Discussion Papers In Economics
And Business

Public Research Spending in an Endogenous Growth Model

Kunihiko KonishiDiscussion Paper 13-26

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPublic Research Spending in an Endogenous Growth Model

Kunihiko KonishiDiscussion Paper 13-26

October 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPublic Research Spending in an Endogenous Growth Model∗
Kunihiko Konishi†

Abstract
This study constructs a variety expansion growth model with public research spending,in which public researchers raise the productivity of private R&D. We show that the relationship between public research spending and the growth rate follows an inverted U-shape.
This is because public research spending increases private R&D productivity, but crowds
out labor input to private R&D. It is also shown that the welfare-maximizing level of public research spending is below the growth-maximizing level. With regards to tax policy, azero-proﬁt tax maximizes both growth and welfare. Finally, the study analyzes the stabilityof the steady state, showing that the equilibrium is indeterminate when the government’s
revenue source depends on asset income tax.

Keywords: Public expenditure, Endogenous growth, Innovation, Indeterminacy
JEL classiﬁcation: E62, O41∗
We would like to thank Koichi Futagami, Yoshiyasu Ono, Shinsuke Ikeda, Tatsuro Iwaisako, Noritaka Maebayashi, Asuka Oura, and the participants at the 2013 Autumn Annual Meeting of the Japanese Economic
Association in the University of Kanagawa for their helpful comments and suggestions. This study was supported in part by Grants for Excellent Graduate Schools, MEXT, Japan. Of course, any remaining errors are
our responsibility.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, JAPAN;
E-mail: nge008kk@student.econ.osaka-u.ac.jp

1

Introduction

It is widely recognized that productive government spending plays an important role in economic growth. Arrow and Kurz (1970) were the ﬁrst to study this issue using growth models.
They focused on infrastructure, such as highways, airports, railroads, and electrical facilities,
and introduced the notion of productive public capital. However, in their model, growth is determined by exogenous factors. In terms of endogenous growth models, Barro (1990) presented
the seminal model that includes productive government spending. He assumes that public services raise the productivity of private ﬁrms. Under this setting, the social rate of return on
private capital becomes constant, and the long-run growth rate is determined endogenously.
As a result, we have been able to investigate the relationship between productive government
spending and the long-run growth rate. While Barro (1990) treats government spending as aﬂow variable, many studies have extended the Barro model by introducing the stock of public capital and examining the eﬀects of public capital (see Futagami et al., 1993; Fisher andTurnovsky, 1998).1,2
Although many studies analyze the eﬀects of productive government spending on economic
growth, the Schumpeterian growth models developed by Romer (1990), Grossman and Helpman (1991), and Aghion and Howitt (1992) focus mainly on the activities of private R&D ﬁrms
motivated by monopoly proﬁt. However, many empirical studies argue that the public researchsector is an important factor in economic growth. For instance, Mansﬁeld (1991) ﬁnds that 10%of the innovations in the United States between 1975 and 1985 could not have been developed,
or could have developed with great delay, without academic research. Caloghirou et al. (2001)
analyzed over 6,000 research joint ventures in 42 nations that received funding from the European Commission during the period 1983-1996. They found that 65% of research joint ventures
involved one or more universities. Based on the Swedish Community Innovation Survey, Lööf
and Broström (2008) found that university collaboration has a positive inﬂuence on the innovative activity of large manufacturing ﬁrms. On the other hand, few theoretical studies haveconsidered public research policy from a macroeconomic perspective. Glomm and Ravikumar(1994) present a model in which the stock of technological knowledge depends upon public research, but the model does not allow for private R&D. In contrast, Park (1998) considers both
1
Futagami et al. (1993) developed an endogenous growth model with productive public capital, and show
diﬀerent results to those of Barro (1990). Since their model includes two stock variables, it has transition
dynamics. The growth-maximizing tax rate is the same as that of Barro (1990), but the welfare-maximizing tax
rate is lower than that of Barro (1990). Fisher and Turnovsky (1998) suppose that public capital has a congestioneﬀect. They show there is a trade-oﬀ between the degree of congestion and the substitutability between public
and private capital in production.
2
An extensive survey of the literature is provided by Irmen and Kuehnel (2009).

1

public and private research. Public research indirectly contributes to market production by
inﬂuencing the knowledge accumulation of private R&D. However, the study is more concernedwith open economy issues and international spillovers than public research policy. In orderto analytically examine the growth and welfare eﬀects of public research spending, this study
incorporates public researchers who raise the productivity of private R&D in a variety expansion model, following the work of Grossman and Helpman (1991). The government expends
public research spending and interest from government debt, but raises funds from taxes on
asset income, consumption, and corporate proﬁt.3
In this study, we obtained the following three main results. First, the relationship between
public research spending and the growth rate follows an inverted U-shape. That is, growthmaximizing public research spending exists. The inverted U-shaped relationship occurs because
public research spending raises the productivity of private R&D, but crowds out the labor input
to private R&D. In addition, only the tax on proﬁt has a negative eﬀect on growth because
an increase in the tax rate decreases monopoly proﬁt, which is the rate of return for setting
up new ﬁrms. Thus, a zero-proﬁt tax maximizes growth and the government should ﬁnanceexpenditure by taxing asset income and consumption.
Second, the welfare-maximizing level of public research spending is lower than the growthmaximizing level. In this study, welfare is driven by households’ consumption expenditure andthe growth of diﬀerentiated goods (i.e., love of variety). While public research spending and thegrowth rate follow an inverted U-shape, households’ consumption expenditure decreases with anincrease in public research spending. This is because higher public research spending implies aheavier tax burden on households, so reducing their consumption expenditure. Therefore, there
is a trade-oﬀ between households’ consumption expenditure and the growth of diﬀerentiatedgoods. This trade-oﬀ leads to the welfare-maximizing level of public research spending being
below the growth-maximizing level. With regards to tax policy, a zero-proﬁt tax maximizes
welfare because a decrease in proﬁt tax increases growth, which increases welfare.
Third, an increase in asset income tax raises the possibility of indeterminacy, whereas anincrease in government debt, consumption, and proﬁt tax decreases the possibility of indeterminacy.4 Some related studies have also pointed out the possibility that indeterminacy depends
on ﬁscal policies (e.g., Guo and Harrison, 2008; Kamiguchi and Tamai, 2011). Guo and Harrison (2008) show that the presence of productive government spending and the distortion by
3

