capital at the period when the shock occurs. At t = 1, private capital investment decreases due to thecrowding-out e¤ect while a decrease in public investment is caused by an increase in interest payment ofpublic bonds. The impact on public capital accumulation is weaker than the impact on private capital
accumulation. Thus, public-to-private capital ratio slightly increases at t = 1. At the next period, theinterest payment increases as well as the interest rate. Public investment has declined for two periods.
In contrast, growth rate of private capital is weakly recovered from the bottom at t = 2. Accordingly,
economic growth rate has reduced for two periods: from t = 0 to t = 1, a 1%-point increase in
reduces 1.3% point in economic growth rate ( 0 = 1:338 to 1 = 1:325). Finally, economic growth rate
converges to its new stationary level ( = 1:324). Debt-…nanced public investment decreases economic
growth rate by 1.4%-point.
Scenario 2. With ( ; ) = (0:2; 0:6), the initial values are x0 = 0:803 , r0 = 0:843, and 0 = 2:429(i.e., 0 = 1:429). Then, we have r0 < 0 and the economy is dynamically ine¢ cient. Similar to
Scenario 1, the policy shocks occur at t = 1. Figure 6 illustrates the dynamic e¤ects of the policy
shocks.
1%-point increase in income tax rate: Increased tax enhances private and public capital accumulation with high productivity e¤ects of public capital at t = 1. Public-to-private capital ratio rises
because the e¤ect on public capital accumulation dominates that on private capital accumulation.
Although the crowding-out e¤ect and an increasing in interest payment arise, these negative e¤ects on
capital accumulation are weak. Therefore, the policy shock raises economic growth rate (and growth
rates of private and public capital) to its new stationary level at almost one push. Based on thecalculated results, economic growth rate increases 44.1%-point from t = 0 to t = 1 ( 0 = 1:429 to
1 = 1:870) and 43%-point in the long-term ( = 1:859).
1%-point increase in debt …nancing ratio: The numerical simulation results are similar to thoseof 1%-point increase in the income tax rate. The di¤erence between the impacts of di¤erent policy
shocks depends on the magnitude of the crowding-out e¤ect and interest payment. A rise in increases
these two negative e¤ects on growth in two ways. A rise in (issuing bonds) directly increases interest
payment and strengthens the crowding-out e¤ect, whereas a rise in income tax rate temporally reduces
interest payment through a decrease in post-tax interest rate. The calculation shows that economic
growth rate rises by 35.2%-point (to 1 = 1:781) from t = 0 to t = 1 and by 35% in the long-run( = 1:779). Hence, the impact of 1%-point increase in the debt …nancing ratio on economic growthis smaller than its impact of 1%-point increase in the income tax rate.

6

Conclusion

The literature on …scal policy and economic growth has numerous studies examining how public investment, …nancing methods, and …scal rules mutually a¤ect macroeconomic outcomes. In particular,
the recent literature has focused on the growth e¤ects of debt-…nanced public investment. Contrary
to theoretical …ndings, empirical studies have found positive growth e¤ects of the debt-…nanced public
investment. Without positive growth e¤ects, debt …nancing has not been theoretically recommended,though such …nancing has been adopted in practice. Hence, this paper aims to elucidate growth e¤ects
of debt-…nanced public investment and examine the possibility of choosing borrowing rules by existing
generations endogenously.
The answer to the …rst research question is clear. If interest rate exceeds economic growth rate(i.e., the economy is dynamically e¢ cient), then debt …nancing for public investment decreases theequilibrium growth rate. In contrast, if the interest rate is below to the economic growth rate (i.e., theeconomy is dynamically ine¢ cient), then debt-…nanced public investment may raise the equilibrium
growth rate. Particularly, debt-…nanced public investment has a positive e¤ect on equilibrium growthrate when public capital has su¢ ciently large productivity e¤ect. Naturally, public debt and economic
growth could have a monotonic or non-monotonic inverted-U shaped relationship.

