# Dynamic Analysis of Two Policy Lags in a Kaldorian Model

Authors: Eiji Tsuzuki
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We examine the effects of policy lags on local economic stability using a Kaldorian model. This study analyzes two cases: the case of a monetary policy with a time lag and the case of a policy with both fiscal and monetary lags. Similar to the case of fiscal policy lags examined in a previous study, monetary policy lags have destabilizing effects on economic stability. However, in the case of the existence of both fiscal and monetary policy lags, there is a possibility that a monetary policy lag can stabilize an economy.


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## Body

## 1. Introduction

In this study, we investigate the effects of policy lags (i.e., fiscal and monetary policy lags) on the stability of an economy. As the analytical framework, we employ a traditional Keynesian macrodynamic model. In general, two types of time lags have been studied: distributed and fixed lags. The present study focuses on the fixed lag (the time-to-build model developed by Kalecki [1] is the origin of economic models that consider a fixed lag; Szydłowski [2, 3] develops Kalecki’s [1] model to incorporate the factors of economic growth; economic models that introduce a distributed lag include Fanti and Manfredi’s [4] dynamic IS-LM model and Yoshida and Asada’s [5] Keynes-Goodwin model; Yoshida and Asada also examine the case of a fixed policy lag).One of the representative traditional Keynesian macrodynamic models is the dynamic IS-LM model developed by Schinasi [6] and Sasakura [7]. In a previous study, De Cesare and Sportelli [8] develop a dynamic IS-LM model with a tax collection lag. Furthermore, Matsumoto and Szidarovszky [9] use the dynamic IS-LM model to compare the case of a fixed lag with that of a distributed lag in tax collections and demonstrate that a larger stable region can be established in the case of a fixed lag compared with a distributed lag. Both of these studies evidently show that a policy lag restricts the ability of traditional fiscal policies to stabilize an economy.The Kaldorian model originated in the studies of Kaldor [10] and Chang and Smyth [11] is another representative model of the traditional Keynesian macrodynamic model. Asada and Yoshida [12] introduce a fiscal policy lag into this model and demonstrate that an increase in the responsiveness of a fiscal policy can lead to local instability, as is the case with the dynamic IS-LM model.Recently, economic models that consider two time lags have been developed. For example, Zhou and Li [13] propose a dynamic IS-LM model with two capital accumulation lags (Zhou and Li’s [13] model can be considered as an extension of Cai’s [14] model, which is a dynamic IS-LM model with only one capital accumulation lag). Sportelli et al. [15] present a similar model with two time lags in the public sector: government expenditure and tax collection lags. Moreover, Matsumoto and Szidarovszky [16] propose a nonlinear multiplier-accelerator model with investment and consumption lags. These studies demonstrate that the steady state fluctuates between stability and instability with an increase in a certain policy lag; that is, multiple stability switches can occur.In the present study, we first propose a Kaldorian model that only introduces the monetary policy lag, which can be considered as the monetary policy lag version of Asada and Yoshida’s [12] model, and perform a local stability analysis. Furthermore, we propose a model that considers the existence of both fiscal and monetary policy lags. For the analysis of this model, we employ the mathematical method developed by Gu et al. [17]. Their method enables us to represent a stability crossing curve (a curve that separates stable and unstable regions on a lag parameter plane) as a perfectly parameterized form. Before the advent of Gu et al.’s [17] method, one could only represent one lag as a function of another lag. Using their method, we can present a more exact figure of a stability crossing curve. Few studies have employed this method for economic analysis.This study proceeds as follows. Section2 presents a dynamic system that represents a model economy, while Section 3 examines the local stability in the case of a monetary policy with a lag. Subsequently, Section 4 examines the case wherein both fiscal and monetary policy lags exist. Finally, Section 5 presents our conclusion.


