The Conditions for Indeterminacy in Two
Types of Balanced Budget
Rules − Reconsidered

Fujio Takata

February 2018
Discussion Paper No.1806

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANThe Conditions for Indeterminacy in Two
Types of Balanced Budget
Rules−Reconsidered ∗Fujio Takata †February 11, 2018

Abstract
When governments levy taxes on labor income on the basis of abalanced budget rule, and if a countercyclical income tax is adopted,
two steady states in an economy exist, of which one can cause indeterminacy paths and a saddle path. Conversely, without a countercyclicalincome tax, indeterminacy disappears, as does one steady state, even
if a balanced budget rule is adopted. This holds regardless of labor
supply adjustment.

∗

I thank Professor Jeﬀrey Herrick for checking my English in this paper. Naturally, I
accept full responsibility for any errors.
†
Institutional aﬃliation: Graduate School of Economics, Kobe University. e-mail address: jikairf@clock.ocn.ne.jp.

11

Introduction

The literature on indeterminacy is substantial. Many scholars, like Benhabib
and Farmer, deal with this hot topic from the viewpoint of naturally produced
economic outcomes, such as externality and monopolistic competition. But
we are interested in artificial determinants such as fiscal policy. Along thisline, Guo and Harrison (2004) argues that fiscal policies alone can cause
indeterminacy to disappear. The foundation for this is based on a balanced
budget rule with fixed tax rates. In general, that rule can be classified into
two categories: one in which government spending is fixed and tax rates arecorrespondingly determined in order to be equal to that fixed spending, theother in which tax rates are fixed and government spending is equal to therevenue determined under these fixed tax rates. Below, we denote the former
as balanced budget rule I, the latter as balanced budget rule II.
Following Schmitt-Grohé and Uribe (1997), Guo and Harrison (2004)
argues that under the second category this indeterminacy never occurs, while
under the first category it can occur. However, when they make this assertion,
they base it on two economies, one with indivisible labor, one with divisible
labor. A decisive diﬀerence between these two economies exists: the former
has full participation in the labor market, while the latter always has partial
participation in the labor market. Since Schmitt-Grohé and Uribe (1997)
deals with an economy with indivisible labor, which derives from Hansen(1985) and Rogerson (1988), uncertainty remains in the labor market. Thisphenomenon apparently aﬀects their logic. If one seeks an explanation forthe disappearance of indeterminacy, one should keep other factors unchanged,
except for the fact postulated as a cause. In this context, Guo and Harrison
should have assumed and analyzed only an economy with divisible labor. We
analyze their logic from this viewpoint.
In what follows, our model is introduced in the next section. In section 3,
the model is converted to dynamics and a linearization system is established.
Subsequently in section 4, two diagrams in relation to economic movements
are depicted. Finally, section 5 is devoted to conclusions.

2

The model

Guo and Harrison (2004) oﬀers a model with a diﬀerent assumption from thatof Schmitt-Grohé and Uribe (1997), as follows:its maximization problem is2

expressed as

∫ ∞
max

Ct ,Ht

−ρt

e
0

(
)
AHtγ+1logCt −
dt
γ+1

(1)

and assumes the budget constraint as
K̇t = (1 − τh ) Ht wt + (1 − τk ) (rt − δ) Kt + Tt − Ct .
Here, Ct means individual household consumption, Ht hours worked, γ theinverse of the intertemporal elasticity of substitution in the labor supply, andρ the subjective discount rate.
Furthermore, Kt means the household’s capital stock, wt the real wage
rate, rt the real rental rate of capital, τh the labor income tax rate andτk the capital income tax rate. Both tax rates are constant. δ means thedepreciation rate of capital and Tt the lump-sum transfers.
At this stage, it is crucial that Guo and Harrison (2004) assumes an
economy with divisible labor, contrary to that in Schmitt-Grohé and Uribe(1997). The former implies an economy with intensive arrangements in thelabor supply, while the latter implies one with extensive arrangements in thelabor supply. In this situation, Guo and Harrison insist that in the former
economy indeterminacy disappears because of balanced budget rule II alone,while in the latter economy indeterminacy can appear due to balanced budget
rule I alone.
In general, in order to discover a cause, one should form a hypothesis
associated with the cause, in this case, the diﬀerence between the balanced
budget rules, with other factors unchanged. Otherwise, other factors may
cause the diﬀerence in the phenomenon which the hypothesis assumes. Guo
and Harrison (2004) deals with indeterminacy problems based on diﬀerent
economies. In order to avoid this defect, it is better to discuss the economies
with the same properties except for the balanced budget rule alone. Though
Guo and Harrison (2004) deals with an economy with divisible labor andbalanced budget rule II, it should demonstrate the appearance of indeterminacy in an economy with divisible labor and balanced budget rule I, like
Schmitt-Grohé and Uribe (1997). Guo and Harrison mention a model withindivisible labor only for analytical tractability, but the two economies definitely are diﬀerent.1 To highlight this, Takata (2017) oﬀers a demonstration
with detailed analyses, stating what conditions can cause the indeterminacy
in a divisible labor economy.
1

See footnote 6 in Guo and Harrison (2004).

