Discussion Paper No.254

Two Time Lags in the Public Sector:
Macroeconomic Stability and Complex Behaviors

Eiji Tsuzuki

June 2015

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Two Time Lags in the Public Sector: Macroeconomic
Stability and Complex Behaviors
Eiji Tsuzuki∗

Abstract
This study develops a macroeconomic model that considers two time lags in the
public sector—a government expenditure lag and a tax collection lag—and examines
the effects of these lags on local stability of the steady state. According to previous
studies, a sufficiently large expenditure lag causes economic instability. However,
we show that a tax collection lag can have a stabilizing effect on the steady state. In
addition, we develop an analysis of global dynamics to demonstrate that an increasein a tax collection lag can yield complex behaviors.

JEL Classification: E12; E30; E62
Keywords: Keynesian macrodynamic model, fiscal policy lag, delay differential equations, stability analysis

1

Introduction

Recently, many studies have examined the effects of time lags on macroeconomic stability
using traditional Keynesian models. For instance, Sportelli and Cesare (2005) introduce
a tax collection lag into the dynamic IS-LM model developed by Schinasi (1981) andSasakura (1994), which is a traditional Keynesian model, and examine the local andglobal dynamics of the system. The standard dynamic IS-LM model with no policy lag
comprises three equations that represent the goods market, monetary market, and budget
constraints of the consolidated government. These equations form an ODE (ordinary
Faculty of Economics, Chiba Keizai University, Chiba, Japan; Tel.: +81-43-253-9745; Fax: +81-43254-6600; E-mail: tsuzukie5@gmail.com∗

1differential equations) system. The introduction of a time lag transforms this systemfrom an ODE to a DDE (delay differential equations) system.1
Generally, models with a time lag can be categorized into two types: fixed lag modeland distributed lag model. Fanti and Manfredi (2007) develop a dynamic IS-LM modelwith a distributed tax collection lag, whereas Sportelli and Cesare (2005) analyze the caseof a fixed lag. Both these studies demonstrate that a time lag evidently causes complex
behaviors, including chaos, and that a traditional fiscal policy is likely to be ineffective.
Moreover, Matsumoto and Szidarovszky (2013) compare the case of a fixed lag with thatof a distributed lag in tax collections. They demonstrate that a larger stable region canbe established in the case of a fixed lag compared with a distributed lag.
Another type of traditional Keynesian macrodynamic model that incorporates a capital accumulation equation in place of the disequilibrium adjustment function of the monetary market, which is often termed the Kaldorian model, has been proposed. This model
originated from Kaldor (1940) and its primary characteristic is found in the assumptionof an S-shaped configuration of the investment function. Chang and Smyth (1971) reconstruct the Kaldorian model to form an ODE system. Asada and Yoshida (2001) introduce
a fixed government expenditure lag into the model proposed by Chang and Smyth (1971)and show that an increase in the responsiveness of a fiscal policy could lead to economic
instability.
Further, Gabisch and Lorenz (1989) propose a hybrid model of the standard dynamic
IS-LM model and the Kaldorian model, which involves both functions of capital accumulation and disequilibrium adjustment in the monetary market. Cai (2005) and Neamţu,
Opriş, and Chilarescu (2007) introduce a fixed capital accumulation lag and a fixed tax
collection lag, respectively, into this hybrid model and comprehensively discuss the occurrence of a Hopf bifurcation.
Moreover, Zhou and Li (2009) and Sportelli, Cesare, and Binetti (2014) propose macrodynamic models with two fixed time lags. Zhou and Li (2009) develop Cai’s (2005) model
to include two capital accumulation lags. In addition, Sportelli, Cesare, and Binetti (2014)
present a dynamic IS-LM model with two time lags in the public sector: a government
expenditure lag and a tax collection lag. These studies demonstrate that the steady statesfluctuate between stability and instability as a certain lag increases.
1

Schinasi (1981) does not consider disequilibrium of the monetary market. Sasakura (1994) develops
Schinasi’s (1981) model by introducing a disequilibrium adjustment function of the monetary market.
Sasakura’s (1994) model is now used as a benchmark of the dynamic IS-LM model.

