The Appearance of Indeterminacy Paths-An
Economy with the Balanced Budget Rule andwithout Depreciation
Fujio Takada
February , 2020
Discussion Paper No.2001

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANThe Appearance of Indeterminacy Paths – An
Economy with the Balanced Budget Rule andwithout Depreciation
Fujio Takata
Graduate School of Economics, Kobe University;
1-1, Rokkodai-cho, Nada-ku, Kobe, 657-8501, Japan,
jikairf@clock.ocn.ne.jp.
February 6, 2020

Abstract
This paper deals with multiple movement patterns in an economy
with infinite capital duration. It demonstrates what conditions are required for indeterminacy. Moreover, itshows that the capital durationis unrelated with the movement patterns.
JEL classification: C61; C62; E20; E32
Keywords: Two movement patterns; Balanced budget rule;
Labor income taxation ; Divisible labor11Introduction

The theme that sun spots might aﬀect real economies is interesting. Jevonspreviously discussed the influence of sun spots on agricultural production
and business cycles. His assertions were naturally dismissed by orthodox
economists in his day, but his ideas are not unrooted fairy stories today, because we are facing the phenomenon of global warming, which will influence
our economies and business cycles.
In order to analyze sun spot problems, first of all, we need to elucidate
how indeterminacy is structured. Some factors have been found, such asexternal factors or monopolistic competition1 , increasing returns to scale,
externality, and taxation.2 Moreover, Kamihigashi (2002) deals with this issue in terms of externality and nonlinear discounting with respect to utility.
These studies seek the causes of indeterminacy in economic conditions, thatis, naturally produced economic outcomes. However, we focus on indeterminacy based on fiscal policies, such as labor income taxation and the balanced
budget rule. Since we can easily manage these fiscal policies intentionally,
we can induce concrete and desirable policies if necessary.
We adopt Schmitt-Uribe (1997) as our base-line model, and deduce aclear necessary condition for indeterminacy. Although our model features an
economy without depreciation, we obtain the same conclusions as those in an
economy with depreciation. In short, it becomes obvious that depreciation
is not related to movement patterns in an economy, in our context. But
Wen (2001) discusses the problem in terms of externality and depreciation.
In the next section, we deal with our base-line model. In section 3,
we deduce the necessary conditions for the indeterminacy in our economy.
Moreover, we verify the same conditions as those in an economy with depreciation. The fourth section is devoted to a conclusion.

2

The base line model
We describe a baseline model based on Schmitt-Uribe (1997) as∫ ∞e−ρt [log ct − AH] dt,max
ct ,Ht(1)0

subject toK̇t = ut Kt + (1 − τt )wt Ht − ct ,1
2

Benhabib and Farmer (1994).
Guo and Lansing (1998).

2t ∈ ℜ+ .

(2)

Here, we denote the meaning of notations; ℜ+ is the set of positive
reals; ρ indicates a subjective discount rate of utility, H labor inputs; and A
a marginal disutility of labor, respectively. Moreover, ct indicates consumption, Kt capital stock, τt labor wage income tax rates, ut real rental rates ofcapital, and wt real wage rates in the t period, respectively.
However, the assumptions in (1) and (2) have strong features. First,
Schmitt-Uribe (1997) adopts the same postulations concerning labor inputs
as those in Hansen (1985). Representative household agents are supposed
to choose whether to work or not in every time period. Thus, their disutilities from entering the labor market are naturally positive, while thosefrom quitting the market are null. In other words, the labor arrangements
involve extensive margins, which means that in the labor supply uncertainty
exists. Therefore, (1) shows the present value of an expected utility. Second, because Hansen (1985) assumes two cases, entering and quitting thelabor market, the disutilites corresponding the two cases have two values,
and one of them is supposed to be null. This leads to the fact that the expected disutility function really consists of one argument. In short, Hansen(1985) normalizes the disutility function so that when the agents quit thelabor market, their disutilities are null. Third, Hansen (1985) assumes anarbitrary constant labor input in every time period, when the representative agents dicide to work. As a result, the utility function partly consistsof linear labor inputs.
But judging from common sense, due to fatigue from long labor or fromaging, people can not naturally oﬀer unlimited labor, regardless of incentives
such as real wages.
Accepting what we have said above, if depreciation rates are supposed
to be null, the model of Schmitt-Uribe (1997) can be modified as in (1) and(2).
Furthermore, we postulate the balanced budget rule, as described below.
G = τt wt Ht .

