Discussion Papers In Economics
And BusinessIndeterminacy and utility-generating government
spending under balanced-budget fiscal policies
Takeo Hori and Noritaka Maebayashi

Discussion Paper 13-13

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPAN

Indeterminacy and utility-generating government
spending under balanced-budget fiscal policies
Takeo Hori and Noritaka Maebayashi

Discussion Paper 13-13

July 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPAN

Indeterminacy and utility-generating government
spending under balanced-budget fiscal policies∗
Takeo Hori†

Noritaka Maebayashi‡

Abstract
We reexamine indeterminacy and utility-generating public spending under balancedbudget rules in a simple one-sector growth model. The introduction of consumptiontax (subsidy) as well as subsidies for savings and labor modify indeterminacy conditions in the existing studies. We show that if consumption subsidies and income
taxes exist, indeterminacy occurs even when private and public consumption are
Edgeworth substitutes and public spending and leisure are separable in the utility
function. Indeterminacy also occurs even when they are weak Edgeworth complements if consumption tax and subsidies for savings and labor are present. Surprisingly, when they are strong Edgeworth complements, the stronger external eﬀects
of public consumption tend to lower the possibility of equilibrium indeterminacy inthe presence of consumption tax.

JEL classification: E32, E62
Keywords: utility-generating public spending, indeterminacy, balanced-budget rule∗

We are responsible for any remaining errors.
Corresponding author: College of Economics, Aoyama Gakuin University, 4-4-25, Shibuya, Shibuyaku, Tokyo 150-8366, JAPAN; Tel: +81-3-3409-8863; E-mail: t-hori@aoyamagakuin.jp‡
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
JAPAN; e-mail: gge010mn@mail2.econ.osaka-u.ac.jp
†

1

1

Introduction

It is well known that in the presence of certain of externalities or market incompleteness, awide variety of growth models exhibit equilibrium indeterminacy. It has been shown that
production and/or consumption externalities may be sources of indeterminacy.1 Other
possible sources of indeterminacy are public goods or services because they are typicalgoods that yield external eﬀects. In a model where public spending ﬁnanced by income
taxes exerts positive external eﬀects on the productivity of ﬁrms, Guo and Harrison (2008)
show that indeterminacy may arise if the external eﬀects of public spending are suﬃciently
strong.2,3The external eﬀects of public spending are not limited to productivity and can extend to the utility of households, which may be a source of indeterminacy. Fernández
et al. (2004) and Guo and Harrison (2008) introduce utility-generating public spending ﬁnanced by income tax with constant tax rates into otherwise standard one-sector
real business cycle (RBC) models and examine the conditions for indeterminacy underbalanced-budget rules. Assuming that private and public consumption are Edgeworth
substitutes, Fernández et al. (2004) show that if public consumption and leisure are nonseparable in the utility function, indeterminacy might arise. On the other hand, in a model
where public consumption and leisure are separable, Guo and Harrison (2008) show that
if private and public consumption are Edgeworth substitutes, indeterminacy never arises,
whereas the model economy exhibits equilibrium indeterminacy only when private andpublic consumption are Edgeworth complements and the external eﬀects of public consumption are suﬃciently strong.4 These studies show that under balanced-budget rules,
1

See Benhabib and Farmer (1994), Benhabib and Perli (1994), Benhabib and Nishimura (1998) andMino (2001), for studies on production externalities. Alonso-Carrera et al. (2008), Chen and Hsu (2007)
and others study indeterminacy and consumption externalities.
2
Extending the model of Guo and Harrison (2008), Kamiguchi and Tamai (2001) show that if productive public spending is ﬁnanced by consumption tax, indeterminacy does not arise.
3
Cazzavillan (1996), Raurich (2003) and Palivos, Yip and Zhang (2003) examine conditions for equilibrium indeterminacy in endogenous growth models with productive government spending.
4
Lloyd-Braga et al. (2008) study the same issue in a segmented asset market economy model of the
Woodford (1986) type, rather than standard one-sector RBC models. Under Edgeworth complementarity
between private and public consumption, they also show that with constant income tax rates, indeterminacy arises if the external eﬀects of public consumption are suﬃciently strong. In addition, they showthat if the consumption tax rate responds negatively to the tax base, indeterminacy is possible in asegmented asset markets framework.

