ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E07-1

Fiscal and Monetary Policy in an Endogenous
Growth Model with Public CapitalbyToshiki Tamai

February 2007

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Fiscal and Monetary Policy in an Endogenous
Growth Model with Public CapitalToshiki Tamai∗
January 30, 2007

Abstract
This paper describes an endogenous growth model in which the government finances its public investment using both income taxation andseigniorage. Arguments presented in this paper show that monetary policy always has positive impacts on economic growth and the inflation rate.
Furthermore, we show that the growth-maximizing tax rate on income isless than the output elasticity of public capital on the BGP, a suitable
result compared to those of empirical studies.

Keywords: Fiscal policy; monetary policy; economic growth; welfare
JEL classification: H54; O11; O23∗ Address: Graduate School of Economics, Nagoya University, Furocho, Chikusa-ku,
Nagoya, 464-8601, Japan E-mail: o060539r@mbox.nagoya-u.ac.jp (Toshiki Tamai)

11Introduction

An extensive body of literature examines endogenous growth models with government expenditure. Barro (1990) presents a pioneering approach in which
productive government expenditure engenders sustained per-capita growth in
the long run. In that model, productive government expenditure positively affects the production of final goods and generates endogenous growth in the longrun.1 Futagami et al. (1993) extend the Barro model by assuming that public
capital has a positive effect on aggregate production. They subsequently examine whether the results derived by Barro (1990) are achieved in their model.
In particular, one strand of recent literature compares the different meansof financing public investment in the growth process. For example, within an
endogenous growth model with public capital, Greiner and Semmler (2000) examine the effects on growth of an increase in public investment that is financedby tax and government debt. On the other hand, few studies have addressed theeffect of money-financed public investment in public capital. Public investment
financing involves the use of both seigniorage and income taxation. Indeed,
Basu (2001) shows that public investment positively correlates to seigniorage.
This empirical evidence might indicate that governments finance public investment through seigniorage. Therefore, investigation of fiscal and monetary policy
interactions might provide some pertinent insights.
In a framework of endogenous growth, some studies have examined the effectsof tax-financed and money-financed government expenditures. Espinosa-Vega
1 Aschauer (1989) shows that public capital positively affects the output of final goods.

2and Yip (1999) introduce financial factors to compare the effects of moneyfinanced and tax-financed increases in non-productive government expenditure
on inflation and the economic growth rate. In contrast to non-productive government expenditures, another strand of recent literature centers upon analysesof productive government expenditure (e.g., Ferreira (1999) and Hung (2005)).
Ferreira (1999) develops a model in which government investment in public
education is financed entirely by an inflation tax (seigniorage). He concludes
that inflation is desirable, provided that seigniorage-financed investment in public education facilitates accumulation of private capital. Hung (2005) develops
endogenous growth models with productive government expenditure financed
using an income tax and seigniorage.2 He derives the composition of productive
government expenditure financing, showing that the demand for money plays
an important role in reducing the optimal income tax rate.
The purposes of this paper are to investigate the effects of public investment
financed by an income tax and seigniorage on economic growth rate, inflation
rate, and welfare. The spirit of the arguments presented here is common toexisting studies on fiscal and monetary policy interactions. One departure fromthe existing literature on fiscal and monetary policy interactions is the assumption that the stock of public investment contributes to the production of final
goods. Productive public inputs are stock variables in reality. Public capitalsuch as roads, highways, ports, and airports are representative examples. In