3

So, first, we will convert Guo and Harrison (2004) to a form comparablewith that demonstration.
(1) is modified as
∫ ∞
max

(
−ρt

e

Ht1−χ
logCt −
1−χ

)
dt.

(2)

K̇t = (1 − τ ) Ht wt + ut Kt − Ct .

(3)

Ct ,Ht0Here, A is assumed to be 1, γ is denoted as -χ.

Here, τh = τ , τk = 0, rt = ut , δ = 0 and Tt = 0. The modification describedabove implies that capital tax is omitted, capital is not depreciated and that
there are no lump-sum transfers. These assumptions are for simplicity andto focus attention on labor and labor income taxation.
Second, there are many identical firms, which are competitive. The
representative firm produces Yt utilizing a Cobb-Douglas production function, which has a constant-to-scale property, as follows: Yt = F (Kt , Ht ) =Ktα Ht1−α , where 0 < α < 1 implies the share of capital in output. The firmaims to maximize its profit and consequently wt = (1 − α)ktα and ut = αktα−1hold. Additionally, we assume the output price is 1.
Third, the government spends all labor income taxes, so government
spending fully depends on labor income taxes. This features balanced budget
rule I. We can describe the Hamiltonian L as
)
(
Ht1−χ
−ρt
+ µt (Kt ut + (1 − τ )Ht wt − Ct ) .
(4)
L=elog Ct −1−χBased on (4), we obtain the first-order conditions as[](1 − τ )wt − χ1
Ht =
,
Ct

(5)

Ċt
= ut − ρ.
(6)Ct(5) shows the labor supply, (6) the consumption Euler equation, respectively.
The transversality condition can be expressed as
kt e−ρtlim= 0.
t→∞ Ct4

3

Dynamics

Our dynamic system can be expressed as two simultaneous diﬀerential equations.
(χ((1 − α)τ − 1) + α)e(α−1)λt + χ((α − 1)(τ − 1))1/χ e
λ̇t =
α−χ

(χ−1)δtχe(α−χ)λt
χ

−ρ

,

(7)

and
δ̇t = αe(α−1)λt − ρ.

(8)Here, λt = log(kt ), δt = log(Ct ).2
This diﬀerential system has a unique fixed point (λ∗ , δ ∗ )3 . The result is( )log αρ
∗
λ =
,
α−1
(
)
(9)
(ρ)
(1−(1−α)τ )χ
log
α
log
(1−α)(1−τ
)
α
+
.
δ∗ =
α−1
χ−1
At this stage, we intend to establish a linear approximation system around(9). The Jacobian matrix J is showned as
(
)
0 a12
J=
a21 a22 .
∂f1
∂f2
∂f2
, a21 =
, and a22 =
.
∂λ
∂δ
∂λ
Moreover, a12 , a21 and a22 are respectively values evaluated at (9).
We calculate a12 , a21 and a22 in order:

Additionally, we postulate that a12 =a12 = (−1 + α)ρ,
a21 =
a22 =
2
3

( ρ ) (χ − 1)((α − 1)τ + 1)
(α − χ)α,( ρ ) (1 − α)τ (α − (2 − α)χ) + (α − 2)(α − χ)αχFor deduced processes in the above, see Appendix I.
For a proof, see Appendix II.

5.

We would like to know the signs of the products of eigenvalues of J, which
can be identified by |J|.
( ρ ) (χ − 1)((α − 1)τ + 1)
|J| = −a12 · a21 = −(−1 + α)ρ ·
<0
(10)
α
(α − χ)Since the determinant of J implies the product of two eigenvalues of J, we
can know that one eigenvalue is positive, the other negative and that oursystem has a saddle path. Indeterminacy obviously disappears in the case ofbalanced budget rule II.