2

preference structure is crucial for indeterminacy in standard one-sector RBC models withutility-generating public spending.
We also introduce utility-generating public spending into a standard one-sector RBC
model where public consumption and leisure are separable and then reexamine conditions
for indeterminacy under balanced-budget rules. In contrast to Fernández et al. (2004)
and Guo and Harrison (2008), we allow the government to ﬁnance its spending by consumption tax as well as income taxes. Subsidies for savings, labor and consumption are
also considered. All of these tax (subsidy) rates are assumed to be constant. We showthat even when private and public consumption are Edgeworth substitutes and public
consumption and leisure are separable in the utility function, indeterminacy arises in thepresence of consumption subsidies as well as income taxes. The indeterminacy conditions under Edgeworth complementarity between private and public consumption are as
follows. Even when the external eﬀect of public spending is not strong, indeterminacy
occurs if consumption tax and subsidies for savings and labor are present. When theexternal eﬀects of public spending are strong, the ﬁnancing of public spending is crucial.
If public spending is mainly ﬁnanced by income taxes, indeterminacy arises whereas it
does not arise if the revenue from consumption tax is a main source of public spending.
As the consumption tax rate increases, the occurrence of indeterminacy becomes unlikely.
In addition, unlike most existing studies on indeterminacy and externalities, we show thatas the external eﬀects of public spending become stronger, the model economy tends tobe more unlikely to exhibit indeterminacy under Edgeworth complementarity with strong
externalities. Our results suggest that in addition to preference structure, the designs ofﬁscal policy are crucial for the stability of the economy.
The rest of the paper is organized as follows. Section 2 presents the model. Conditions
for indeterminacy are examined in Section 3. Concluding remarks are in Section 4.

2

The Model

We follow the model presented in Subsection 3.2 of Guo and Harrison (2008). Departingfrom Guo and Harrison (2008), we consider consumption tax (or subsidy) and subsidies

3

for savings and labor as well as income taxation. Guo and Harrison (2008) focus only on
the case where private and government consumption are Edgeworth complements. We
examine the case where they are Edgeworth substitutes as well as the case where they
are Edgeworth complements.
Consider a competitive economy that consists of ﬁrms, households and the government.
Time is continuous and is denoted as t ≥ 0. The representative ﬁrm produces a single
ﬁnal good by using the following technology:
Yt = AKt α Lt 1−α , A > 0 and α ∈ (0, 1),(1)where Yt is the output, Kt is capital input and Lt is labor input at time t. Throughproﬁt maximization, the rental rate of capital, rt , and the wage rate, wt , are equal to themarginal products:( )α−1Kt
Yt= αA,
rt = α
Kt
Lt
( )α
Yt
Kt
wt = (1 − α) = (1 − α)A
.
Lt
Lt

(2a)
(2b)

The utility function of the representative household is given by
∫ ∞{
U=
0

}
(Ct θC Gt θG )1−σ
− BLt e−ρt dt, B, θC , θG , σ > 0 and σ ̸= 1,
1−σ

(3)

where ρ > 0 is subjective discount rate, Ct is private consumption, Lt is the labor supply,and Gt is the public spending of the government. The assumption θG > 0 implies thepresence of external eﬀects of public spending. σ is the inverse of the intertemporalelasticity of substitution in eﬀective consumption, Ct θC Gt θG . To ensure strict concavity
with respect to Ct , we assume θC (1 − σ) − 1 < 0. If σ = 1, then Ct and Gt are separable
and thus, the presence of Gt in (3) does not aﬀect the dynamic properties of the model.
Hence, we restrict our attention to σ ̸= 1. Assuming σ < 1, Guo and Harrison (2008)
consider only the case where Ct and Gt are Edgeworth complements. We examine thecase where Ct and Gt are Edgeworth substitutes (σ > 1), and the case where they are4