2Unlike in Sportelli, Cesare, and Binetti (2014), this study uses the Kaldorian macrodynamic model to investigate the interaction of two time lags in the public sector. Therefore,
our model can be considered as introducing a tax collection lag into Asada and Yoshida’s(2001) model. We examine two cases where a fiscal policy is active and where it is passive. An active fiscal policy strongly responds to the national income, whereas a passive
fiscal policy is less responsive to the national income. In addition, we perform a stability analysis employing a mathematical method developed by Gu, Niculescu, and Chen
(2005). This method enables us to present an exact figure of a stability crossing curve—acurve that separates stable and unstable regions on a parameter plane. Few studies have
employed this method for economic analysis.2
This study proceeds as follows: Section 2 presents a dynamic system that represents
a model economy. Section 3 examines the local dynamics around the steady state. Subsequently, Section 4 examines the global dynamics. Section 5 presents our conclusion.

2

We shall refer other Keynesian macrodynamic models that consider a time lag as follows. The time-tobuild model developed by Kalecki (1935) is the basis of economic models with a fixed time lag. Szydlowski(2002, 2003) develops this model into models with economic growth. Moreover, Yoshida and Asada (2007)
examine the effects of a lag in government expenditure (where they examine both distributed and fixedlags) using the so-called Keynes–Goodwin model. Further, Asada and Matsumoto (2014) introduce a
distributed lag of monetary policy implementation into the Keynesian equilibrium model proposed by
Asada (2010). Asada’s (2010) model comprises a monetary policy rule and an expectation adjustment
function. A fixed lag version of this model is proposed by Tsuduki (2015). Furthermore, Matsumoto andSzidarovszky (2014) develop a nonlinear multiplier-accelerator model with investment and consumption
lags. Finally, Bellman and Cooke (1963) provide a helpful introductory textbook of delay differential
equations (i.e., differential-difference equations).

3

2

The model

2.1

Dynamic system

The model economy comprises the following equations:Ẏ (t) = α[C(t) + I(t) + G(t) − Y (t)]; α > 0,

C(t) = c[Y (t) − T (t)] + C̄; 0 < c < 1; C̄ > 0,

(1)
(2)

T (t) = τ Y (t − θ2 ) − T̄ ; 0 < τ < 1; T̄ ≥ 0,

(3)

I(t) = I(Y (t), K(t), r(t)); IY > 0; IK < 0; Ir < 0,

(4)

K̇(t) = I(Y (t), K(t), r(t)),(5)G(t) = β[Ȳ − Y (t − θ1 )] + Ḡ; β > 0; Ȳ > 0; Ḡ > 0,

(6)

M(t)/P (t) = L(Y (t), r(t)); LY > 0; Lr < 0,

M(t) = γ[Ȳ − Y (t)] + M̄ ; γ > 0; M̄ > 0,
P (t) = P (Y (t)); PY > 0,

(7)
(8)
(9)

where Y = real national income (output); C = real private consumption; I = real private
investment; G = real government expenditure; T = real income tax; K = real capitalstock; M = nominal money supply; P = price level; r = nominal interest rate; α =adjustment speed of the goods market; c = marginal propensity to consume; C̄ = baseconsumption; τ = marginal tax rate; T̄ = real subsidy; β = responsiveness of government
expenditure to national income (i.e., activeness level of the fiscal policy); Ȳ = target
level of real national income; Ḡ = target level of real government expenditure; γ =responsiveness of nominal money supply to national income (i.e., activeness level of themonetary policy); M̄ = target level of nominal money supply; t = time; θ1 = government
expenditure lag; and θ2 = tax collection lag.
Equations (1) and (2) represent a disequilibrium adjustment function of the goods
market and a consumption function, respectively. Equation (3) is a tax collection functionthat represents income tax T as a function of past national income Y (t − θ2 ). It may be
more general to formulate T as a function not only of a past income but also of the present
income denoted by Y (t). However, this change does not affect the nature of our argument;
hence, we simply assume that T is a function only of Y (t − θ2 ). Equations (4) and (5)represent an investment function and a capital accumulation function, respectively. For
simplicity, we assume that capital depreciation does not exist. Equation (6) represents a4