(3)

Here, G indicates government expenditure. Since the aggregate demand
equalizes the aggregate supply,
ct + G + Kt = F (Kt , Ht ) = Ktα Ht1−α(4)prevails. Additionally, F indicates a production function. 0 < α < 1 is aconstant.

3

3

The movements of our economy

We proceed to analyze our system and to introduce two movement types.
In light of the budget constraint in (2), as the first conditions of optimization of (1), we obtain the following system:
A = Λt (1 − τt ) wt ,

(5)

Λ̇t
= ρ − ut ,
Λt

(6)

1
− G,
Λt

K̇t = F (Kt , Ht ) −

(7)

and
G = Ht τt wt .

(8)Additionally, Λ = 1/c.

3.1

The labor wage incomesFirst, we explore how G is financed as τ s are determined. As a preparation for these analyses, we define sii and sh , and deduce some important
relationships, as below.
Ksii == H α−1 K 1−α =
F

(

K
H

)1−α
=

and
Hw
sh =
= wH α K −α = w
F

(

K
H

α
sk
1 − sh
=
=
,
ρ
ρ
ρ

)−α

= 1 − α.

Thus, we can express G as
(
G=

sh
sii

)

(
τK =

wH
K

)
τK

Based on (9), we intend to explore G.
( )
( )
sh
∂K
∂G
sh
G
G
∂K
K+
τ×
=
=
+
×
,
∂τ
sii
sii
∂τ
τ
K
∂τ
where,
∂K
K
F Ksh
−
=
∂τ
F (1 − τ )sh + F ρsii 1 − τ
4

(9)

(10)

=

Ksh
K
−
(1 − τ )sh + ρsii 1 − τprevails. Furthermore, after some calculations, the rate of change of G
regarding τ can be deduced as
((
)
)
∂G
1
τ (1 − τ )sh=G− τ + (1 − τ ) ,∂ττ (1 − τ )(1 − τ )sh + ρsii
which results in()−2τ (sh + ρsii ) + (sh + ρsii ) + τ 2 sh
∂G
=G×
.
∂τ
τ (1 − τ ) ((sh + ρsii ) − τ sh )(11)

Simplyanalyzing (11), it becomes obvious thatin the range 0 < τ <√√1− 1−sh1− 1−sh, G(τ ) increases, while in the range< τ < 1, it decreases.
sh
sh

3.2

The steady states

Our economy is composed of (5) through (8), in which two equations
showing how to move and two equations showing how to keep constant areincluded.
Second, in this system, we need to demonstrate the existence of steadystates.
From (6),( )∗ ( ) 1
Kρ − 1−α=(12)Hαprevails. (6) is derived from the fact that in the steady states, the marginal
production of capital equals the time preference of utilities. Since the capital
equipment regarding labor,(k/H)∗ , is determined and K ∗ is predetermined,
H is determined as( ) 1∗∗ ρ 1−α
H =K
,
(13)
α
then K as(1 − τ ∗ ) w∗].
K∗ = [(14)() ( H )∗∗A ρ + FH K
,
1
(1
−
τ
H
K
Moreover, τ is determined as
G = Hτ w = τ w

∗

(

K∗
k∗

5

)
,

∗

k =

(

K∗
H

)
.

(15)

3.3

Linearization
Here, we transform the above variables as follows:( )Λt∗λt = log Λt − log Λ = log
,
Λ∗
(
)
Kt
∗
kt = log Kt − log K = log
,
K∗
)
(
Ht
,
ht = log Ht − log H ∗ = log
H∗
(τ )
t
τ̃t = log τt − log τ ∗ = log ∗ .
τThen, our system, (5) through (8), can be converted as
A = Λt (1 − τt ) wt = (1 − α)Λt (1 − τt ) Ht−α Ktα ,Λ̇t= ρ − ut = ρ − αHt1−α Ktα−1 ,Λt11K̇t = F (Kt , Ht ) −− G = −G + Ht1−α Ktα − ,ΛtΛt
G = Ht τt wt = (1 − α)τt Ht1−α Ktα .