Edgeworth complementarity (σ < 1).5 In contrast to Fernández et al. (2004), Gt and Lt
are separable in (3).
The budget constraint of the household is given byK̇t = (1 − τr )rt Kt − δKt + (1 − τw )wt Lt − (1 + τc )Ct ,(4)

where τr , τw and τc (∈ (−1, 1)) are the tax (or subsidy) rates on capital income, labor
income and consumption, respectively. When τx > (<)0 holds (x = r, w and c), τx
denotes the tax (subsidy) rate. The capital depreciation rate is denoted as δ > 0.
The household maximizes (3) subject to (4) and the no-Ponzi game condition, whichyields the following ﬁrst order conditions:θC Ct θC (1−σ)−1 Gt θG (1−σ) = Λt (1 + τc ),(5a)

B = Λt (1 − τw )wt ,(5b)Λ̇t = {ρ + δ − (1 − τr )rt }Λt ,(5c)where Λt is the co-state variable associated with (4).
Under the balanced-budget rule, the budget constraint of the government is given by
Gt = τr rt Kt + τw wt Lt + τc Ct . Public spending, Gt , must be strictly positive in equilibrium.
Using (2a) and (2b), we rewrite the budget constraint of the government as
Gt = τ Yt + τc Ct ,(6)

where τ ≡ ατr + (1 − α)τw . The good market equilibrium condition is given byK̇t = Yt − δKt − Ct − Gt .
5

(7)

Many researchers have tested whether private and government consumption are complements or
substitutes. However, empirical evidence is mixed. Aschauer (1985), Bean (1986), and Kormendi and
Meguire (1995) ﬁnd evidence for substitutability. Karras (1994) ﬁnds evidence for complementarity, andNi (1995) ﬁnds that results are sensitive to model speciﬁcations. More recently, Hamori and Asako (1999)
and Chiu (2001) ﬁnd substitutability for Japan and Taiwan, respectively. Fleissig and Rossana (2007)
ﬁnd Morishima substitutes between government and private consumption from US data. In contrast,
Ohkubo (2003) ﬁnds that the two types of consumption may be complements or may be unrelated fromJapanese quarterly data. Fiorito and Kollintzas (2004) ﬁnd that they are complements from data for
European countries.

5

Using (1), (2a), (2b), (5b), (5c), (6) and (7), we can reduce the equilibrium conditions
to (5a) and the following four equations:{}Λ̇t = ρ + δ − (1 − τr )αA (Kt /Lt )α−1 Λt ,(8a)K̇t = (1 − τ )AKt α Lt 1−α − δKt − (1 + τc )Ct ,(8b)

Gt = τ AKt α Lt 1−α + τc Ct ,(8c)

B = Λt (1 − τw )(1 − α)A (Kt /Lt )α .

(8d)When τr = τw > 0, τc = 0 and σ < 1 hold, our model reduces to the one examined in
Guo and Harrison (2008).
To derive the steady state equilibrium in which Λt , Kt , Ct , Lt , and Gt are constantover time, let us denote K̂ ≡ K ∗ /L∗ , Ĉ ≡ C ∗ /L∗ , and Ĝ ≡ G∗ /L∗ .6 From (8a), (8b),(8c), Λ̇t = 0 and K̇t = 0, K̂, Ĉ and Ĝ are given by{K̂ =(1 − τr )αA
ρ+δ

1
} 1−α

,

(9a)

{(1 − τ )AK̂ α−1 − δ}K̂Ĉ =,1 + τc(9b)Ĝ = τ AK̂ α + τc Ĉ.

(9c)

To ensure Ĝ > 0 (or equivalently, G∗ > 0), at least one of τ and τc must be positive.
From (8d), we have Λ∗ = B{(1 − τw )(1 − α)AK̂ α }−1 . Apparently, K̂, Ĉ, Ĝ and Λ∗ are
uniquely determined. We rearrange (5a) as
θC Ĉ θC (1−σ)−1 ĜθG (1−σ) L∗ (1−σ)(θC +θG )−1 = (1 + τc )Λ∗ .
When (1 − σ)(θC + θG ) − 1 < (>)0 holds, the left-hand side (LHS) of the above equation
monotonically decreases (increases) from ∞ (zero) to zero (∞) as L∗ increases from zero
to ∞. Irrespective of whether σ < 1 or σ > 1 holds, there exists a unique L∗ . Then, K ∗ ,C ∗ and G∗ are also unique and there exists a unique steady state.