fiscal policy reaction function with a government expenditure lag. Equation (7) represents
the monetary market equilibrium condition, where the left-hand side denotes real money
balance and the right-hand side denotes a demand function for money. In this study, we
ensure that the adjustment of the monetary market is rapid, and therefore, the balanceof demand and supply of this market is always maintained. Equation (8) represents amonetary policy reaction function. Finally, Equation (9) represents an aggregate supply
function, by which the price level is determined.
In the case of no tax collection lag (i.e., θ2 = 0), the system compounded from Equations (1)–(9) essentially becomes similar to that of Asada and Yoshida (2001). However,
the existence of a positive θ2 significantly complicates the dynamic property of the system,
thereby resulting in a major change in the economic implication of time lags.

2.2

Summarizing the equations

In this section, we summarize Equations (1)–(9) in a two-dimensional dynamic system.
Substituting Equations (8) and (9) into Equation (7) and solving for r, we obtainr(t) = r(Y (t)),(10)

where rY = −(γP + PY M + P 2 LY )/P 2 Lr > 0.
Substituting Equation (3) into Equation (2) and substituting Equation (10) into Equation (4), we obtain
C(t) = cY (t) − cτ Y (t − θ2 ) + C̄ + cT̄ ,I(t) = I(Y (t), K(t), r(Y (t))).

(11)
(12)

Finally, substituting Equations (6), (11), and (12) into Equation (1) and substituting
Equation (12) into Equation (5) yields the following system of differential equations with
two time lags:Ẏ (t) = α[I(Y (t), K(t), r(Y (t))) − (1 − c)Y (t) − βY (t − θ1 ) − cτ Y (t − θ2 )+ C̄ + cT̄ + β Ȳ + Ḡ],K̇(t) = I(Y (t), K(t), r(Y (t))).

5

(13)

2.3

Linearization

To analyze the local dynamics of System (13), we linearize the system around the steadystate (Y ∗ , K ∗ ) and obtain˙Ŷ (t) = α[{A1 − (1 − c)}Ŷ (t) − β Ŷ (t − θ1 ) − cτ Ŷ (t − θ2 ) + IK K̂(t)],˙K̂(t) = A1 Ŷ (t) + IK K̂(t),(14)

where Ŷ (t) = Y (t) − Y ∗ , K̂(t) = K(t) − K ∗ , and A1 = IY + Ir rY . By necessity, thecoefficients of these equations are evaluated at the steady state.
Assuming the exponential functions Ŷ (t) = C1 eλt and K̂(t) = C2 eλt (where C1 andC2 are arbitrary constants, and λ denotes the eigenvalue) as the solutions of the above
system and substituting these into System (14), we obtain
"
#"
# " #
λ − α{A1 − (1 − c)} + αβe−θ1 λ + αcτ e−θ2 λ −αIKŶ (t)0=.
−A1
λ − IK K̂(t)0For non-trivial solutions to exist for this system, the determinant of the left-hand sidematrix, denoted by ∆(λ), must equal zero; i.e.,∆(λ) = λ2 − [IK + α{A1 − (1 − c)}]λ − α(1 − c)IK

+ αβ(λ − IK )e−θ1 λ + αcτ (λ − IK )e−θ2 λ = 0= p0 (λ) + p1 (λ)e−θ1 λ + p2 (λ)e−θ2 λ = 0,

(15)

where
p0 (λ) = λ2 + b1 λ + b2 ,b1 = −[IK + α{A1 − (1 − c)}],b2 = −α(1 − c)IK ,p1 (λ) = αβ(λ − IK ),p2 (λ) = αcτ (λ − IK ).