2 In his paper, that productive government expenditure is called public capital. However, it
resembles the setting of Barro (1990) rather than public capital in the sense defined by Arrow
and Kurz (1970) and Futagami et al. (1993)3

examining public capital, we are envisioning particular economic experiences inwhich public capital is essential for growth promotion.
Our main results can be summarized as follows: First, we show that thegrowth-maximizing tax rate on income is less than the output elasticity of public
capital when public investment is financed by income taxation and seigniorage,
and that a rise in the money creation rate has always had a positive impact on
the economic growth rate. Seigniorage plays an important role in reducing the
growth-maximizing tax rate on income when public investment is financed byincome taxation and seigniorage. Second, we demonstrate that a raised income
tax rate reduces (raises) the inflation rate if the income tax is lower (higher)
than the growth-maximizing one; in addition, an increase in the money creation
rate raises the inflation rate under a plausible assumption.
This paper is organized as follows: Section 2 presents a description of ourmodel. Section 3 solves the model and characterizes the transitional dynamics.
Section 4 describes an investigation of the effects of fiscal and monetary policy
on economic growth and inflation. Section 5 presents discussion of additional
issues. Finally, Section 6 concludes this paper.

2

The model

Consider a closed, single-good economy with identical and fully rational individuals and perfectly competitive firms. Final goods are producible using private
capital and public capital. The production function for each firm is formulated4asy = F (k, g) = ξk 1−α g α ,(1)where y is the real output, g is the real public capital, k is the real private capital,and ξ > 0 is the productivity parameter.3 A representative firm maximizes its
profit π = y − rk for given g. Then, the factor price of private capital is givenasr = f 0 (x) = (1 − α)ξxα ,(2)where x ≡ g/k.
The representative households in this economy live infinitely. The number ofhouseholds is normalized to unity; there is no population growth. The objectivefunction of the household isZ ∞U=u(c, m) exp(−ρt)dt,(3)0where c is the per-household consumption, m is the real money holding, ρ isthe subjective discount rate of ui > 0, and uii < 0 (i = c, m).4 The household
maximizes Eq. (3) subject toȧ = (1 − τ )(ra + π) − c − [(1 − τ )r + γp ]m + h.

(4)

3 Throughout the discussion in this paper, capital letters are used for nominal variables;
small letters are used to denote real variables.
4 This type of utility function, presented by Sidrauski (1967), is functionally equivalent tothe assumption of liquidity costs (see Feensra (1986)).

5

In the above, a ≡ k + m is the total available financial wealth, τ is the income
tax rate, γp ≡ ṗ/p is the inflation rate, and h is the amount of a lump-sumtransfer. Thereby, the Hamiltonian associated with the optimization problem
for the household can be set asmax H = u(c, m) + q [(1 − τ )(ra + π) − c − {(1 − τ )r + γp }m + h] ,c,mwhere q is the costate variable. Solving the optimization problem, the optimum
rule for real money holding and the dynamic path of consumption are given asum= (1 − τ )r + γp − δ,
uc
·
¸
uc
ucm ṁċ=−(1 − τ ) r − ρ +.
cucc · cuc m

We now assume that u(c, m) is logarithmic:u(c, m) = log c + η log m.

In the above, η > 0 represents the taste for real money holding. Therefore, we
obtain the money demand function and the evolution of consumption:ηc,(1 − τ )r + γp(5)ċ
= (1 − τ ) r − ρ.
c

(6)

m=

This government apportions tax revenues and seigniorage between public

6

investment and a lump-sum transfer (if it is available); it maintains the money
creation rate and the rates of income tax at constant rates θ and τ , respectively(i.e., Ṁ /M ≡ θ). Therefore, the flow budget constraint for the government canbe written asġ = θm + τ y − h = (θ − z)m + τ y.

(7)

We assume that government also maintains h/m at a constant rate, z, and thatθ ≥ z for analytical simplicity.
Finally, we obtain, using m ≡ M/p,γm = θ − γp .

(8)Indeed, we can find the resource constraint, y = c + ġ + k̇, using Eq. (2), Eq.
(4), Eq. (7), and Eq. (8).