Many theoretical studies have investigated the ﬁscal policy of productive public spending with debt ﬁnancing.
For example, see Bruce and Turnovsky (1999), Greiner and Semmler (2000), Ghosh and Mourmouras (2004),
Futagami et al. (2008), Yakita (2008), Maebayashi et al. (2013), and Morimoto et al. (2013).
4
In this study, indeterminacy means we cannot select a continuum of equilibrium trajectories that all converge
on the same steady state.

2taxes aﬀect the possibility of indeterminacy. This implies that ﬁnancing government spending
using distortionary taxes might bring about equilibrium indeterminacy. Then, Kamiguchi andTamai (2011) show that income tax ﬁnancing has a greater inﬂuence on indeterminacy thanthe presence of productive government spending. It seems that the ﬁndings in these studies are
similar to those of this study. However, these studies use real business cycle models, not an
endogenous growth model, and so do not consider the eﬀect of government debt. In fact, few
studies examine the relationship between the debt policy rule and equilibrium indeterminacy.
Futagami et al. (2008) shows that, in a closed economy model of endogenous growth with public
services and a debt policy, the high growth steady state can be equilibrium indeterminate when
the long-run debt-private capital ratio is suﬃciently high. Morimoto et al. (2013) show that, ina small open economy model of endogenous growth with public capital and a debt policy, equilibrium indeterminacy arises when the long-run debt-GDP ratio is suﬃciently high.5 In contrast
to these studies, this study shows that equilibrium indeterminacy arises when government debt
is suﬃciently small.
The rest of this paper is organized as follows. Section 2 establishes the model used in thisstudy. Section 3 derives the equilibrium and dynamics of the economy. Section 4 examines
the stability of the steady state. Section 5 analyzes the policy eﬀect on economic growth andSection 6 investigates the policy eﬀect on welfare. Finally, Section 7 concludes the paper.

2

Model

There is a unit continuum of identical households. Each household supplies one unit of inelastic labor supply. The factor market is perfectly competitive, while the goods market ismonopolistically competitive, as explained below. The households have perfect foresight.

2.1

Households

The households maximize the following lifetime utility:
U0 ≡

∫ ∞

e−ρt log Ct dt,(1)0

where Ct represents an instantaneous utility derived from the consumption of a composite goodand ρ > 0 is a rate of time preference. Ct is given by
] ε
[∫ Nt
ε−1
ε−1
ε
,
Ct =
ct (j) dj

(2)

0
5

Maebayashi et al. (2013) show that, in a closed economy model of endogenous growth with public capitaland a debt policy, equilibrium indeterminacy never arises.

3

where ct (j) denotes the consumption of good j and Nt denotes the number of available varieties.
We assume that ε > 1. ε is the elasticity of substitution between any two products. Denoting∫N
the expenditure of the households as Et = 0 t Pt (j)ct (j)dj, we obtain the demand function forgood j as follows:

Pt (j)−ε Et
ct (j) = ∫ Nt,
1−εdi
0 Pt (i)where Pt (j) is the price of good j, and PD,t is the price index, deﬁned as
(∫ Nt
) 1
1−ε
1−ε
.
PD,t =
Pt (i) di

(3)

(4)

0

Then, the maximization problem for households is as follows:
max

U0

subject to Ȧt = (1 − tA )rt At + Wt − (1 + tE )Et ,where At , rt , and Wt represent the households’ asset holding, rate of return on assets, and wage
rate. The wage rate is the numeraire, and thus Wt = 1. The government taxes asset income
at rate tA ∈ [0, 1) and consumption at rate tE ≥ 0. Substituting (3) into (2), we obtain theindirect sub-utility function as follows:
Ct =

Et
.
PD,t

(5)

The maximization subject to the intertemporal budget constraint yields the following Euler
equation:Ėt= (1 − tA )rt − ρ.
Et

2.2

(6)

Firms

This subsection considers producer behavior. Producers undertake two distinct activities. They
create blueprints for new varieties of diﬀerentiated goods, and manufacture the diﬀerentiatedgoods that have been created by R&D.
We assume that each diﬀerentiated good is produced by a single ﬁrm because the good is
inﬁnitely protected by a patent. We further assume that one unit of labor input produces one
unit of a diﬀerentiated good. The ﬁrm manufacturing good j (ﬁrm j) maximizes its after-tax
proﬁt:()(1 − tπ )Πt (j) = (1 − tπ ) Pt (j)Xt (j) − Xt (j) ,(7)

where Xt (j) and tπ ∈ [0, 1) represent the output for ﬁrm j and the tax rate on proﬁt. Thenﬁrm j charges the following price:
Pt (j) =

ε
= P.
ε−1
4

(8)

Therefore, all goods are priced equally. Pricing rules (8) and (3) yield Xt (j) = Xt = (ε −1)Et /εNt . Then, the per brand operating proﬁts are as follows:
Πt =

Et
.
εNt

(9)