Equation (15) is a characteristic equation of System (14). The significant feature of thisequation is the existence of the exponential terms (e−θ1 λ and e−θ2 λ ).
First, we examine the case with no time lags. When θ1 = θ2 = 0, Equation (15) canbe rewritten as follows:∆(λ) = λ2 + (b1 + α(β + cτ ))λ + b2 − αIK (β + cτ ) = 0,6(16)

which is an ordinary quadratic equation of λ.
Thus, we can state that if b1 + α(β + cτ ) > 0 (i.e., the coefficient of λ from Equation
16 is positive), the real parts of the roots of Equation (16) are negative.3 In contrast, if
b1 +α(β+cτ ) < 0, then the real parts of the roots are positive. Therefore, if b1 +α(β+cτ ) >0, the steady state is locally stable, and if b1 + α(β + cτ ) < 0, it is unstable.
In the discussion below, we assume the following condition:
Assumption 2.1 b1 + α(β + cτ ) > 0.
This assumption implies that if a lag does not exist in the public sector, an economy isstable. Under this assumption, we analyze the effects of the lags (θ1 , θ2 ) on local stability.

3

Local dynamics

The following analysis is performed based on the technique developed by Gu, Niculescu,
and Chen (2005).

3.1

PreconditionsFirst, to apply the technique of Gu, Niculescu, and Chen (2005), some preconditions
should be checked. According to their study, Equation (15) should satisfy the following
conditions:(I) deg(p0 (λ)) ≥ max{deg(p1 (λ)), deg(p2 (λ))};(II) ∆(0) 6= 0;(III) a solution common to all three polynomials p0 (λ) = 0, p1 (λ) = 0, and p2 (λ) = 0
does not exist;(IV) limλ→∞ (|p1 (λ)/p0 (λ)| + |p2 (λ)/p0 (λ)|) < 1.
In our system, Condition (I) is satisfied by 2 > max{1, 1}. Condition (II) is alsosatisfied by ∆(0) = αIK [−(1 − c) − β − cτ ] > 0. Concerning Condition (III), we cancheck as follows: substituting IK into p1 (λ) and p2 (λ), we obtain p1 (IK ) = p2 (IK ) = 0.
3

See Chapter 18 in Gandolfo (2010) for details of the relationship between the roots and coefficientsof a quadric equation.

7

However, p0 (IK ) = −αA1 IK 6= 0. Hence, Condition (III) is satisfied. Finally, Condition(IV) is satisfied by limλ→∞ (|p1 (λ)/p0 (λ)| + |p2 (λ)/p0 (λ)|) = 0.
Now, we examine the effects of lags (θ1 , θ2 ) on the stability of the steady state. The
analysis proceeds as follows:(1) We characterize the points at which the local dynamics can change, i.e., the points
at which the pure imaginary roots appear.4 These points are referred to as the
crossing points.
(2) We depict the sets of the crossing points (which we refer to as the crossing curves)on the θ1 -θ2 plane by using numerical simulation.
(3) We reveal the directions of changes in the signs of the real parts that occur whenlags (θ1 , θ2 ) cross the crossing curves.

3.2

Crossing points

Dividing Equation (15) by p0 (λ), we obtain
1 + a1 (λ)e−θ1 λ + a2 (λ)e−θ2 λ = 0,

(17)

where
p1 (λ)αβ(λ − IK )= 2,
p0 (λ)
λ + b1 λ + b2
p2 (λ)αcτ (λ − IK )a2 (λ) == 2.
p0 (λ)
λ + b1 λ + b2

a1 (λ) =

(18)
(19)

Moreover, we denote a pure imaginary root as λ = vi (where v = imaginary part 6= 0√
and i = −1). Then, the values of v that satisfy Equation (17) can be characterized by
the following lemma:
Lemma 3.1 (Gu, Niculescu, and Chen 2005, Proposition 3.1) For each v satisfying p0 (vi) 6= 0, λ = vi is a solution of ∆(λ) = 0 for some (θ1 , θ2 ) ∈ R2+ if and only if|a1 (iv)| + |a2 (iv)| ≥ 1,

−1 ≤ |a1 (iv)| − |a2 (iv)| ≤ 1.
4

It is ensured from precondition (III) that a zero real root cannot be a root.