19

The answer to the second research question is that the existing generations tends to choose debt
…nancing for public investment through an endogenous determination of …scal policy. The GRPF,
as a perfect debt …nance, is preferred if the economy is dynamically ine¢ cient. Furthermore, debt
…nancing for public investment could be chosen even if the economy is dynamically e¢ cient. Thesu¢ cient condition is that the marginal productivity e¤ect of public capital is su¢ ciently large to raise
capital income.
Our quantitative analysis supports the two main …ndings of our qualitative analysis. Equilibrium
dynamics with debt …nanced public investment could be complicated and chaotic with low productivity.
Calculation based on data shows that equilibrium tax rate and degree of debt …nance are close torealistic values. Persistent economic ‡uctuations are conceivable when the equilibrium degree of debt
…nance is large, though the economy in the baseline case converges to a unique stationary equilibrium.
On the stable growth path, a 1%-point increase in debt …nancing ratio raises approximately 35% pointof economic growth rate as well as the growth rate of private capital and public capital if the growthrate is larger than the interest rate at the initial time. Debt cost is essential to determining the growthe¤ect of debt …nanced public investment.
Future directions of this research may address peculiar issues to the overlapping generations model.
The extensive analyses should include the impact of uncertainty, the role of intergenerational altruism, the e¤ect of early retirement with endogenous labor supply. First, our model has no disparity
between the rates of return on assets without uncertainty. To address the real disparity, the impact of
uncertainty is worthwhile to examine.
Second, this paper mentioned the importance of the growth e¤ects of public investment in theprocess of determining …scal policy endogenously if the authority cares about future generations. Sucha government’s objective function naturally arises from intergenerational altruistic preferences. Then,a self-enforcing commitment of government policy can be treated in the extensive analysis.
Third, one of the bene…ts of public investment for existing generations is the future increase in theirincome through interest and wage rate increase. Endogenous labor supply in the old generation willbe essential to evaluate future bene…ts. This factor a¤ects the growth e¤ects of public investment andendogenous determination of …scal policy through the change in future bene…t of public investment.
These extensions will modify our results and give us further insights into the e¤ects of debt-…nanced
public investment. Our main …ndings will remain to highlight new aspects. This paper provides theanalytical basis for future studies.

20Appendix
A. Proof of Proposition 1(x) x is also
As is continuous in the closed interval [0; x] with (0) > 0 and (x) = 0, (x)continuous in the closed interval [0; x] with (0) = (0) > 0 and (x) = x < 0. There exists atleast one stationary equilibrium that satis…es x = (x) (xt+1 = xt ). Di¤erentiating (x) with respect
to x yields
0(x) = 0 (x) 1 R 0 , x Q xe;where xe is fxj 0 (ex) = 0g. Note that xe<xb holds. With (0) > 0 and 0 (x) > 0 for x 2 [0; xe], wex; x], a unique …xed point exists,have (x) > 0 for x 2 [0; xe]. As (x) < 0 and 0 (x) < 0 for x 2 [esuch as (x) = 0 in the range [ex; x]. Hence, there is a unique stationary equilibrium. If j 0 (x)j < 1,
the stationary equilibrium is locally stable.
We now consider the possibility that is chaotic map in the Li-Yorke sense. The value of at
xt = x
b is
(b
x) =

Since x satis…es

(1

(1
) x
b] A + (1
) (1
)x
b
) A
f[
(1
) x
b] A + (1

[
) (1

) (1

(x) = 0, total di¤erentiation of
0=[

(1

)

,

A @x
=
x @A

(1

x] dA

(1

)x
b g

) (1

(x) = 0 yields)

Adx + (1

) (1
) (1
)x +
(1
) (1
)x

) (1

1

) x

:

dx

< 0:

Hence, we haveA!1)x=(1and A ! 0 ) x ! 1:)

A di¤erentiation of xb with respect to A is

x
A @b
x
b @A

1
1

(A1)

< 0:

Then, we have A ! 1 ) x
b = 0 and A ! 0 ) x
b ! 1. Using these equations, we obtain
A!1)

(b
x) =

(1

) (1

) (1

= (0)

)

0 and A ! A )Equations (A1) and (A2) show that there exists the critical value of A where
(1

) (1

) (1

<

)

(1

(b
x) ! 1:(bx) = x if

:

)

(A2)

(A3)

If (0)xb and (bx) = x, there exists the Li-Yorke chaos (Li and Yorke, 1975). One of thesu¢ cient condition for 0 < (0) < x
b < (bx) = x is Equation (A3) and
(1
) (1
)
x
b > (0) ,
(1
) A

1

1

>

(1

) (1

) (1

)

:

(A4)

> 0;

(A5)

B. Derivation of equations (13a) and (13b)Partial di¤erentiation ofwith respect to

@ (x)
(1 +
=
@

x) A + [(1yields) (121) A + (1 +x) A] xwhere[(1(1) x] A + (1
) (1
) (1
) (1
) A
f[

)x ;
(1

)

x] A + (1

) (1

)x g:

Using Equation (A5), we obtain Equation (13a) from total di¤erentiation of Equation (11) when xt = x:

Partial di¤erentiation of

@x
=
@
1

1

with respect to

is

@ (x)
[(1
=
@

0 (x)

@ (x)
> 0:
@

) (1 + ) x(1 + x) (r

)] x

:

(A6)

The sign of Equation (A6) is indeterminate. Equation (A6) and total di¤erentiation of Equation (11)when xt = x lead to Equation (13b):
1

@x
=
@
1

0 (x)

@ (x)
:
@

C. Proof of Proposition 2
Using Equation (13a), there exists a lower bound of that satis…es= 0. Let bebound. Then, we have lim & x = 0. Taking the limit of (13a) leads to
lim
&

1 @x
x@

as the lower

= +1:

(A7)

Equations (14) and (A7) derive
lim
&

1@
@

=

1

+ (1

1

)

x
1+ x

lim
&

lim
&

1 @x
x@

= +1:

By Equation (13a), an upper bound of exists. Let be as the upper bound. Note thatbe less than unity. Then, lim %= 0 and lim % x = +1 hold. Equation (13a) yields
lim
%

1 @x
x@

=

1

> 0:

1

must

(A8)

Using Equations (15) and (A8) provide
lim
%

1@
@

=

1
1

+ (1

) lim
%

1 @x
x@

lim
%

x 1 @x=
1+ xx@

1+
1< 0:

These results show that there exists a tax rate to maximize the equilibrium growth rate for 0 << 1.