(16)
(17)
(18)
(19)

We need to linearize the above four formulae around the steady states deduced.
First, we obtain the following relationship based on (16).
−α (Ht − H ∗ ) α (Kt − K ∗ ) Λt − Λ∗ τ ∗ (τt − τ ∗ )
+
+
−
= 0,
H∗
K∗
Λ∗
(1 − τ ∗ ) τ ∗which can be converted as−τ∗× τ˜t + sk (kt − ht ) + λt = 0.
1 − τ∗(20)

Moreover, based on (17),λ̇t = ρsh (kt − ht )(21)

prevails, and additionally, (19) is converted asτ̃t + sk (kt − ht ) + ht = 0.

6

(22)

Finally, we proceed to explore how to establish the dynamic structure of(19). First, transforming (18), we obtainG1+ Ht1−α Ktα−1 −.
k˙t = −
Kt
Kt Λt
By a Taylor-series approximation around the steady states, we obtain
[
]
G
1
∗ 1−α
∗ α−1
k˙t =
+
(α
−
1)
(H
)
(K
)
+
kt
K∗
Λ∗ K ∗+(1 − α)ht (H ∗ )1−α (K ∗ )α−1 +

λt
,
∗
Λ K∗

which is crucial and eventually leads to
sh
ρ sc
λt +
ht + ρ k t .
k˙t =
sk
sii

(23)

At this stage, we can say that our system consists of two equations, in which
one, (20), has four arguments, λt , kt , ht and τt , while the other, (22), has
three arguments, kt , ht and τt , and two diﬀerential equations, in which one,(23), consists of three arguments, λt , kt , and ht , while the other, (21), has
three arguments, λt , kt and ht .
Here, we eliminate τt and ht from the two equations. Then, by substituting the two variables into the two simultaneous diﬀerential equation
systems, we obtain two simultaneous equation systems with two arguments,
kt and λt .
We demonstrate the procedures below. Based on (20) and (22), we
obtain(1 − τ ∗ ) λt
sk kt
ht =
+
.
(24)
∗
sk − τ
sk − τ ∗We should notice that the above relationship is derived from two crucial
properties in the assumptions in Schmitt-Uribe (1997): First, the marginal
disutility of households is constant; second, the governments follow the balanced budget rule. Consequently, the following system prevails:ρλt (1 − τ ∗ ) sh ρkt τ ∗ sh−,λ̇t = −sk)− τ ∗( sk − τ ∗∗(1 − τ ) sh(25)
ρλt sc +∗∗ρ (1 − τ ) kt
sk − τk˙t =+,sk
sk − τ ∗which obviously shows movements in our economy. So, let us explore (25)intensively. For convenience, we consider the right side in (25) as a matrix
with 2 columns and 2 rows, and we denote J.
7

After some analyses, the trace of J is described as a function with anargument, τ ∗ .
ρ (τ ∗ − 1) (sh − 1)trace(J) =.
sk − τ ∗k −1)
We denote this as Φ(τ ), which has properties such as dΦ/dτ = ρ(sh(τ−1)(s>−sk )20, Φ(0) = ρ and Φ(1) = 0. Additionally,{limτ →sk Φ(τ ) → −∞ from above,limτ →sk Φ(τ ) → ∞from below

prevail.
Next, the determinant of J, Det(J), is expressed as
(
)(
)
√
√
1−sh +1hρ2 s2h (τ − (1 − sh )) τ − 1− s1−sτ−shhDet(J) =.
2
sk (sk − τ )Similarly, we consider Det(J) as a function with an argument, τ ∗ .
In our context, Det(J) = 0 can have three solutions regarding τ , but thesolution sequence must be as follows: 3√
1 − 1 − sh
< 1.
(26)
0 < 1 − sh <shIn order to elucidate (26), we replace Det(J) with
(
)(
)
√
√
1−sh +1hρ2 s2h τ − 1− s1−sτ−shhΠ(τ ) = −,sk (sk − τ )which has properties such as{2
Π(0) = − ρs2sh < 0,kΠ(1) = −ρ2 < 0.
Additionally,{limτ →sk Π(τ ) → ∞limτ →sk Π(τ ) → −∞from above,from below

prevail.
3

Notice that at τ =τ ∗ = τ below.