The no-arbitrage condition is given by(1 − tA )rt = (1 − tπ )Πt v̇t
+ ,
vt
vt

(10)

where vt denotes the value of a ﬁrm.6
Next, we consider the technology involved in developing a new good.7 R&D ﬁrms create
blueprints and expand the varieties of goods available for consumption. We assume that R&D
activity needs labor input. Because the knowledge that has already been produced includes allthat is needed for invention, greater knowledge means further invention. Since the knowledge
is non-rival and non-excludable, an expansion in the number of varieties reduces the labor
input. Furthermore, we incorporate public researchers into the model. Public research activities
enhance the productivity of private R&D through a variety channel (for example, publications,scientiﬁc reports, conferences, research joint ventures, and university collaboration). We assume
the following production function for R&D:
Ṅt = f (Gt )Nt LR,t ,(11)

where Gt and LR,t represent the number of public researchers and the amount of labor devoted
to R&D. In addition, we postulate that f satisﬁes the following conditions:f (0) > 0 and f ′ (Gt ) > 0 and f ′′ (Gt ) < 0.
We assume that ﬁrms freely enter into the R&D race. Therefore, the free entry condition is
given byvt =1
⇔ Ṅt > 0.
f (Gt )Nt

(12)

Equation (12) shows that an increase in Gt decreases vt . The reasoning is as follows. Greater
R&D productivity creates an excess supply of blueprints. In a competitive market, the Walrasianadjustment mechanism reduces the patent value, vt .

2.3

Government

The government taxes asset income, consumption, and corporate proﬁt. We assume that therespective tax rates are held constant over time, and that the government maintains its debt,
67Note that proﬁt tax can be translated as dividend tax.
See Grossman and Helpman (1991) for more details of the R&D process.

5

B̄, at the constant level. Therefore, the government ﬁnances public research spending and theinterest on the debt. As a result, it satisﬁes the budget constraint
Gt + rt B̄ = tA rt At + tE Et + tπ Πt Nt .

(13)

Note that B̄ = 0 implies a balanced budget.8,93

Equilibrium

3.1

Dynamics

Labor is used for production, private R&D, and the employment of public researchers. The
labor market equilibrium becomes
Nt Xt + LR,t + Gt = 1.

(14)

From (12), the asset market equilibrium is as follows:
At = B̄ + Nt vt = B̄ +1.
f (Gt )

(15)

From (9), (12), and (14), the no-arbitrage condition (10) becomes:(tπ )f ′ (Gt )(1 − tA )rt = 1 −Ġt .
f (Gt )Et − f (Gt )(1 − Gt ) −εf (Gt )

(16)

By using (9), (13), (15), and (16), we obtain{}(f ′ (Gt )
tπ )
1 (
tπ )
Ġt = 1 −f (Gt )Et − f (Gt )(1 − Gt ) +tE +Et − Gt ,f (Gt )
ε
∆t
ε

(17)

where ∆t ≡ tA /{(1 − tA )f (Gt )} − B̄. From (12), (16), and (17), Euler equation (6) becomes:
{
}
(
Ėt
1
tπ )
=
G t − tE +Et − ρ.
(18)
Et
∆t
ε
Equations (17) and (18) formulate the autonomous dynamic system with respect to Et and Gt .

8
Our main results do not change if the government uses a debt policy rule following that of Futagami et al.
(2008). In that case, the budget constraint of the government is as follows:

Gt + rt Bt = tA rt At + tE Et + tπ Πt Nt + Ḃt .
The government adjusts its debt according to the following rule:Ḃt = −ϕ(Bt − B̄),
where B̄ and ϕ denote the target level of government debt and the adjustment coeﬃcient of the rule, respectively.
9
If B̄ < 0, the government lends to households and earns interest.

6

3.2

Steady state

In this section, we examine the steady state of the economy. Henceforth, we omit time index,
t. Imposing Ė = Ġ = 0 in (17) and (18), respectively, results in
(

tπ )
tE +
E = G − ρ∆,
ε
(
tπ )
ρ= 1−
f (G)E − f (G)(1 − G).
ε

(19)
(20)

By eliminating E from equations (19) and (20), we have()()
tE + tεπtA
1
1 tE + tεπ
1 + tEB̄ =++−G ≡ Λ(G).
1 − tA
1 − tεπ f (G) ρ 1 − tεπ
1 − tεπ(21)

As shown in Figure 1, the intersection of the LHS and RHS of (21) determines the steady statevalue, G∗ .10 The steady state value E ∗ is obtained from (19), as follows:
{
}
1
ρ
∗
∗
E =
(1 − G ) +.
f (G∗ )
1 − tεπFigure 1(22)

Figure 2

From (11), (14), and (22), the growth rate at the steady state is given by
( )∗
}
Ṅ
1 {
∗
γ ≡
=
(1 − tπ )f (G∗ )(1 − G∗ ) − (ε − 1)ρ .
N
ε − tπ(23)

Moreover, we assume a positive growth rate at G∗ = 0, as follows:γ ∗ |G∗ =0 =}1 {(1 − tπ )f (0) − (ε − 1)ρ > 0.
ε − tπ10 ∗x represents the steady state value of variable x.

7

(24)

Diﬀerentiating γ ∗ with respect to G∗ , we obtain}∂γ ∗1 − tπ { ′ ∗
∗
∗
=
f
(G
)(1
−
G
)
−
f
(G
)
.
∂G∗
ε − tπ(25)

From the assumption of f , ∂γ ∗ /∂G∗ is decreasing in G∗ .11 Assuming that f ′ (0) > f (0), there
exists public research spending that satisﬁes f ′ (G∗ )(1 − G∗ ) − f (G∗ ) = 0. We deﬁne G∗ as Gg ,which yields the following relation:∂γ ∗R 0 ⇔ G∗ Q Gg .
∂G∗

(26)

Therefore, we can illustrate the graph of (23) as shown in Figure 2. As shown by (26), therelationship between the growth rate and G∗ follows an inverted U-shape, and the growthmaximizing public research spending is Gg . From Figure 2, the growth rate becomes 0 if G∗ is
suﬃciently large. In this case, R&D is not undertaken since a suﬃciently large value of G∗ makeslabor input to private R&D zero. Here, Ĝ is deﬁned as γ ∗ |G∗ =Ĝ = 0. In Figure 1, we assume thatthe growth rate is positive in a balanced budget (B̄ = 0), that is, G∗ |B̄=0 < Ĝ. Furthermore,
to focus on the positive growth rate and G∗ ≥ 0, we impose the following condition:
Λ(Ĝ) < B̄ ≤ Λ(0).
These results are summarized in the following proposition:

Proposition 1
There is a positive growth steady state in this economy if Λ(Ĝ) < B̄ ≤ Λ(0). In addition, if
f ′ (0) > f (0), there exists a positive growth-maximizing level of public research spending, Gg .