8

(20)
(21)

We denote the set of v > 0 that satisfy conditions (20) and (21) as Ω, which is termed
as the crossing set.5 For any given v ∈ Ω, the sets (θ1 , θ2 ) satisfying Equation (17) (eachof which corresponds to a crossing point) must satisfy the following relationships (Figure1).
∓δ1 = arg(a1 (iv)e−ivθ1 ) + 2mπ;
±δ2 = arg(a2 (iv)e−ivθ2 ) + 2nπ;m = 0, 1, 2, · · · ,n = 0, 1, 2, · · · ,

(22)
(23)

where δ1 , δ2 ∈ [0, π].
Ima2 (iv)e−ivθ2

δ2
0

@
I
@
@
@ a1 (iv)e−ivθ1
@
@
δ1 @@

1

Re

Figure 1: Triangle formed by 1, |a1 (iv)|, and |a2 (iv)| on the complex plane
Incidentally, on the complex plane, a multiplication of amplitudes becomes a sum ofparts; therefore, we obtainarg(a1 (iv)e−ivθ1 ) = arg(a1 (iv)) − vθ1 ,arg(a2 (iv)e−ivθ2 ) = arg(a2 (iv)) − vθ2 .

Figure 1 also demonstrates that the following relationships hold:Im(a1 (iv))−1
arg(a1 (iv)) = tan,
Re(a1 (iv))

Im(a2 (iv))−1
arg(a2 (iv)) = tan.
Re(a2 (iv))
5

(24)
(25)

(26)
(27)

Pure imaginary roots are always conjugated. Therefore, we can assume v > 0 without a loss ofgenerality.

9Moreover, after some manipulation, Equations (18) and (19) derive the following expression:Im(a2 (iv))
b1 vIK + v(b2 − v 2 )Im(a1 (iv))==.
Re(a1 (iv))
Re(a2 (iv))
b1 v 2 − IK (b2 − v 2 )(28)

Thus, using Equations (24)–(28), Equations (22) and (23) can be rewritten as follows:2θ1 =K +v(b2 −v )tan−1 ( bb11vI) ± δ1 + 2mπ
v2 −IK (b2 −v2 )

v

,

(29)

,

(30)

2

θ2 =

K +v(b2 −v )) ∓ δ2 + 2nπ
tan−1 ( bb11vIv2 −IK (b2 −v2 )v

where the interior angles of the triangle denoted by δ1 and δ2 are given by the cosine
theorem as follows:
1 + |a1 (iv)|2 − |a2 (iv)|2−1
δ1 = cos
2|a1 (iv)|
!
2 2
2
2
2
2
2
(b
−
v
)
+
(b
v)
+
(αβI
)
+
(αβv)
−
(αcτ
I
)
−
(αcτ
v)
2
1
K
K
p
p
= cos−1,2 (αβIK )2 + (αβv)2 (b2 − v 2 )2 + (b1 v)2
1 + |a2 (iv)|2 − |a1 (iv)|2−1
δ2 = cos
2|a2 (iv)|
!
2 2
2
2
2
2
2
(b
−
v
)
+
(b
v)
−
(αβI
)
−
(αβv)
+
(αcτ
I
)
+
(αcτ
v)
2
1
K
K
p
p
= cos−1.
22222 (αcτ IK ) + (αcτ v) (b2 − v ) + (b1 v)2Equations (29) and (30) characterize the sets of the crossing points (θ1 , θ2 ) ∈ R2+ .
Depending on the signs of δ1 and δ2 , we can define two types of crossing points, denotedby L1 (m, n) and L2 (m, n), as follows:2

L1 (m, n) :

θ1 =
θ2 =

K +v(b2 −v )tan−1 ( bb11vI) + δ1 + 2mπ
v2 −IK (b2 −v2 )v2

K +v(b2 −v )) − δ2 + 2nπ
tan−1 ( bb11vIv2 −IK (b2 −v2 )

v

,

,

2

L2 (m, n) :

θ1 =
θ2 =

K +v(b2 −v )tan−1 ( bb11vI) − δ1 + 2mπ
v2 −IK (b2 −v2 )v2

K +v(b2 −v )tan−1 ( bb11vI) + δ2 + 2nπ
v2 −IK (b2 −v2 )

v

,

.