4

Diagrams

Our aim is to identify the reason for the appearance and disappearance ofindeterminacy in budget rules I and II. So, we utilize diagrams, in which
the determinant and the trace of the Jacobian matrix in both economies are
drawn, when τ is chosen.
First, in the case of balanced budget rule II, we have established thedeterminant as( ρ ) (α − 1)ρ(χ − 1)((α − 1)τ + 1)
DJ (τ ) = −,α(α − χ)which has properities such as
d Dj (τ ) ( ρ ) (1 − α)(α − 1)ρ(χ − 1)
=
> 0,
dτ
α
(α − χ)( ρ ) (1 − α)ρ(χ − 1)
< 0, andDJ (0) =αα(α − χ)(1 − α)ρ2 (χ − 1)< 0.
α−χOn the other hand, the trace Tj (τ ) has been deduced as(ρ) 1
Tj (τ ) =[(α − 1)((2 − α)χ − α)τ + (α − 2)(α − χ)]α χDJ (1) =which has properties such as
dTj (τ ) ( ρ ) 1=(α − 1)((2 − α)χ − α) < 0,
dτ
α χ
6

1 (ρ)(α − 2)(α − χ) > 0 and
χ α
)
(
1
Tj (1) = ρ −α − + 2 > 0.
χTj (0) =Based on the above analyses, we can show the diagrams below.

Tj (0)•Tj (τ )• Tj (1)
•

0

1

τ

• Dj (1)Dj (0) •Dj (τ )
Figure 1: In a range of 0 < τ < 1, a unique saddle path occurs.
Second, in the case of balanced budget rule I, we obtained figure 2.

7

Dj (τ ∗ )Tj (τ ∗ )τ ∗ = α−χ1−χ

τ ⋆⋆•1 τ ∗0Dj (τ ∗ )Tj (τ ∗ )Figure 2:
0 < τ∗ <

Given

α−χ
< τ ∗ < τ ⋆⋆ , indeterminacy paths exist; when1−χα−χ, or τ ⋆⋆ < τ ∗ < 1, a saddle path occurs.
1−χ

Figure 2 shows a possibility of indeterminacy. In addition, in Figure 1,
τ is an arbitrary tax rate imposed by the government, while in Figure 2,
τ ∗ shows the corresponding values at steady states when the government
8determines its spending.4

5

Conclusions

We demonstrated that in a economy with balanced budget rule I, indeterminacy can appear, while in a economy with rule II, it does not. Our demonstration is based on analyses focusing on whether tax rates are given or not.
Guo and Harrison (2004) briefly says, ”Suppose that the future return on
capital is expected to increase. Without taxes, indeterminacy cannot occur
since a higher capital stock is associated with a lower rate of return underconstant returns-to-scale. However, a balanced-budget rule with countercyclical income taxes can cause the after-tax return on capital to rise, thus
validating agents’ initial optimistic expectations.” This statement, as far as
it goes, is true.
However, Guo and Harrison discuss diﬀerent economies, that is, one withdivisible labor and the other with indivisible labor. They deploy divisible
economy, then compare the result with the indivisible economy analyzed by
Schmitt–Grohé and Uribe (1997). To make their conclusion accurate, theyneed to analyze an economy with divisible labor alone, as we demonstrated
in this paper.5 But if they conjecture that both economies have the samefeatures in relation to movement, they should demonstrate it.6

6

Appendices

6.1

Appendix I

We establish our final system, which consists of two variables, λ and δ. Basedon (3) and the definition of kt , we can deduce the following:k̇t = −

Ḣt ktCt−+ kt ut + (1 − τ )wt
Ht
Ht

4

See Takata (2017) for details.
See Takata (2017).
6
See Mulligan (1999).
59

which is transformed to()
Ċt ẇtkt−()
Ct w tCt−1/χk̇t = −
− Ct
+ kt ut + (1 − τ )wt .
χ(1 − τ )wt
Here, (5) was utilized. Moreover, substituting the following relationships
ut = αktα−1 , wt = (1 − α)ktα ,Ċt= αktα−1 − ρ,Ct

we finally obtain
χ−1

α

k̇t
χ((α − 1)(τ − 1))1/χ Ct χ ktχ
=
kt

−1

+ (α(−τ )χ + α + (τ − 1)χ)ktα−1 − ρ.
α−χMoreover, when we replace log (kt ) with λt and log (Ct ) with δt , we reach (7).
(8) obviously holds.

6.2

Appendix II

Next, we proceed to (9). In order to obtain the steady state, we respectively
set λ̇ = 0 in (7) and δ̇ = 0 in (8). This manipulation means that we makethe right side in (7) and that in (8) equal to null. Then, we solve these
two equations for λ and δ. In these processes, it is convenient to transformλ(χ−1)
δ(χ−1)
variables such as e χ = q and e χ = y. Finally, we can obtain (9).

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Discussion Paper, Kobe University, No. 1702.

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