We next study the detailed relationship between the growth rate and public research spending at the steady state. From (11), the growth rate, γ ∗ = f (G∗ )L∗R , is determined by f (G∗ )and L∗R . Diﬀerentiating γ ∗ with respect to G∗ yields
∗
∂γ ∗
′
∗
∗
∗ ∂LR
=
f
(G
)L
+
f
(G
)
.
R
∂G∗
∂G∗

While the eﬀect of the productivity of R&D is positive, f ′ (G∗ ) > 0, the eﬀect of labor input,∂L∗R /∂G∗ , is ambiguous. From (14), labor input in private R&D at the steady state is given by∗∗∗
L∗R = 1 − G∗ − ε−1
ε E . Diﬀerentiating LR with respect to G yields∂L∗R
ε − 1 ∂E ∗= −1 −
.
∗
∂G
ε ∂G∗
11

Diﬀerentiating ∂γ ∗ /∂G∗ with respect to G∗ yields}∂2γ∗
1 − tπ { ′′ ∗
∗
′
∗
=
f
(G
)(1
−
G
)
−
2f
(G
)
< 0.
∂G∗2
ε − tπ8(27)The ﬁrst term represents a direct crowding-out eﬀect on private R&D labor input, and thesecond term represents the eﬀect of labor demand on production, which depends on households’
consumption expenditure. To more precisely investigate the eﬀect of the second term, we
diﬀerentiate E ∗ with respect to G∗ , as follows:
{
}
∂E ∗
1
f ′ (G∗ )=−1 − ρ< 0.
∂G∗(f (G∗ ))2
1 − tεπ(28)

Thus, E ∗ is decreasing in G∗ . Since higher public research spending implies a heavier tax
burden on households, an increase in public research spending reduces households’ consumption
expenditure. Then, we examine the value of G∗ that crowds out private R&D labor input. From(27) and (28), we obtain

{
}
∂L∗R
1
f ′ (G∗ )=−(1 − tπ ) + (ε − 1)ρ
.
∂G∗
ε − tπ(f (G∗ ))2

From the assumption of f , f ′ (G)/(f (G))2 is decreasing in G.12 We deﬁne G̃ as f ′ (G̃)/(f (G̃))2 =(1 − tπ )/(ε − 1)ρ. Then we obtain the following relation:∂L∗R
R 0 ⇔ G∗ Q G̃.
∂G∗When G∗ < G̃, the eﬀect of labor demand on production exceeds the direct crowding-out
eﬀect; and ∂L∗R /∂G∗ > 0 holds.13 In contrast, when G∗ > G̃, the direct crowding-out eﬀect is
suﬃciently large; so ∂L∗R /∂G∗ < 0 holds. Next, we compare Gg to G̃. Substituting G∗ = G̃into (25) yields∂γ ∗= f ′ (G̃)LR |G∗ =G̃ > 0.
∂G∗ G∗ =G̃
Therefore, from (26), we have G̃ < Gg .
We summarize these results as follows. If G∗ < Gg , the eﬀect of an increase in R&D
productivity exceeds the crowding-out eﬀect, and thus an increase in G∗ raises the growthrate. However, if G∗ > Gg , the crowding-out eﬀect exceeds the eﬀect of an increase in R&D
productivity. In this case, an increase in G∗ decreases the growth rate. From these results, thegrowth rate and G∗ follow an inverted U-shape.
Finally, we have to consider the condition of Gg > 0 that corresponds to f ′ (0) > f (0).
From the above discussion, if the productivity eﬀect is larger than the crowding-out eﬀect at
G∗ = 0, there exists a positive growth-maximizing level of public research spending. On theother hand, if the crowding-out eﬀect is larger than the productivity eﬀect at G∗ = 0, publicresearch spending crowds out private R&D input, that is, there is no need for public research
spending.
{}
Diﬀerentiating f ′ (G)/(f (G))2 with respect to G yields f ′′ (G)f (G) − 2(f ′ (G))2 /(f (G))3 < 0.
13
Note that G̃ can be a negative value. In this case, there is no crowding-in eﬀect.
12

9

4

Stability

We examine the local stability at the steady state. Approximating (17) and (18) linearly in theneighborhood of the steady states, we obtain
( ) (
)(
)
Ġ
JGG JGEG − G∗=.
JEG JEE
E − E∗
Ė
Here, Jij (i, j = G, E) denotes entities in the Jacobian matrix of this system (see Appendix A
for more detail):{}tAf ′ (G∗ )f (G∗ ),JGG = − ′ ∗ ∗ 1 + ρ
f (G )∆
1 − tA (f (G∗ ))2{(}(
f (G∗ )
tπ )
tπ )
∗
∗
JGE = ′ ∗ ∗,1−f (G )∆ + tE +f (G )∆
ε
ε
{
}
E∗
tA
f ′ (G∗ )JEG = ∗ 1 + ρ,∆
1 − tA (f (G∗ ))2(tπ ) E ∗JEE = − tE +,ε ∆∗where ∆∗ = tA /{(1 − tA )f (G∗ )} − B̄.14 The eigenvalues of the Jacobian matrix, J, are deﬁned
as λi (i = 1, 2). Here, λ1 and λ2 are the roots of the characteristic equation, λ2 − (JGG +JEE )λ + JGG JEE − JGE JEG = 0. To check the stability, we investigate the sign of detJ =JGG JEE − JGE JEG as follows:[]tA f ′ (G∗ )f (G∗ )E ∗∗.
detJ = − ′ ∗ ∗ f (G ) + ρ
f (G )∆
1 − tA f (G∗ )Because the sign of the square bracket is positive, the sign of detJ is determined by the signof ∆∗ . When ∆∗ > 0, the sign of detJ is negative. We then have λ1 > 0 and λ2 < 0. In thiscase, the steady state is saddle-point stable. Since E and G are ﬂow variables, the equilibriumis indeterminate.
On the other hand, when ∆∗ < 0, the sign of detJ is positive. In this case, we must examine
the sign of trJ = JGG + JEE , as follows:
[
]
(
1 (
tA
tπ ) ∗tπ ) f (G∗ )1
trJ = − ∗ 1 −+ρ
+ tE +E .
∆εf ′ (G∗ )1 − tA f (G∗ )εSince ∆∗ < 0 implies trJ > 0, the eigenvalues of J have positive real parts. Thus, the steadystate is unstable, that is, the equilibrium is determinate. From these results, we can state thefollowing proposition:

14

If ∆∗ = 0, the economy jumps to the steady state immediately. See Appendix B for more detail.

10

Proposition 2
The equilibrium is indeterminate if ∆∗ > 0, while determinate if ∆∗ ≤ 0.
In this study, G∗ is determined by the government’s budget constraint, that is, G∗ is affected by the ﬁscal variables. Therefore, from the deﬁnition of ∆∗ , the ﬁscal variables inﬂuence
the value of ∆∗ . We next examine the relationship between ∆∗ and the ﬁscal variables. To
begin with, we investigate the eﬀects of the ﬁscal variables on G∗ through changes in the ﬁscal
variables. As shown in Appendix C, we can state the following lemma:

Lemma
The eﬀects of changes in policy variables B̄ and ti (i = A, E, π) are as follows:
∂G∗
∂G∗
< 0,
> 0.
∂ti
∂ B̄

An increase in B̄ raises the interest payment on debt, and G∗ decreases. Meanwhile, anincrease in ti (i = A, E, π) raises government revenue, and G∗ increases. By using the Lemma,
we examine the eﬀects of ∆∗ through changes in the ﬁscal variables. As shown in Appendix D,
we obtain the following relationship:
∂∆∗
∂∆∗
∂∆∗
< 0,
> 0,
< 0. (i = E, π)
∂tA
∂ti
∂ B̄
From these results and Proposition 2, an increase in tA increases the possibility of indeterminacy,
whereas an increase in tE , tπ and B̄ reduces the possibility of indeterminacy. In addition, we
consider the following extreme cases:tA
1− B̄ > 0,
1 − tA f (G∗ )1∆∗ = G∗ > 0,ρ∗
∆ = −B̄ < 0,∆∗ =B̄ ≤ 0, tA , tE , tπ > 0,tE = tπ = 0, B̄, tA > 0,tA = 0, B̄, tE , tπ > 0.

When B̄ = tA = 0 is a special case, because ∆∗ = 0. In this case, we have to reconsider theequilibrium dynamics and stability. Appendix B shows the steady state is unstable, that is, theequilibrium is determinate. In summary, we can state the following corollary:11

Corollary (Local Stability)
When tA is suﬃciently large and tE , tπ , and B̄ are suﬃciently small, the equilibrium is indeterminate. On the other hand, when tA is suﬃciently small and tE , tπ , and B̄ are suﬃciently
large, the equilibrium is determinate.

To investigate the intuitive explanation of indeterminacy, suppose that the initial level of
G is larger than the steady state level of public research spending, G∗ , which increases the
productivity of R&D.15 From (12), the value of a ﬁrm, v, decreases. From (11) and (12), theno-arbitrage condition is given by(1 − tA )r = (1 − tπ )Πf ′ (G)− f (G)LR −
Ġ.
v
f (G)

In this case, the ﬁrst term of the RHS of this condition increases. The second term changes
slightly. This is because an increase in G reduces the labor input to R&D, LR , and this partly
oﬀsets the increase in f (G). If the government does not change the tax rates, Ġ > 0 cannot
satisfy the budget constraint in the long run. Therefore, Ġ < 0 holds, and the third term of the
RHS becomes positive. From these results, an increase in G raises the interest rate, r.16
Using this result, we provide an intuitive explanation of indeterminacy. Suppose that theeconomy is in a steady state and households expect the government to raise G. This expectation
implies that households also expect an increase in r. Households then have a higher incentive tosave. As a result, households’ asset holding, A, increases and their expenditure, E, decreases.
Under these circumstances, we can think of the government budget constraint as:(tπ )G = tA rA − rB̄ + tE +E.
| {z }ε(29)(1−tA )r∆From equation (29), when the government’s interest payment exceeds the asset income tax
revenue, ∆∗ ≤ 0 holds.17 This implies that the government’s revenue source depends on consumption/proﬁt tax ﬁnancing. In this case, since tA is suﬃciently small, the eﬀect of a reductionin E exceeds that of an increase in A, and the government then decreases G. Thus, the households’ expectations are not self-fulﬁlling; that is, the equilibrium is determinate. On the other
hand, ∆∗ > 0 implies that the asset income tax revenue is suﬃcient to ﬁnance the interest payment and the government’s revenue source depends on asset income tax ﬁnancing. In this case,since tA is suﬃciently large, an increase in A will raise G. In this case, households’ expectations
can be self-fulﬁlling and the equilibrium is indeterminate.
15

Note that this increase in G is not caused by the ﬁscal policy change.
Intuitively, when the price of a patent, v, decreases, the demand for the patent becomes higher than therisk-free asset. This raises the interest rate, r, because the no-arbitrage condition must hold.
17
We can derive tA rA − rB̄ = (1 − tA )r∆ from (15) and the deﬁnition of ∆.
16

12

5

Growth-Maximizing Policy

In this section, we investigate the policy maximizing the long-run growth rate. Equation (23)implies that only proﬁt tax is a distorting tax, and that asset income/consumption tax can beequivalent to a lump-sum tax. We diﬀerentiate γ ∗ with respect to tπ , under given G∗ , as follows
}
ε−1 {
∂γ ∗
∗
∗
=−
f
(G
)(1
−
G
)
+
ρ
< 0.
∂tπ G∗ given(ε − tπ )2Thus, a decrease in tπ raises the graph of the RHS of (23). From (25) and (26), the growthmaximizing public research spending is Gg , regardless of tπ . These relations are shown in
Figure 3. By using these results, Proposition 1, and the Lemma, we can now state the following
proposition representing the growth-maximizing policy.