6

We use asterisks to indicate steady state variables.

6

3

Local Stability and (In)determinacy

To examine the local stability of the steady state, we consider a log-linear approximation
around the steady state. Let us deﬁne λt ≡ ln(Λt /Λ∗ ), kt ≡ ln(Kt /K ∗ ), lt ≡ ln(Lt /L∗ ),ct ≡ ln(Ct /C ∗ ), and gt ≡ ln(Gt /G∗ ). The log-linearized equilibrium conditions are[θC (1 − σ) − 1]ct + θG (1 − σ)gt = λt ,(10a)

λ̇t = (1 − α)(ρ + δ)(kt − lt ),{}{}1−τ 1−α1−τ
1−τ ρ+δk̇t =(ρ + δ) − δ kt +(ρ + δ)lt −− δ ct ,1 − τr1 − τr α
1 − τr ααY ∗(1 − α)Y ∗C∗
g t = τ ∗ kt + τl+τtc ∗ ct ,
G
G∗
G

(10b)

0 = λt + αkt − αlt ,

(10c)
(10d)
(10e)

where Y ∗ ≡ AK ∗ α L∗ 1−α is the output level in the steady state equilibrium. The derivations of (10a)–(10e) are in Appendix. Using (9c), (10a), (10d) and (10e), we eliminate lt
and ct from (10b) and (10c), which yields the following dynamic system:




λ̇t
k̇t

=





− (1−α)(ρ+δ)
α


0

m21

m22




λt

,

(11)

kt

where
[
]
τY ∗
1 − θG (1 − σ) 1−α1−τ ρ+δ
1 − τ (1 − α)(ρ + δ)
α G∗
m21 =
−
−δ
,
∗
1 − τrα2
1 − τr α(θC + θG τCGC∗ )(1 − σ) − 1[]
1−τ ρ+δ
(θC + θG )(1 − σ) − 1
m22 =
−δ
.
∗
1 − τr α(θC + θG τCGC∗ )(1 − σ) − 1
The system, (11), includes only one predetermined variable, kt . One of the eigenvaluesof the coeﬃcient matrix is apparently negative, − (1−α)(ρ+δ)< 0. If the other eigenvalue,αm22 , is positive, the steady state is saddle stable and locally determinate. If m22 isnegative, the steady state exhibits local indeterminacy. If θG = 0 holds, m22 is positive.7
Without the external eﬀects of Gt in (3), equilibrium indeterminacy never arises.
(1−α)(1−τw )1−τ
1−τ ρ+δIf θG = 0, we have m22 = 1−τ
+α > α,α −δ. Because of τr < 1, τw < 1, we have 1−τr =1−τrr
which ensures m22 > 0.
7

7

3.1

The Case of Edgeworth Complements

This subsection considers the local stability under σ < 1.
3.1.1

Stability under Strong Externalities

We ﬁrst consider the case where θG is large enough to satisfy (θC + θG )(1 − σ) − 1 > 0.
(∗)
Equilibrium indeterminacy arises only when θC + θG τcGC∗ (1−σ)−1 < 0 or, equivalently,(θC + θG )(1 − σ) − 1
τc C ∗ < τ Y ∗1 − θC (1 − σ)(13)holds. If the reverse inequality holds in (13), the steady state is locally determinate.
If public spending and consumption subsidy are ﬁnanced by income taxes (τ > 0 andτc ≤ 0), (13) is satisﬁed and hence the steady state is locally indeterminate. However,
if public spending and subsidies for saving and labor are ﬁnanced by consumption tax(τ ≤ 0 and τc > 0), (13) is violated and the steady state is locally determinate. The sourceof public spending is crucial for the stability of the economy. When public spending isﬁnanced mainly by consumption tax, equilibrium indeterminacy does not arise. If themain source of public spending is income taxation, indeterminacy tends to occur.
To obtain further results, assuming τ > 0 and τc > 0, we rearrange (13) to obtain1 − θC (1 − σ)
θG <
1−σ

{
}
1 + τc(ρ + δ)τ
1+≡ Γ.
τc ρ + (1 − α)δ − αρτr − (ρ + δ)(1 − α)τw

(14)

In deriving the above inequality, we use (9a), (9b) and (9c). It is shown that ρ + (1 −α)δ − αρτr − (ρ + δ)(1 − α)τw > ρ + (1 − α)δ − αρ − (ρ + δ)(1 − α) = 0 because of τr < 1and τw < 1. As shown in Figure 1, as τc increases from zero to +∞, Γ monotonicallydecreases from +∞ to Γ̂ > 0, where Γ̂ is1 − θC (1 − σ)
Γ̂ ≡
1−σ

{
1+

(ρ + δ)τ
ρ + (1 − α)δ − αρτr − (ρ + δ)(1 − α)τw}.