√
1− 1−sh, the labor income revenue is maximized.
sh8

We expressClearly, (26) implies
that when 0 < τ < 1 − sh , Det(J)< 0, while√√
1− 1−sh1− 1−sh1 − sh < τ <, Det(J) > 0, and that when< τ < 1,shsh
Det(J) becomes negative.
In order to establish this, sk or sh must satisfy a condition, which can beobtained by solving the previous inequity, (26). The solution is a necessary
condition for indeterminacy. The condition in terms of α is approximately
expressed as
0 < α < 0.56984.
The above condition requires a low share of capital in output. The reason(26) is a necessary condition becomes obvious by examining the relationships
between trace(J) and Det(J).
Figure 1 shows the necessary and suﬃcient√ conditions for indeterminacy,
1− α∗the set {α, ρ, A, G} that satisfies α < τ < 1−α . In this situation, we should
notice that τ ∗ depends on the set with all four elements, α, ρ, A, and G.
Although τ ∗ is aﬀected by α, the value of τ ∗ must be larger than α and√smaller than (1 − α)/(1 − α), for arbitrary ρ, A and G. This implies theexistence of a necessary condition for indeterminacy in terms of α, which
was deduced.
Here, we cite examples.
Case (a): A=1; α = 0.4; ρ = 0.01; G=1.
In this case, τ ∗ = 0.273 holds. So a saddle path occurs, because (1 −√α)/(1 − α) = 0.612.
Case (b): A=1; α = 0.4; ρ = 0.01; G=1.5.
In this situation, τ ∗ = 0.497 holds, so indeterminacy occurs.
Based on the information analyzed so far, we can depict Det(J) andtrace(J) as follows:9Π(τ ∗ )Φ(τ ∗ )

τ∗ = α√ατ ∗∗ = 1−1−α•1 τ ∗0Π(τ ∗ )
Φ(τ ∗ )

Figure 1:
Given α < τ ∗ < τ ⋆⋆ , indeterminacy paths exist;∗
0 < τ < α, or τ ⋆⋆ < τ ∗ < 1, a saddle path occurs.

4

when

Conclusion

We have discussed indeterminacy problems, assuming a labor arrangement with extensive margins. Households decide to work or not over whole
lifetime periods. On the other hand, governments follow the balanced bud10

get rule, while firms make decision to gain maximum profits in every period.
As a result, their decisions are shortsighted and passive. Their behavior contrasts directly with that of households, resulting in a focus on households,
especially in terms of labor supply.
We have demonstrated that if governments adopt a balanced budget
rule, indeterminate paths can occur, in an economy with capital composedof infinite duration, and that a necessary condition in terms of the ratio of
capital distribution to outputs accounts for the phenomenon.
In line with this, if we seek the anticipations of economic agents as one
source of sun spots as in Farmer (2002), our anticipations will be self-fulfilled.

11References[1] Attanasio, P. O.,. (2015), “Frank Ramsey’s Mathematical Theory of
Saving,” The Economic Journal, Vol. 125, No. 583, 269-277.
[2] Benhabib, J. and Farmer, R. F., (1994), “Indeterminacy and Increasing
Returns,” Journal of Economic Theory, 63, 19-41.
[3] Farmer, E. A., (2002), Macroeconomics of Self-fulfilling Prophecies, 2ndedition (Cambridge, MA: MIT Press)[4] Hansen, G. D., (1985), “Indivisible Labor and the Business Cycle,”
Journal of Monetary Economics, 16, 309-327.
[5] Kamihigashi, T., (2002), “Externality and Nonlinear Discounting: Indeterminacy,” Journal of Economic Dynamics and Control, 26, 141-169.
[6] Mino, K., (2017). Growth and Business Cycles with Equilibrium Indeterminacy (Tokyo: Springer) Advances in Japanese Business andEconomics, 13.
[7] Mulligan C., (1999), “Microfoundations and Macro Implications of Indivisible Labor,” NBER Working Paper, No. 7116.
[8] Ramsey, F. P., (1928), “A Mathematical Theory of Saving,” The Economic Journal, 38, No. 152 (Dec.), 543-559.
[9] Rogerson R., (1988), “Indivisible Labor, Lotteries and Equilibrium,”
Journal of Monetary Economics, 21, 3-16.
[10] Schmitt-Grohé, S. and Uribe, M., (1997), “Balanced-Budget Rules,
Distortionary Taxes and Aggregation Instability,”Journal of Political
Economy, Vol. 105, No. 5, 976-1000.
[11] Takata, F., (2017),“Two Steady States and Two Movement Patterns
under the Balanced Budget Rule-An Economy with Divisible Labor,”
Discussion Paper No.1702, January 2017, Kobe University.
[12] Wen, Y., (2001), ”Understanding Self-Fulfilling Rational Expectations
Equilibria in Real Business Cycle Models,”Journal of Economic Dynamics and Control, 25, 1221-1240.

12