D. Proof of Proposition 3
From Equation (13b), we have
1 @x
=
x@
(1

(1
) (1 + ) x (1 + x) (r):) (1 + ) [(1) (1)(r + ) ] (1 + x)

22

(A9)

Equation (A9) becomes
1 @x
x@
1 @x
x@

lim

&0

lim

%1

=
=

(1 + ) x+ + (1
r
:
r+

(r
) (1

)
)

;

(A10)
(A11)

Using Equations (15) and (A9), we obtain
@
@

@x
x@
[(1
) + ] ( + ) + (1) (1) (1 + ) [(1
) (1

= (1
=

)

(1

where

) (r
)

)
;
(r + ) ]

(A12)

x
:
1+ x

Equations (A10)–(A12) yield
lim

&0

lim

%1

1@
@
1@
@

( + ) x + (1(1 + ) (1
r
1
r+

=
=

) (r
)
;
)
x
x
:
1+x
1+x

(A13)
(A14)

If the economy is dynamically e¢ cient, Equations (A12) and (A13) lead to@@< 0 for 0

1:

If the economy is dynamically ine¢ cient, then the sign of Equation (A12) is ambiguous, though thatof Equation (A9) is positive. There exists the growth-maximizing level of for 2 (0; 1) if Equations(A13) and (A14) have a positive and negative signs, respectively. To ensure the existence of ? 2 (0; 1),
the following is required:
xB <

(1

)

B

( B+ )

rB

(1
G+

and

)
(1

G

rG

)( G

rG )

< xG :The su¢ cient condition indicates that public-to-private capital ratio under a balanced budget scheme
is su¢ ciently small and that under the GRPF is su¢ ciently large. By Equation (A9), it can be replaced
as the condition that the marginal e¤ect of debt …nancing on x is su¢ ciently large.

E. Proof of Proposition 5
As shown in Appendix C, the lower bound and upper bound of exist and thenEquations (18a) and (A7), the limit of Equation (18a) as ! is[ + (1 + 2 )]
(1
+ lim
&
1

)
x

@x
= +1:
@

By Equations (18a) and (A8), the limit of Equation (18a) as!

becomes

lim
&

lim
%

@W0
=
@

@W0
=
@

+ (1 + 2 )
(1
+ lim
%
1

These results show that there exists

@x
=
@

+

= f j@W0 =@ = 0; <

< g.

23

)
x

+ (1 + )12 ( ; ). Using< 0:

F. Proof of Proposition 6For= 1 and= 0, Equations (11) and (A10) yield

lim x = xB =&0For

= 1 and

(1

) (1

)

and lim

1 @x&0 x @

=

(1

) [(1
)
(1
) (1

]+
)

:

(A15)

= 1, Equations (11) and (A11) provide
lim x = xG =
%1

(1

)

and lim

1 @x%1 x @

=

(1

) [(1
(1

)

]

:

(A16)

] + g;(A18))

The limits of Equation (18b) with Equations (A15) and (A16) lead to@Wt@x= sgn lim
= sgn f(1
&0 @
@
@x
@Wt
= sgn lim
= sgn [(1sgn lim%1 @%1 @sgn lim&0

If the economy is dynamically ine¢ cient, we have (1
1 @x
(1
=
x@
(1

) [(1

)

)

]:

)

>

(A19)

and

) (1 + ) x (1 + x) (r)> 0 for 0 <) (1 + ) + (1 + r) (1 + x)

< 1:

(A20)

Using Equations (18b) and (A15)–(A20), the following equation holds:
@Wt
@Wt
@Wt
> 0; where lim
> lim
> 0:
&0 @
%1 @
@
Therefore, = 1 (GRPF) is chosen.
If the economy is dynamically e¢ cient, (1) < holds. Using Equations (A15)–(A19), wehave(1) (1 + ) 1
@Wt
@Wt
R0, R
and lim
< 0:
lim
%1 @
&0 @
(1
) (1 + )When is su¢ ciently large, there exists
where @Wt =@ = 0 and
2 (0; 1). On the other hand,= 0 (the balanced budget) is preferable to GRPF or debt …nancing when is su¢ ciently small.

24

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27