Proposition 3
If f ′ (0) > f (0), the policy mix of tπ = 0 and B̄ = Λ(Gg ) maximizes the long-run growth rate.

Figure 3

Proposition 3 states that the government should not levy corporate proﬁts and ﬁnanceits spending through asset income/consumption taxation. This is because proﬁt tax has anegative growth eﬀect by reducing the rate of return to setting up new ﬁrms. Furthermore,

13Λ(Gg ) is increasing in tA and tE . This implies that larger government debt needs a larger asset
income/consumption tax rate.
In the Barro model, when the government imposes tax only on household income, there
exists a growth-maximizing government spending level. Meanwhile, when the government isable to use consumption or lump-sum tax, the growth rate is increasing in productive government spending. In other words, there is no growth-maximizing government spending level.18
This implies that Proposition 3 is diﬀerent to Barro’s results. The reasoning is as follows. In
the Barro model, when the government raises productive government spending, the production of the ﬁnal output increases. Because the government transforms the ﬁnal output into a
productive input, an increase in the available resources can be devoted to further productive
government spending. Thus, if the government can use a non-distortionary tax, the growthrate is increasing in government spending.19 On the other hand, in this study, public research
spending requires labor input. Since the total labor supply is constant over time, larger publicresearch spending crowds out private R&D input, as discussed in Subsection 3.2. Therefore,
there exists a growth-maximizing public research spending level, even if the government is able
to use a non-distortionary tax.

6

Welfare-Maximizing Policy

In this section, we investigate the welfare level of the steady state. In the steady state, (2), (5),and (8) yieldlog C = log E ∗ +(ε − 1) 1
1γ ∗ t + log
N0ε−1 .
ε−1
ε1

Without any loss of generality, we set (ε − 1)N0ε−1 /ε = 1. If indeterminacy occurs, the eﬀectof the welfare-maximizing policy is ambiguous. To focus on the determinate equilibrium, we
impose the following condition using the deﬁnition of ∆∗ and (D.1).

Condition : Determinate equilibrium
tA ≤

B̄
1
f (G∗ ) + B̄or tE +

18tπ
(1 + tE )G∗≥.
ρεf (G∗ ) + 1

A detailed explanation is provided by Irmen and Kuehnel (2009).
If the government’s revenue source depends on only a distortionary income tax, an increase in productive
government spending can decrease the after-tax private marginal product of private capital through a necessary
increase in the distortionary income tax. Therefore, there exists a growth-maximizing government spending level.
1914Note that this condition implies that ∆∗ ≤ 0. Since the economy jumps to the steady state,
the welfare level at the steady state is calculated by
U∗ =

1
1
log E ∗ + 2γ∗.
ρρ (ε − 1)(30)

Diﬀerentiating U ∗ with respect to G∗ , we obtain∂U ∗1 ∂E ∗
1
∂γ ∗
=
+
.
∂G∗
ρE ∗ ∂G∗ ρ2 (ε − 1) ∂G∗(31)

From the fact that the relationship between γ ∗ and G∗ is an inverted U-shape (see Figure 2)
and E ∗ is decreasing in G∗ (see Equation (28)), we can depict equation (30) in (G∗ , U ∗ ) space,as shown in Figure 4. If the growth eﬀect, ∂γ ∗ /∂G∗ , is suﬃciently large, U ∗ and G∗ follow an
inverted U-shape (see Figure 4-1).20 In contrast, if the growth eﬀect is suﬃciently small, U ∗ isdecreasing in G∗ (see Figure 4-2).

Figure 4-2 : the growth eﬀect is small

Figure 4-1 : the growth eﬀect is large

We next consider the welfare-maximizing policy. From (22), (23), and (30), only the proﬁttax is a distorting tax and the asset income/consumption tax is equivalent to a lump-sum tax,
as discussed in Section 5. Thus, diﬀerentiating U ∗ with respect to tπ , under given G∗ , we obtain1∂U ∗=Φ(G∗ ),∂tπ G∗ given ρ2 (ε − tπ )2
where
Φ(G) ≡ ρ(ε − 1 − tπ ) − f (G)(1 − G).
Then, we investigate the sign of Φ(G∗ ) under Ṅ > 0 (i.e., 0 ≤ G∗ < Ĝ). Appendix E showsthat the sign of Φ(G∗ ) is negative when 0 ≤ G∗ < Ĝ. Thus, the sign of ∂U ∗ /∂tπ G∗ given is20

We derive this condition later.

15

also negative. Analogous to the discussion of the growth-maximization policy, a decrease in the
proﬁt tax raises the graph of the RHS of (30) (see Figure 4-1) and the optimum can be attainedby setting the proﬁt tax to zero.
Finally, we derive the welfare-maximizing public research spending level. By setting tπ = 0
and using (25), (28), and (31), we obtain
{
}
{
} 1
1
f ′ (G∗ )
1
∂U ∗
′
∗
∗
∗
=
−1
−
ρ
+
f
(G
)(1
−
G
)
−
f
(G
)
≡ Ω(G∗ ).
∂G∗
ρE ∗(f (G∗ ))2ρ2 ε(ε − 1)ρ(32)