3Dynamic system

This section specifically examines the long-run general equilibrium and investigates the determinacy of BGP. First, we construct a dynamic system. Herein,
λ and µ are defined respectively as c/m and k/m. Using (1), (2), and (4)-(8),λ̇ = [ηλ − (ρ + θ)]λ,(9)µ̇ = [α(1 − τ )ξxα − θ + ηλ]µ − (λ + θ − z),(10)ẋ = [τ − (1 − τ )x]ξxα + [(1 + x)(θ − z) + λx]/µ.

(11)

7

Equations (9)-(11) constitute a system of three equations in λ, µ, and x. We
define a stationary equilibrium as a solution of the system constituted by Eqs.
(9)-(11). In a stationary equilibrium, all of c, k, m, and g grow at the same rateγ ∗ , i.e., ċ/c = k̇/k = ṁ/m = ġ/g = γ ∗ , so-called BGP.
We now establish the existence, uniqueness, and stability of the stationary
equilibrium as follows.
Proposition 1 There exists a unique stationary equilibrium; a convergent pathto such an equilibrium is locally determinate if ρ + γp∗ > 0.
(Proof) In a stationary equilibrium, we have λ̇ = µ̇ = ẋ = 0. From Eq. (9), Eq.
(10), and Eq. (11),λ∗ = (ρ + θ)/η,(12)

1α(1 − τ )ξxα − θ + ηλ
=
,
µ
λ+θ−z

(13)

(1 − τ )ξxα+1 − τ ξxα1=.
µθ − z + (λ + θ − z)x(14)

Using Eq. (12), Eq. (13), and Eq. (14), P (x) is defined as

P (x) = τ ξxα−1 +α(θ − z)(1 − τ )ξxα−1− (1 − α)(1 − τ )ξxα
λ∗ + θ − z
ρ(θ − z)+ ρ.
+ ∗(λ + θ − z)x∗

8

(15)

The function, P , has the following properties: P (0) = ∞, P (∞) = −∞, andP 0 (x) = −(1 − α)τ ξxα−2 − α(1 − α)(1 − τ )ξxα−1−α(1 − α)(θ − z)(1 − τ )ξ(x∗ )α−2
ρ(θ − z)− ∗< 0.
λ∗ + θ − z(λ + θ − z)(x∗ )2(16)

Therefore, there exists x∗ such that P (x∗ ) = 0. Substituting x∗ for x in Eq.
(13), we obtain µ∗ . Linearizing the dynamic system of Eqs. (9)–(11) aroundthe stationary equilibrium, the linearized system is given as λ̇   J11

 

 
 µ̇  =  J  21

 

 
ẋ
J31


0
J22
J32



0   λ − λ∗ 






J23   µ − µ∗ 
,




J33
x − x∗

(17)

where

J11 = ηλ∗ = ρ + θ > 0,
J21 = ηµ∗ − 1,
J22 = α(1 − τ )ξ(x∗ )α − θ + ηλ∗ = α(1 − τ )ξ(x∗ )α + ρ > 0,
J23 = α2 (1 − τ )ξ(x∗ )α−1 µ∗ > 0,
J31 = x∗ /µ∗ > 0,
J32 = −[(θ − z)(1 + x∗ ) + λ∗ x∗ ]/(µ∗ )2 = [τ ξ(x∗ )α−1 − (1 − τ )ξ(x∗ )α ]x∗ /µ∗ ,J33 = −(1 − α)τ ξ(x∗ )α−1 − α(1 − τ )ξ(x∗ )α − (θ − z)/(µ∗ x∗ )= ατ ξ(x∗ )α−1 − (1 − τ )ξ(x∗ )α + ρ.

9The characteristic polynomial of the above linearized system Eq. (17) isχ3 − trJχ2 + ∆χ − detJ = 0,

(18)

where

trJ ≡ J11 + J22 + J33 = ατ ξ(x∗ )α−1 + 2ρ + θ − γ ∗ = ατ ξ(x∗ )α−1 + 2ρ + γp∗ ,∆ = J11 J22 + J11 J33 + J22 J33 − J23 J32 ,anddetJ ≡ |J| = J11 (J22 J33 − J23 J32 ).
Because J11 > 0, the sign of detJ depends on J22 J33 − J23 J32 :J22 J33 − J23 J32 = −ρ[(1 − α)(1 − τ )ξ(x∗ )α − ρ]− α(1 − τ )ξ(x∗ )α [(1 − α)(1 − τ )ξ(x∗ )α ] + ρατ ξ(x∗ )α−1 .