Apparently, Γ̂ monotonically increases with τr and τw . When τr = τw = 0, we haveθG > Γ̂ because (θC + θG )(1 − σ) − 1 > 0. When both τr and τw tend to one, Γ̂ tends to+∞(> θG ). Then, when τr > 0 and τw > 0 are suﬃciently small, θG > Γ̂ holds. There8

exists a unique τ̂c > 0 such that if τc < (>)τ̂c , we have θG < (>)Γ. If the government sets
τc larger than τ̂c , it can eliminate equilibrium indeterminacy. When τr and τw (> 0) are
suﬃciently large, θG < Γ̂ holds and we have θG < Γ for any τc > 0. Then, the government
cannot stabilize the economy by setting a high consumption tax rate. We obtain the next
proposition.
Proposition 1 Suppose 0 < σ < 1 and (θC + θG )(1 − σ) − 1 > 0.
1. If τ ≤ 0 and τc > 0, the steady state is locally determinate.
2. If τ > 0 and τc ≤ 0, the steady state is locally indeterminate.
3. Assume τ > 0 and τc > 0. (a) Given τr and τw , if θG > Γ̂, there exists a uniqueτ̂c > 0 such that if τc < τ̂c , the steady state is locally indeterminate, whereas it islocally determinate if τc > τ̂c . (b) Given τr and τw , if θG < Γ̂, the steady state islocally indeterminate for any τc > 0.
[Figures 1 and 2]
Apparently, τ̂c decreases with θG (see Figure 1). As the external eﬀects of Gt become
stronger, the range of τc where indeterminacy arises becomes narrower and hence indeterminacy is more unlikely to occur. Assuming τr = τw = τ , Figure 2 shows this resultgraphically on the (τ, τc ) plane.8 Equilibrium indeterminacy occurs in the shaded areas
while the steady state is locally determinate in the other regions. Apparently, as theexternal eﬀects increase, the shaded area becomes narrower and hence the steady state
tends to exhibit determinacy for wider ranges of tax rates.
The last result may be somewhat counterintuitive because most existing studies onindeterminacy and externalities show that as external eﬀects become stronger, indeterminacy tends to occur. The intuition behind Proposition 1 helps our understanding. We
focus on the case where τ > 0 and τc > 0 hold. Suppose that agents expect future public
spending to increase. Because of Edgeworth complementarity, the marginal utility of future private consumption is expected to increase. The household has an incentive to save
8

The parameter values in Figure 2 are A = 1, α = 0.36, δ = 0.1, σ = 0.3, ρ = 0.05, B = 1, andθc = 1.3.

9

more, which stimulates capital accumulation. Future output will be larger, which will
have positive eﬀects on future public spending. At the same time, capital accumulation
will raise the future wage rate. From (5b), future Λt will decrease. Because (5a) must
hold in equilibrium, future private consumption will increase, causing external eﬀects andincreasing future public spending further. If τc is large enough to satisfy τc > τ̂c , the external eﬀects of future private consumption on future public consumption are strong, andhence, future public spending increases more than expected. As a result, (5a) is violated,which cannot happen in equilibrium. The expectation is not self-fulﬁlling, and hence, thesteady state is locally determinate. As θG increases, the LHS of (5a) responds more toincreases in future public spending, and hence, (5a) tends to be violated. Then, as theexternal eﬀects increase, the steady state tends to exhibit determinacy for wider ranges oftax rates. When τc is not too large (τc < τ̂c ), increases in future private consumption have
only negligible eﬀects on future public spending. Thus, future public spending increases
as much as expected. The expectation is self-fulﬁlling, and hence, indeterminacy arises.
3.1.2