From Figure 4, when welfare-maximizing public research spending exists, the sign of ∂U ∗ /∂G∗at G∗ = 0 becomes positive (i.e., the growth eﬀect is suﬃciently large at G∗ = 0). Hence, theexistence condition of positive welfare-maximizing public research spending becomes as follows:
{
}
(
)
1
f ′ (0)1′Ω(0) =−1 − ρ
+f (0) − f (0) > 0.
(33)
ρ
1 + f (0)(f (0))2
ρε(ε − 1)
When ε and ρ are suﬃciently small, and f ′ (0) is suﬃciently large, Ω(0) > 0 holds (see Appendix
F for more detail). Let us deﬁne Gw by Ω(Gw ) = 0. In this case, ∂U ∗ /∂G∗ R 0 holds whenG∗ Q Gw (see Figure 4-1), and thus Gw represents the welfare-maximizing public research
spending. However, if Ω(0) ≤ 0, then ∂U ∗ /∂G∗ < 0 holds (see Figure 4-2). In this case, G∗ = 0
maximizes the welfare. Furthermore, we can compare the welfare-maximizing level of publicresearch spending with the growth-maximizing level. Substituting G∗ = Gg into (32) yields{}
f ′ (Gg )∂U ∗1−1 − ρ
< 0.
=∂G∗ G∗ =Gg
ρE ∗(f (Gg ))2
Therefore, we can see that the welfare-maximizing level of public research spending is lower
than the growth-maximizing level. In summary, we can state the following proposition:

Proposition 4
If Ω(0) > 0, the policy mix of tπ = 0 and B̄ = Λ(Gw ) maximizes welfare. In addition, thewelfare-maximizing level of public research spending is below the growth-maximizing level.

In line with Proposition 3, Proposition 4 states that the government should not levy a tax
on corporate proﬁt and ﬁnance its spending through asset income/consumption taxation. From(30), welfare is driven by household consumption expenditure and the growth of diﬀerentiatedgoods. Thus, to maximize the growth eﬀect, the tax rate on proﬁt should be zero.
In the Barro model, when the government only imposes a tax on household income, thegrowth-maximizing policy is equivalent to the welfare-maximizing policy.21 However, in this21
If the government can use consumption or lump-sum tax, welfare is increasing in government spending aswell as the growth rate, as discussed in Section 5. That is, there is no welfare-maximizing government spending
level.

16

study, the welfare-maximizing public research spending, Gw , is lower than the growth-maximizing
public research spending, Gg . As mentioned above, higher public research spending decreases
the household consumption expenditure. Therefore, there exists a trade-oﬀ between household consumption expenditure and the growth of diﬀerentiated goods. When the government
increases public research spending to maximize the growth rate, this harms household consumption expenditure, and thus the welfare-maximizing public research spending level is below thegrowth-maximizing level.

7

Conclusion

In this study, we developed an R&D-based growth model to examine the eﬀects of publicresearch spending on private R&D activities. We ﬁnd that a zero-proﬁt tax maximizes growth
and welfare at the steady state. Furthermore, the growth-maximizing public research spending
level is above the welfare-maximizing level. We also ﬁnd that the equilibrium is indeterminate
when government debt, consumption, and proﬁt tax are suﬃciently small and asset income tax
is suﬃciently large.
Finally, this study is a ﬁrst-generation R&D-based growth model that exhibits scale eﬀects.
For future research, it would be interesting to consider the non-scale growth model.22

AppendixA.

Derivation of the Jacobian matrix
From (17) and (18), we obtain
[
{
}]
tπ )
tπ )
f (G) (
1 (Ġ = ′
1−f (G)E − f (G)(1 − G) +
tE +
E−G ,
f (G)
ε
∆
ε
[ {
}
]
(
1
tπ )
Ė = EG − tE +E −ρ .
∆εApproximating these equations linearly in the neighborhood of the steady states, the following
elements of the Jacobian matrix yield[]tπ ) ′ ∗ ∗f (G∗ ) (
′
∗
∗
∗
1−
f (G )E − f (G )(1 − G ) − f (G )JGG = ′ ∗f (G )ε[]{(} tf (G∗ )tπ ) ∗f ′ (G∗ )
A
∗
∗
+ ′ ∗
−∆
+
t
+
E
−
G
,
E
f (G )(∆∗ )2ε
1 − tA (f (G∗ ))222

See Jones (1995) for a more detailed discussion of scale eﬀects in R&D-based growth models.

17

{
}
f (G∗ ) (
tπ )
1 (
tπ )
∗
1
−
f
(G
)
+
t
+
,
E
f ′ (G∗ )
ε
∆∗
ε
[
]
{
(
E∗
tπ ) ∗ } tAf ′ (G∗ )
∗
∗
E
JEG =
∆
+
G
−
t
+
, and
E
(∆∗ )2ε
1 − tA (f (G∗ ))2JGE =(tπ ) E ∗JEE = − tE +.
ε ∆∗From (20), we obtain(

1−tπ ) ′ ∗ ∗f (G )E − f ′ (G∗ )(1 − G∗ ) − f (G∗ ) = 0.
ε

(A.1)

By using (19) and (A.1), we can rewrite Jij (i, j = G, E) as follows:{}tAf ′ (G∗ )f (G∗ ),JGG = − ′ ∗ ∗ 1 + ρ
f (G )∆
1 − tA (f (G∗ ))2{(}(
f (G∗ )
tπ )
tπ )
∗
∗
JGE = ′ ∗ ∗1−f (G )∆ + tE +,f (G )∆εε{}E∗f ′ (G∗ )tA
JEG = ∗ 1 + ρ,∆
1 − tA (f (G∗ ))2(tπ ) E ∗JEE = − tE +
.
ε ∆∗

B.

Dynamics and stability when ∆∗ = 0
From the deﬁnition of ∆∗ , the case where ∆∗ = 0 implies that B̄ > 0, tA > 0 or B̄ = tA = 0.