(19)Using Eq. (7) and ġ/g = γ ∗ ,ατ ξ(x∗ )α−1 = αγ ∗ −α(θ − z)
.
µ∗ x∗

(20)

Combining Eq. (20) with Eq. (19),J22 J33 − J23 J32 = −ρ(1 − α)γ ∗ − α(1 − τ )ξ(x∗ )α (γ ∗ + ρ) −10α(θ − z)
< 0.
µ∗ x∗

Then, we obtain detJ < 0. These conditions show that the characteristic polynomial Eq. (18) has two unstable roots and one stable root. Because both λand µ are jumpable variables and x is a state variable, the converging path tothe stationary equilibrium is uniquely determined. Q.E.D.

Under Proposition 1, the economic growth rate is given asγ ∗ = (1 − α)(1 − τ )ξ(x∗ )α − ρ.

(21)

Thus, the economic growth rate depends on the policy variables: τ , θ, and z.
In a stationary equilibrium, Eq. (8) is rewritten asγp∗ = θ − γ ∗ .

(22)The inflation rate, γp∗ , is positive (γp∗ > 0) if the money creation rate, θ, is higher
than the economic growth rate, γ ∗ (θ > γ ∗ ). On the other hand, the inflation
rate is negative (γp∗ < 0) if the money creation rate is lower than the economic
growth rate (θ < γ ∗ ).

4

Policy analysis

In this section, we investigate the macroeconomic effects of fiscal and monetary
policy. First, we examine the effects of government policy on the economic
growth rate. Second, we analyze the effects of government policy on the inflation11

rate.
To analyze the growth effects of government policy, it is sufficient to derive
the effects of changes in the income tax rate and money creation rate on ratios
of public capital to private capital, x. Differentiating Eq. (15) yields

1 + (1 − α)x∗ − α(θ − z)/(λ∗ + θ − z)τ ∂x∗
≡ ²1 = −τ ξ(x∗ )α−2 > 0, (23)∗x ∂τP 0 (x∗ )θ ∂x∗(ρ + α(1 − τ )ξ(x∗ )α )θλ∗
≡
²
=
−
> 0.
2
x∗ ∂θ
(λ∗ + θ − z)2 P 0 (x∗ )(x∗ )2(24)

Using Eq. (21), Eq. (23), and Eq. (24), we obtain
·
¸
µ
¶
∂γ ∗
α ∂x∗
α(γ ∗ + ρ)τ∗ α= (1 − α)ξ(x ) (1 − τ ) ∗
−1 =
²1 −
R 0,
∂τ
x ∂τ
τ
α(1 − τ )
(25)
∂γ ∗
∂x∗
α(γ ∗ + ρ)
= α(1 − α)(1 − τ )ξ(x∗ )α−1=
²2 > 0.
∂θ
∂θ
θ

(26)

From Eq. (25) and Eq. (26), we have the following proposition.
Proposition 2 The growth-maximizing tax rate on income is given asτ=α(γ ∗ + ρ) − (1 − α)(θ − z)ρ
.
γ ∗ + ρ + (θ − z)(1 − α)γ ∗Furthermore, a rise in money creation rate, θ, has a positive effect on the economic growth rate when θ > z.