In the next section, based on the study of Asada and Yoshida (2001), we illustrate theexamples of L1 (m, n) and L2 (m, n) by using numerical simulations.
10

3.3

Numerical simulations

Following Asada and Yoshida’s (2001) study, we assume the investment function as follows:
p
p
p
400
I(Y (t), K(t), r(Y (t))) =
−
0.01
Y
(t)
−
0.5K(t)
−
10γ(
Y
(t)
−
Ȳ ).
1 + 12e−0.1(Y (t)−400)

Further, we set the parameter values as follows: α = 0.9; c = 0.625; τ = 0.2; Ȳ = 400;
C̄ + cT̄ + Ḡ = 200; and γ = 8.6. Under these specifications, the steady-state values of
System (13) are given by (Y ∗ , K ∗ ) = (400, 61.138).
In the following discussion, we compare two cases: the case of an active fiscal policy
with that of a passive fiscal policy.
3.3.1

Example 1

When β = 4.1, which represents a relatively active fiscal policy, the crossing set Ω isgiven by v ∈ (3.6506, 3.8716) (Figure 2). For v ∈ Ω, we can depict L1 (m, n) and L2 (m, n)
as shown in Figure 3, where m = 0, 1, 2 and n = 0, 1, 2. The dotted curves represent
L1 (m, n), and the solid curves represent L2 (m, n). These curves are referred to as the
crossing curves.
3.3.2

Example 2

When β = 0.1, which represents a passive fiscal policy, the crossing set Ω is given byv ∈ (0.2636, 0.5120) (Figure 4). In this case, the crossing curves L1 (m, n) and L2 (m, n)
can be depicted for v ∈ Ω as shown in Figure 5. The starting points of both curves
L1 (m, n) and L2 (m, n) (i.e., the points corresponding to v = 0.2636) are given by theupper connecting points of the circles.
Next, we examine how the real parts of the roots change when lags (θ1 , θ2 ) cross the
crossing curves.

3.4

Direction of crossing

We reveal the direction in which the roots cross the imaginary axis when the value of θ1
increases. It is determined by the sign of dReλ/dθ1 |λ=iv (where v∈Ω) . If dReλ/dθ1 |λ=iv > 0,
the roots cross the imaginary axis from left to right with an increase in θ1 (which indicatesdestabilization). In contrast, if dReλ/dθ1 |λ=iv < 0, the roots cross the imaginary axis
11

3
2.5
2
1.5
1
|a (iv)|+|a

Ω

0.5

1

2

|a (iv)|-|a
1

0

2

(iv)|
(iv)|

-0.5
-1
-1.5
0

2

4

6

8

10

v

Figure 2: Crossing set Ω (β = 4.1)

4.5
4

L 2 (0,2)

L (1,2)L (2,2)L (0,2)L 1 (1,2)

L (2,2)L (0,1)

L (1,1)

L (1,1)

L (0,1)

L (1,1)

L (2,1)

L (0,0)L (1,0)L (2,0)

2

2

3.5
1

1

3
2.5

2

2

θ

2

2

2
1.5

1

1

1

1
2

2

2

0.5
L (0,0)L (1,0)

1

0
0

0.5

L (2,0)

1

1

1.5

2

θ

2.5

1

3

3.5

1

Figure 3: Crossing curves (β = 4.1)

12

4

4.5

1.5
Ω
1

0.5

|a (iv)|+|a
1

2

(iv)|

0
|a (iv)|-|a
1

2

(iv)|

-0.5

-1
0

2

4

6

8

10

Figure 4: Crossing set Ω (β = 0.1)50
L (2,2)45

2

L (1,2)L (0,2)40

21

L (1,2)1

35

L (2,2)
1

θ

2

30
25

L (2,1)2

20
L (1,1)
2

15

L (0,1)1

L (2,1)1

10L 1 (1,1)
5

L (1,0)L (2,0)

2

2

0
0

10

20

θ

30

40

1

Figure 5: Crossing curves (β = 0.1)