Stability under Weak Externalities

Next, we observe that in contrast to Guo and Harrison (2008), even when θG is suﬃciently
small such that it satisﬁes (θC +θG )(1−σ)−1 < 0, indeterminacy arises if consumption tax(∗)
is present. Indeterminacy arises only when θC + θG τcGC∗ (1 − σ) − 1 > 0 or, equivalently,θG (1 − σ)τc C ∗ > G∗
1 − θC (1 − σ)(15)holds. Because of (θC +θG )(1−σ)−1 < 0, we have 0 < θG (1−σ)/{1−θC (1−σ)} < 1. The
above inequality holds only when τc C ∗ > G∗ holds. The last inequality implies τc > 0 andτ < 0. In other cases, indeterminacy never arises. To obtain further results, we assume
τc > 0 and τ < 0 and rearrange (15) as θG > Γ, where Γ is deﬁned as in (14). Because∗

C (1−σ) GΓ is equal to 1−θ1−σ, Γ must be positive. When τ < 0, Γ increases from −∞ to Γ̂τc C ∗as τc increases from zero to +∞ (see Figure 3). We assume α < 1/2 to ensure Γ̂ > 0.9 If
θG < Γ̂ holds, there exist τ c (> 0) and τ c (> τ c ) such that θG > Γ > 0 holds for τc ∈ (τ c , τ c )Γ̂ has the same sign as the following equation: ρ + (1 − α)δ − αρτr − (ρ + δ)(1 − α)τw + (ρ + δ)τ =ρ + (1 − α)δ + αδτr > ρ + (1 − 2α)δ > 0. The assumption of α < 1/2 ensures the last inequality.
910

and θG < Γ holds for τc > τ c . If θG > Γ̂ holds, we have θG > Γ > 0 for τc > τ c . We obtainthe following proposition.
Proposition 2 Suppose α < 1/2, 0 < σ < 1 and (θC + θG )(1 − σ) − 1 > 0.
1. If τc ≤ 0 and τ > 0, or if τ ≥ 0 and τc ≥ 0 hold and at least one of them holds withstrict inequality, the steady state is locally determinate.
2. Assume τ < 0 and τc > 0. (a) When θG < Γ̂ holds, there exist unique τ c andτ c (0 < τ c < τ c ), given τr and τw . The steady state is locally indeterminate ifτc ∈ (τ c , τ c ) (0 < τ c < τ c ), whereas it is locally determinate if τc > τ c . (b) When
θG > Γ̂ holds, there exists a unique τ c , given τr and τw . If τc > τ c , the steady stateis locally indeterminate. In both cases, if τc < τ c , G∗ < 0 holds.
[Figures 3 and 4]
Apparently, τ c increases with θG , while τ c is independent of θG . Unlike the case whereθG satisﬁes (θC + θG )(1 − σ) − 1 > 0, as the external eﬀects of G increase, indeterminacy ismore likely to occur. Figure 4 presents this result.10 Indeterminacy occurs in the shaded
areas.11 As θG increases, indeterminacy occurs for wider ranges of tax rates.
The intuition behind Proposition 2 is as follows. Assume τ < 0 and τc > 0. Again,
suppose that agents expect future public spending to increase. As discussed following
Proposition 1, the expectation stimulates capital accumulation, thus increasing future
output, and bearing has negative eﬀects on future public spending. At the same time,
because of increases in the future wage rate, future private consumption will increase,thus bearing positive external eﬀects on future public spending. When τc is larger thanτ c , the positive eﬀect on future public spending dominates the negative one. However,if τc is too large (τc > τ c ), future public spending increases more than expected, whichcannot be equilibrium. If τc < τ c , future public spending increases as much as expected.
The expectation is self-fulﬁlling, and indeterminacy arises.
10

The parameter values in Figure 4 are A = 1, α = 0.36, δ = 0.1, σ = 0.3, ρ = 0.05, B = 1, andθc = 1.1.
11
In the areas to the left of the shaded areas, G∗ < 0 holds. In the other regions, the steady state islocally determinate.