First, we examine the case where B̄ > 0, tA > 0. We deﬁne G∆ as f (G∆ ) = tA /(1 − tA )B̄, that
is, G∆ satisﬁes ∆ = 0. When G = G∆ , (9) and (13) yield E∆ ≡ G∆ /(tE + tεπ ). Therefore,Ġ = Ė = 0 holds, and the economy jumps to the steady state immediately. This result seems
to imply that there are multiple steady states; G∗ and G∆ . However, G∆ cannot hold theno-arbitrage condition unless G∗ = G∆ . We rearrange equation (21) as follows:()
tE + tεπ 11 tE + tεπtA
1
1 + tE
=
+
B̄ −
−
G
.
1 − tA f (G)1 − tεπ f (G) ρ 1 − tεπ
1 − tεπ
|
{z
}

(B.1)

−∆

As shown in Figure B.1, the intersection of the LHS and RHS of (B.1) determines G∗ . WhenG∗ ̸= G∆ , the following inequality holds:)
tE + tεπ1
1 ( tE + tεπ
1 + tE+−G> 0.
∆
1 − tεπ f (G∆ ) ρ 1 − tεπ
1 − tεπThus, this inequality implies that the no-arbitrage condition cannot hold.
Second, we investigate the case where B̄ = tA = 0. From (9) and (13), we obtain
(
tπ )
E.
G = tE +
ε
18

(B.2)

Figure B.1

Figure B.2

Equations (6), (16), and (B.2) yield the following autonomous dynamic system, with respect toG:()
1 − tεπf ′ (G)G Ġ
1+=f (G)G − f (G)(1 − G) − ρ ≡ ψ(G).
f (G) GtE + tεπ
Note that ψ(0) < 0 and ψ(G) is increasing in G (or U-shaped). As shown in Figure B.2, thesteady state is unstable, that is, the economy jumps to the steady state immediately.

C.

Proof of Lemma
We examine the eﬀects of the ﬁscal variables on G∗ through changes in the ﬁscal variables.

The eﬀect of B̄ can easily be examined by using (21). When B̄ increases, the horizontal line
goes up, as shown in Figure C.1. Thus, G∗ decreases.

Figure C.2

Figure C.1

19

Diﬀerentiating Λ(G) with respect to tA , tE , and tπ , we obtain
1
1
∂Λ
=
> 0,
2
∂tA G given (1 − tA ) f (G)
}
∂Λ
1 { 1
1
=
+
(L
−
G)
> 0,
∂tE G given 1 − tεπ f (G) ρ}∂Λ
1
1 + tE { 1
+
(L
−
G)
> 0.
= (
)
∂tπ G given ε 1 − tπ 2 f (G) ρεWhen the tax rates ti (i = A, E, π) increase, the graph of the RHS of (21) goes up (see FigureC.2). Therefore, G∗ increases.

D.

Eﬀects of ∆∗ through changes in ﬁscal variables
Diﬀerentiating ∆∗ with respect to ti (i = E, π) yields∂∆∗tA
f ′ (G∗ ) ∂G∗=−. (i = E, π)∂ti
1 − tA (f (G∗ ))2 ∂ti

By using the Lemma, we obtain ∂∆∗ /∂ti < 0 (i = E, π). To study the eﬀect of B̄ and tA , werewrite ∆∗ using (22), as follows:
[ (
]
(
1
tπ )
tπ )
1
∗
∗
∆ =
− tE +− tE +
ρ+ (1 + tE )G .
εf (G∗ )ερ(1 − tεπ )(D.1)

Diﬀerentiating ∆∗ with respect to B̄ and tA , we obtain
[(
] ∗
∂∆∗
1
tπ ) f ′ (G∗ )
∂G
=
tE +
ρ
+ (1 + tE )
,
tπ
∗
2
ε
(f (G ))∂ B̄∂ B̄
ρ(1 − ε )[(] ∗tπ ) f ′ (G∗ )
∂∆∗
1
∂G
=
tE +
.
ρ
+ (1 + tE )
t
∗
2
π
∂tA
ε
(f (G ))∂tA
ρ(1 − ε )From the Lemma, ∂∆∗ /∂ B̄ < 0 and ∂∆∗ /∂tA > 0 both hold.

E.

Sign of Φ(G∗ )We investigate the sign of Φ(G∗ ) under Ṅ > 0 (i.e., 0 ≤ G∗ < Ĝ). Diﬀerentiating Φ(G∗ )with respect to G∗ , we obtain
Φ′ (G∗ ) = −f ′ (G∗ )(1 − G∗ ) + f (G∗ ) Q 0 ⇐⇒ G∗ Q Gg .
Therefore, Φ(G∗ ) and G∗ follow a U-shape. Assumption (24) yields f (0) > (ε − 1)ρ/(1 − tπ ),and thus we obtain
Φ(0) < −ρtπ − ρ(ε − 1)

20

tπ
< 0.
1 − tπThe deﬁnition of Ĝ yields
f (Ĝ)(1 − Ĝ) =(ε − 1)ρ
.
1 − tπ(E.1)

By using (E.1), we obtain
Φ(Ĝ) = −ρtπ (ε − tπ )< 0.
1 − tπThus, the sign of Φ(G∗ ) is negative when 0 ≤ G∗ < Ĝ.

F.

Condition of Ω(0) > 0
From (33), the necessary condition for Ω(0) > 0 is f ′ (0) > f (0). In this case, ∂Ω(0)/∂ε < 0apparently holds. Diﬀerentiating Ω(0) with respect to ρ and f ′ (0) yields
[
](
)
1
1
∂Ω(0)
=− (
f ′ (0) − f (0) < 0,)2 +∂ρρε(ε − 1)
f (0) + ρ∂Ω(0)−ρ2 ε(ε − 1) + (f (0))2 + ρf (0){} .
=∂f ′ (0)ρε(ε − 1) (f (0))2 + ρf (0)From the assumption in (24), we obtain −ρ2 ε(ε−1)+(f (0))2 +ρf (0) > 0. Thus, ∂Ω(0)/∂f ′ (0) >0 holds. From these results, when ε and ρ are suﬃciently small and f ′ (0) is suﬃciently large,Ω(0) > 0 holds.

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22