1350

from right to left with an increase in θ1 (which indicates stabilization). For convenienceof calculation, we observe the sign of Re(dλ/dθ1 )−1 |λ=iv instead of that of dReλ/dθ1 |λ=iv .
Differentiating Equation (17) with respect to θ1 , we obtain ′ dλ
a1 (λ)e−θ1 λ − a1 (λ)e−θ1 λ θ1 + a′2 (λ)e−θ2 λ − a2 (λ)e−θ2 θ2= a1 (λ)e−θ1 λ λ,dθ1or equivalently



dλ
dθ1

−1

=

a′1 (λ)e−θ1 λ + a′2 (λ)e−θ2 λ − a2 (λ)e−θ2 λ θ2 θ1− ,a1 (λ)e−θ1 λ λ
λ

(31)

where
αβp0(λ) − αβ(λ − IK )(2λ + b1 ),p0 (λ)2αcτ p0 (λ) − αcτ (λ − IK )(2λ + b1 )
a′2 (λ) =.
p0 (λ)2a′1 (λ) =3.4.1

Example 1

Suppose that β = 4.1. In this case, describing the real part of Equation (31) as a functionof v ∈ Ω, we can derive Figure 6, where the dotted curves are the functions evaluated on
curve L1 (m, n), and the solid curves are the functions evaluated on curve L2 (m, n).
Figure 6 shows that Re(dλ/dθ1 )−1 |λ=iv > 0 holds for all cases in Figure 3. Therefore,at least two imaginary roots with positive real parts emerge when θ1 crosses the crossing
curves from left to right.
Now, a curve formed by connecting curves Lj (0, n) (where j = 1, 2; n = 0, 1, 2) istermed as m0 (an enlarged representation of this curve is proposed in Figure 7). Then,
we can make the following proposition:
Proposition 3.1 For lags (θ1 , θ2 ) lying to the left of curve m0, the steady state is locallystable. However, for lags (θ1 , θ2 ) lying to the right of curve m0, the steady state is
unstable.
Based on this proposition, we can state the following: In the case of θ1 < 0.384, thesteady state is locally stable irrespective of the value of θ2 , i.e., if a government expenditure
lag is sufficiently small, a tax collection lag does not affect economic stability. Moreover, inthe case of θ1 ∈ (0.384, 0.412), the steady state fluctuates between stability and instabilityas θ2 increases. Thus, a tax collection lag can contribute toward stabilizing an economy.
14L (0,0) and L
1

2

(0,0)

L (0,1) and L
1

2

(0,1)

L (0,2) and L
1

2

(0,2)

0.1

1

Re(d λ/d θ )

-1

0.1
0.05

0.05

0

0.05

0
3.7
3.8
L (1,0) and L
1

2

3.9
(1,0)

0
3.7
3.8
L (1,1) and L
1

2

3.9
(1,1)

3.7
3.8
L (1,2) and L
1

2

3.9
(1,2)

0.1

1

Re(d λ/d θ )

-1

0.1
0.05

0.05

0

0.05

0
3.7
3.8
3.9
L 1 (2,0) and L 2 (2,0)

0
3.7
3.8
3.9
L 1 (2,1) and L 2 (2,1)

3.7
3.8
3.9
L 1 (2,2) and L 2 (2,2)
0.1

1Re(d λ/d θ )

-1

0.1
0.05

0.05

0

0.05

0
3.7

3.8

3.9

0
3.7

v

3.8

3.9

3.7

3.8

v

3.9

v

Figure 6: Direction of crossing (β = 4.1)

4.5
4
3.5
3

θ

2

2.5
Stable
2
1.5
1
0.5
0
0

0.1

0.2

0.3
θ

0.4

1

Figure 7: Curve m0

15

0.5

0.6

3.4.2

Example 2

When β = 0.1, the direction of crossing is determined by Figure 8. Figures 8 and 5
demonstrate the following proposition:
Proposition 3.2 In Figure 5, the regions enclosed within curves L1 (m, n) and L2 (m, n)(i.e., regions inside the circles) are unstable, whereas the others are stable.