11

3.2

The Case of Edgeworth Substitutes

This subsection will demonstrate that in contrast to Fernández et al. (2004) and Guo andHarrison (2008), even when Ct and Gt are Edgeworth substitutes (σ > 1) and Gt and Lt
are separable in (3), indeterminacy arises if consumption subsidy is present. When σ > 1,(∗)
indeterminacy arises only when θC + θG τcGC∗ (1 − σ) − 1 > 0 or, equivalently, (15) holds.
Because of θG (1−σ)/{1−θC (1−σ)} < 0, (15) holds only when τc < 0 and τ > 0. In othercases, indeterminacy never arises. To obtain further results, we assume τc < 0 and τ > 0
and then rearrange (15) as θG > Γ. Again, Γ must be positive in equilibrium because∗

C (1−σ)C (1−σ) Gof Γ = 1−θ1−σ. When τ > 0, Γ increases from 1−θ1−σ< 0 to +∞ as τc (< 0)τc C ∗

increases from -1 to zero (see Figure 5). Then, there exist τ c and τ c (−1 < τ c < τ c < 0)such that θG > Γ > 0 holds for τc ∈ (τ c , τ c ) and θG < Γ holds for τc > τ c . The next
proposition is obtained.
Proposition 3 Suppose σ > 1.
1. If τ < 0 and τc > 0, or if τ ≥ 0 and τc ≥ 0 hold and at least one of them holds withstrict inequality, the steady state is locally determinate.
2. Assume τ > 0 and τc < 0. Given τr and τw , there exist unique τ c and τ c (−1 <τ c < τ c < 0). The steady state is locally indeterminate if τc ∈ (τ c , τ c ), whereas it islocally determinate if τc > τ c . If τc < τ c , G∗ < 0 holds.
Apparently, τ c increases with θG while θG does not aﬀect τ c . As the external eﬀects
of G increase, indeterminacy is more likely to occur. In the shaded areas of Figure 6,
indeterminacy occurs.12 As θG increases, the steady state tends to exhibit indeterminacy
for wider ranges of tax rates.
[Figures 5 and 6]
The intuition of the above results is as follows. Assume τ > 0 and τ c < τc < 0.
Suppose that agents expect future public spending to increase. Because of Edgeworth
12

The parameter values in Figure 6 are A = 1, α = 0.36, δ = 0.1, σ = 2, ρ = 0.05, B = 1, and θc = 1.
We have G∗ < 0 in the areas to the left of the shaded areas. In the other regions, the steady state islocally determinate.

12

substitutability, the future marginal utility of private consumption will decrease. Capital
accumulation is discouraged and future output will decrease, which will have negativeeﬀects on future public spending. At the same time, the future wage rate decreases andthus, future Λt increases (see (5b)). Because (5a) must hold in equilibrium, future private
consumption will decrease, which will have positive eﬀects on future public spending.
When τc is smaller than τ c (< 0), the positive eﬀect on future public spending dominates
the negative one.13 Hence, the expectation is self-fulﬁlling, and indeterminacy arises.

4

Concluding Remarks

We reexamine indeterminacy and utility-generating public spending in a simple one-sector
growth model. We allow the government to ﬁnance its spending by consumption tax aswell as income taxes. Subsidies for savings, labor supply, and consumption are also considered. We show that even when private and public consumption are Edgeworth substitutes
and public consumption and leisure are separable in the utility function, indeterminacy
can arise with consumption subsidies. Indeterminacy conditions under Edgeworth complementarity between private and public consumption are as follows. (i) Even when theexternal eﬀect of public spending is not strong, indeterminacy occurs if public spending
and subsidies for income are ﬁnanced by consumption tax. (ii) When the external eﬀect ofpublic spending is strong, indeterminacy does not arise if the revenue from consumption
tax is a main source of public spending, whereas it arises if public spending is ﬁnanced
mainly by income taxation. (iii) When the external eﬀect of public spending is strong, as
the external eﬀects of public spending become stronger, the economy tends to be moreunlikely to exhibit indeterminacy. These results suggest that not only the preference
structure but also the designs of ﬁscal policy are crucial for indeterminacy conditions.