12

(Proof) From Eq. (25) and ∂γ ∗ /∂τ = 0,α ∂x∗−1 + (1 − τ ) ∗
= 0.
x ∂τ

(27)

Using Eq. (23) and Eq. (25),(α − τ )ξ(x∗ )α−1 −(α(1 − τ )ξ(x∗ )α + ρ)(θ − z)= 0.
(λ∗ + θ − z)x∗(28)

After some manipulations, Eq. (28) yieldsτ=α(γ ∗ + ρ) − (1 − α)(θ − z)ρ
.
γ ∗ + ρ + (θ − z)(1 − α)γ ∗(29)

Q.E.D.

Equation (29) implies that the growth-maximizing tax rate on income is lessthan α if θ > z ≥ 0. This equation indicates that seigniorage plays a key role
in reducing the growth-maximizing tax rate on income when the government
finances its public investment through income taxation and seigniorage. If θ =z = 0, the growth-maximizing tax rate on income is τ = α, which is derived byBarro (1990) and Futagami et al. (1993).
Next we investigate the effects of changes in the income tax rate and money
creation rate on the inflation rate. Using Eq. (22), Eq. (25), and Eq. (26), we

13

have
∂γp∗
∂γ ∗
=−
R 0,
∂τ
∂τ
∂γp∗
∂γ ∗
=1−
.
∂θ
∂θ

(30)
(31)

From Eq. (30) and Eq. (31), we obtain the following proposition.
Proposition 3 First, θ > z is assumed. A rise in the income tax rate, τ ,reduces (raises) the inflation rate when the income tax is lower (higher) than
the growth-maximizing one. On the other hand, a rise in the money creation
rate, θ, raises the inflation rate if ²2 < θ/[α(γ ∗ + ρ)].
(Proof) From Eq. (30) and Proposition 2, the former half of Proposition 3 isderived. Using Eq. (26) and Eq. (31),∂γp∗α(γ ∗ + ρ)
=1−
²2 .
∂θ
θ

(32)

If the second term of Eq. (32) is less than unity, we have ∂γp∗ /∂θ > 0:

1>

∂γp∗
α(γ ∗ + ρ)θ
²2 ⇔ ²2 <⇔> 0.
θα(γ ∗ + ρ)∂θ

Q.E.D.

In a stationary equilibrium, the inflation rate negatively depends on theeconomic growth rate. When the income tax rate is less than the growth-14

maximizing rate, a rise in the income tax rate raises the economic growth rate.
Then, such a change reduces the inflation rate. However, a rise in the income
tax rate raises the economic growth rate if the income tax rate is less than thegrowth-maximizing rate. Consequently, such a policy raises the inflation rate.
The effect of an increase in the money creation rate on the inflation rate has
been broken down into two sets of effects. First is the positive inflation effectgiven by the first term of Eq. (31). Second is the negative inflation effect givenby the second term of Eq. (31). If ²2 < θ/[α(γ ∗ + ρ)], the growth effect of achange in money-supply growth rate is less than unity.5 Consequently, the first
effect always dominate the second effect, and a rise in money creation rate raises
the inflation rate.

5

Discussion

In this section, we discuss additional issues: welfare effects of fiscal and monetary
policy. We restrict our analysis of welfare effects of government policy to astationary equilibrium. Following Greiner and Hanusch (1998), we assume thatthe economy immediately attains a new stationary equilibrium after a change
in the income tax and money creation rate. A main objective of the present
study is to investigate whether growth maximization and welfare maximization
involve identical implications.

5 The assumption, ² < θ/[α(γ ∗ + ρ)], implies that one unit of increase in the economic
2
growth rate is less than one unit of the money creation rate.