L (0,0) and L
1

2

(0,0)

L (0,1) and L
1

2

(0,1)

1

Re(d λ/d θ )

-1

100
0

0
-20

-1
1

Re(d λ/d θ )

(0,2)

-100
-200
2

0.5
(1,0)

0.3
0.4
L (1,1) and L
1

2

0.5
(1,1)

100

0.3
0.4
L (1,2) and L
1

2

0.5
(1,2)

200

20
0

0
-20

0

-100

-40

-200
0.3
0.4
L 1 (2,0) and L

0.5
(2,0)
2

0.3
0.4
L 1 (2,1) and L

0.5
(2,1)
2

100

-1

2

0

-40
1

1

1

20

0.3
0.4
L (1,0) and L

Re(d λ/d θ )L (0,2) and L
200

0.3
0.4
L 1 (2,2) and L

2

0.5
(2,2)

200

20
0

0
-20

0

-100

-40

-200
0.3

0.4
v

0.5

0.3

0.4
v

0.5

0.3

0.4

0.5

v

Figure 8: Direction of crossing (β = 0.1)
Comparing the case of a passive policy (β = 0.1) with that of an active policy (β = 4.1)
within an economically meaningful region of (θ1 , θ2 ) (i.e., θ1 and θ2 take values between
0 and 3), the former achieves a larger stable region. This suggests that an active policy
stance may increase economic instability. This result cannot be derived from a model
without a time lag. Furthermore, as indicated by Figure 5, in the case of β = 0.1, thesteady state fluctuates between stability and instability with increases in not only θ2 but
also θ1 . Therefore, not only tax collection but also government expenditure lags cancontribute towards stabilizing an economy.

16

4

Global dynamics

Thus far, we analyzed the local dynamics of System (13) with regard to the steady state.
In this section, we illustrate phase diagrams to visually confirm the result established in
the previous section and provide an example of global dynamics of the system.
We set the same parameter values as those in the previous section and assume thatβ = 4.1 (This section only examines the case with an active fiscal policy.). Further, we
assume θ1 = 0.4. As indicated by Figure 7, if θ2 is sufficiently small (i.e., θ2 ≤ 0.038),
the steady state is locally stable. However, if θ2 > 0.038, then the dynamics of thesolutions change depending on the value of θ2 (Figure 9). When θ2 = 0.7, a stable
cycle exists and the solutions starting from the initial values of (Y (0), K(0)) = (390, 55)
converge to the cycle. When θ2 = 1.7, the steady state becomes locally stable again, andthe solutions converge to the steady state. Moreover, when θ2 = 3.6, a strange-shaped
attractor emerges, and the solutions exhibit chaotic behaviors.
θ =0.7

θ =1.72

θ =3.6

2

400

350

2

350

350

300

300

250

250

300

K(t)

K(t)

K(t)

250
200

200

200
150

150

100

100

150

100

50
300

400
Y(t)

500

50
300

400

500

Y(t)

50
300

400

500

Y(t)

Figure 9: θ1 = 0.4
This example demonstrates that while an increase in a tax collection lag contributes
toward local stability, it can cause globally complex behaviors.

17

5Conclusion

In this study, we developed the Kaldorian model with government expenditure and tax
collection lags and examined the effects of these lags on local stability by using numericalsimulations. In addition, we also examined global dynamics.
As shown by Asada (1987), under a fiscal policy without a lag, the steady stateis locally stable as long as the government is sufficiently active. However, Asada andYoshida (2001) show that under a policy with a sufficiently large expenditure lag, thesteady state becomes unstable even if the government is sufficiently active. This study
showed that under a policy with government expenditure and tax collection lags, a policy
lag can have a stabilizing effect on the steady state.
Under an active policy stance, if a government expenditure lag exceeds a certain
threshold level, then the steady state becomes unstable. This result is similar to thatin Asada and Yoshida’s (2001) study. However, we further demonstrated that in theneighborhood of the threshold, certain positive values of a tax collection lag can achieve
local stability. Therefore, a tax collection lag can contribute toward economic stability.
Similarly, under a passive policy stance, both tax collection as well as government
expenditure lags can contribute to stabilizing an economy.
We also demonstrated that in an unstable parameter region, limit cycles and complex
behaviors can emerge. Therefore, while an increase in a tax collection lag contributes
toward local stability, it can cause globally complex behaviors.
According to Friedman (1948), policy lags are classified into three types: recognition,
implementation, and diffusion lag. Unlike recognition and diffusion lag, implementation
lag can be considered as adjustable to some extent. Therefore, this study suggests that an
adjustment of the timing of policy implementation can be a means to achieve stabilization.

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