Appendix
After dividing both sides of (5a) by θC C ∗ θC (1−σ)−1 G∗ θG (1−σ) = Λ∗ (1 + τc ), we take thelogarithm to obtain (10a). We can derive (10e) by using (8d) in a similar way. From the13

Note τc < 0. Then, as τc is smaller, the positive eﬀect tends to dominate the negative one.

13

deﬁnition of λt , we have Λt = Λ∗ eλt . Note λt = 0 in the steady state. Around the steadystate, Λ̇ is approximated as Λ̇ = Λ∗ λ̇t . In the steady state, we have ∂Λt /∂λt |λt =0 = Λ∗ .
These relationships hold for Kt , Ct , Lt , and Gt . Then, (8a) is approximated as follows:
]
1 ∂Lt
1 ∂Kt
Λ λ̇t = Λ (1 − α)(ρ + δ)
kt − ∗
lt = Λ∗ (1 − α)(ρ + δ) (kt − lt ) .
K ∗ ∂kt kt =0
L ∂lt lt =0[∗∗

In the ﬁrst equality, we use (9a). Dividing the left- and right-hand sides by Λ∗ yields(10b). Similarly, (8b) is approximated as[]
K ∗ k̇t = (1 − τ )αAK̂ α−1 − δ K ∗ kt + (1 − τ )(1 − α)AK̂ α L∗ lt − (1 + τc )C ∗ ct .
After dividing both sides by K ∗ , we rearrange the resulting equation by using (9a) and (9b)to obtain (10c). Finally, (8c) is approximated as G∗ gt = τ αY ∗ kt + τ (1 − α)Y ∗ lt + τc C ∗ ct .
Dividing both sides by G∗ results in (10d).

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16

Γ

θG ↑
θG

Γ̂
0

τ̂c′

τ̂c

τc

Figure 1: The Case of Edgeworth Complements with Strong Externalities17θ =0.5

θ =0.25cτθ =0.75

G

G

θ =1

G

G

1

1

1

1

0.9

0.9

0.9

0.9

0.8

0.8

0.8

0.8

0.7

0.7

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0.2

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0.2

0.1

0.1

0.1

0.1

0

0
0

0.5
τ

1

0
0

0.5
τ

1

0
0

0.5
τ

1

0

0.5
τ

1

Figure 2: Indeterminacy under Edgeworth Complements with Strong Externalities

18

Γ

Γ̂
θG ↑
θG
0

τc

τ ′c

τc

τc

Figure 3: The Case of Edgeworth Complements with Weak Externalities19θ =0.05
G1

θ =0.1G

1

θ =0.15G1

θ =0.2
G

1

0.9

0.9

0.9

0.9

0.9

0.8

0.8

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0.6

0.5

c

τ

1

0.5

G*<0

0.4

0.4

0.3

0.5

G*<0

0.5

G*<0

0.5

G*<0

0.4

0.4

0.4

0.3

0.3

0.3

0.3

0.2

0.2

0.2

0.2

0.2

0.1

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0.1

0

−1

−0.
5
τ

0

0

−1

−0.5
τ

0

0

−1

−0.5
τ

0

0

−1

−0.5
τ

0

0

−1

θ =0.25
G

G*<0

−0.5
τ

0

Figure 4: Indeterminacy under Edgeworth Complements with Weak Externalities

20

Γ

θG ↑
θG

-1

′

τc

τc τc

0

Figure 5: The Case of Edgeworth Substitutes21τcθ =0.25
Gθ =0.5G0θ =0.75
G

0

−0.1

−0.
2

−0.2

−0.2

−0.2

−0.
3

−0.3

−0.3

−0.3

−0.
4

−0.4

−0.4

−0.4

−0.
5

−0.5

−0.5

−0.5

τ

−0.6

−0.
7

−0.6

−0.7

G*<0

τ

−0.1

τ

−0.1

−0.
6

−0.7

G*<0

−0.
8

−0.8

−0.8

−0.8

−0.
9

−0.9

−0.9

−0.9

−1

0

0.5

1

−1

0

0.5

1

−1

0

0.5

1

−1

Figure 6: Indeterminacy under Edgeworth Substitutes

22

G

−0.6

−0.7

G*<0

θ =1

0

−0.
1

c

τ

0

G*<0

0

0.5

1