15

At the stationary equilibrium, the indirect utility function is given as6U=ln(ρ + θ) + (1 + η)[ln(αγ ∗ + ρ) − ln(ρ + θ + η(θ − z))ρ+(1 + η)γ ∗(1 + η)[ln k(0) − ln(1 − α)]
+
.
ρ2
ρ

(33)

Differentiating Eq. (33) leads to
·
¸
∂U
1+η
1 ∂γ ∗
α
=
+
,
(34)
∂τ
ρ
αγ ∗ + ρ ρ ∂τ
·
µ
¶
¸
∂U
1
η(2 + η)(ρ + θ) − η(θ − z)α
1 ∂γ ∗=−+ (1 + η)+. (35)∂θρ(ρ + θ)(ρ + θ + η(θ − z))αγ ∗ + ρ ρ ∂θFrom Eq. (34), the welfare-maximizing tax rate on income tax is equal to thegrowth-maximizing one on the BGP.
When the government finances its public investment through seigniorage,
public investment always has a positive effect on economic growth (Proposition
2) and also has positive effect on the inflation rate (Proposition 3). A positive
growth effect engenders a positive welfare effect on the BGP. On the other
hand, a positive inflation effect tends to lower welfare on the BGP because arise in the inflation rate reduces real money holding. Then, a welfare-maximizing
growth rate of money creation exists, as denoted by θ∗ . Because sign∂U/∂τ =sign∂γ/∂τ , the welfare-maximizing tax rate on income is given as the growthmaximizing one for given θ. When the government sets the money creation rate

6 See the Appendix for a derivation of Eq. (33).

16

at θ∗ , the welfare-maximizing tax rate on income is given asτ∗ =α(γ ∗ + ρ) − (1 − α)θ∗ ρ.
γ ∗ + ρ + (1 − α)θ∗ γ ∗Consequently, the welfare-maximizing tax rate on income is also less than the
output elasticity of public capital on the BGP.

6

Conclusion

This paper developed an endogenous growth model in which the government
finances its public investment using both income taxation and seigniorage. The
main novelty of this model compared with those found in the literature on
endogenous growth with public capital is that it incorporates monetary policy.
Analyzing the model, we characterized the transitional dynamics and derive
the growth and welfare effects of fiscal and monetary policy. We showed thatmonetary policy always has positive effects on both economic growth and theinflation rate. In addition, we demonstrated that the growth-maximizing tax
rate on income is less than the output elasticity of public capital on the BGP.
Finally, we consider the direction of future research. Based on our model,
the composition of government public investment financing plays an importantrole in determination of the optimal income tax rate. Therefore, it will beinteresting to investigate whether public investment that is financed by public
debt engenders a higher or lower growth rate than that financed by income
taxation and seigniorage. Furthermore, it might be fruitful to analyze the effects

17

of fiscal policy on growth rates of an economy on the transition path. These
topics will be important avenues for future investigations.

18Appendix
On the BGP, Eq. (5), Eq. (6), Eq. (8), Eq. (21), and Eq. (22) yieldc(t) = c(0) exp(γ ∗ t),m(t) =ηηη
c(t).
c(t) = ∗c(t) =(1 − τ )r∗ + γp∗
γ + ρ + γp∗
θ+ρ

(36)
(37)

Furthermore, Eq. (4) at time 0 is written asγ ∗ (k(0) + m(0)) = (1 − τ )y(0) − c(0) − (γp∗ − z)m(0)=γ∗ + ρ
k(0) − c(0) − (γp∗ − z)m(0).
1−α

(38)

After some manipulations, Eq. (38) leads toc(0) =(ρ + θ)(αγ ∗ + ρ)k(0).
(1 − α)(ρ + θ + η(θ − z))(39)

Using Eq. (3), Eq. (36), and Eq. (37),Z ∞U=[(1 + η) ln c(0) − η ln(ρ + θ) + (1 + η)γ ∗ t] exp(−ρt)dt.

0

19

(40)

Substituting Eq. (39) for c(0) in Eq. (40) and integrating Eq. (40) with respect
to t from 0 to infinity, we obtainU=ln(ρ + θ) + (1 + η)[ln(αγ ∗ + ρ) − ln(ρ + θ + η(θ − z))ρ+(1 + η)γ ∗(1 + η)[ln k(0) − ln(1 − α)]
.
+
ρ2
ρ

20

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21

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