## 2. The Model

### 2.1. Basic Equations

The model economy comprises the following equations:(1)Y˙t=αCt+It+Gt-Yt;α>0,(2)Ct=cYt-Tt+C¯;0<c<1;C¯>0,(3)Tt=τYt-T¯;0<τ<1;T¯≥0,(4)It=IYt,Kt,rt;IY>0;IK<0;Ir<0,(5)Gt=βY¯-Yt-θ1+G¯;β>0;Y¯>0;G¯>0,(6)K˙t=IYt,Kt,rt,(7)MtP¯=LYt,rt;P¯>0;LY>0;Lr<0,(8)Mt=P¯γY¯-Yt-θ2+M¯;γ>0;M¯>0,where Y = real national income; C = real private consumption; I = real private investment; G = real government expenditure; T = real income tax; K = real capital stock; r = nominal interest rate; M = nominal money supply; α = adjustment speed of the goods market; c = marginal propensity to consume; C¯ = base consumption; τ = marginal tax rate; T¯ = real subsidy; β = activeness level of fiscal policy; Y¯ = target level of real national income; G¯ = target level of real government expenditure; P¯ = price level; γ = activeness level of monetary policy; M¯ = target level of nominal money supply; t = time; θ1 = fiscal policy lag; and θ2 = monetary policy lag.Equations (1), (2), (3), and (4) represent a disequilibrium adjustment function of the goods market, a consumption function, a tax collection function, and an investment function, respectively. Equation (5) represents a fiscal policy reaction function with a government expenditure lag, while (6) signifies a capital accumulation function. For simplicity, we assume that capital does not depreciate. Equation (7) represents the monetary market equilibrium condition, where the left-hand side denotes real money balance and the right-hand side indicates real money demand (in this study, we assume that the price level P is constant; by assuming P(t)=P(Y(t)) (where PY>0), we can allow for variations in the price level; however, this change does not affect the nature of our argument). Here, we ensure that the adjustment of the monetary market is rapid, and therefore the balance of demand and supply of this market is always maintained. Finally, (8) represents a monetary policy reaction function with a money supply lag.If real money balance is constant (i.e.,γ=0), then the system compounded from (1)–(8) essentially becomes similar to that of Asada and Yoshida [12]. However, the assumptions of γ>0 and θ2>0 significantly complicate the dynamic property of the system, thereby resulting in a major change in the economic implication of time lags.


### 2.2. Summarization

We summarize (1)–(8) in a two-dimensional dynamic system. First, by substituting (8) into (7) and solving for r, we obtain(9)rt=rYt,Yt-θ2,where rY(t)=-LY/Lr>0 and rY(t-θ2)=-γ/Lr>0.Furthermore, by substituting (3) into (2) and substituting (9) into (4), we obtain(10)Ct=c1-τYt+cT¯+C¯,(11)It=IYt,Kt,rYt,Yt-θ2.Finally, by substituting (5), (10), and (11) into (1) and substituting (11) into (6), we obtain the following system of differential equations with two time delays:(12)Y˙t=αIYt,Kt,rYt,Yt-θ2-1-c1-τYt-βYt-θ1+cT¯+C¯+βY¯+G¯,K˙t=IYt,Kt,rYt,Yt-θ2.


### 2.3. Linearization

We assume that steady-state solutionsY∗>0 and K∗>0 that satisfy Y˙(t)=K˙(t)=0 exist and these solutions are continuous with respect to all parameters. In order to analyze the local stability around the steady-state point (Y∗,K∗), we linearize the system and obtain(13)Y^˙t=αA1-1+c1-τY^t-βY^t-θ1+A2Y^t-θ2+IKK^t,K^˙t=A1Y^t+A2Y^t-θ2+IKK^t,where Y^(t)=Y(t)-Y∗, K^(t)=K(t)-K∗, A1=IY+IrrY(t), and A2=IrrY(t-θ2).Assuming the exponential functionsY^(t)=C1eλt and K^(t)=C2eλt, where C1 and C2 are arbitrary constants and λ denotes the eigenvalue, as the solutions of system (13) and substituting these into the system, we obtain (14)λ-αA1-1+c1-τ+αβe-θ1λ-αA2e-θ2λ-αIK-A1-A2e-θ2λλ-IKY^tK^t=00.For nontrivial solutions to exist, the determinant of the left-hand side matrix, denoted byΔ(λ), must be equal to zero; that is,(15)Δλ=λ2-IK+αA1-1+c1-τλ-α1-c1-τIK+αβλ-IKe-θ1λ-αA2λe-θ2λ=0=p0λ+p1λe-θ1λ+p2λe-θ2λ=0,where(16)p0λ=λ2+b1λ+b2,b1=-IK+αA1-1+c1-τ,b2=-α1-c1-τIK>0,p1λ=αβλ-IK,p2λ=-αA2λ.Equation (15) is a characteristic equation of system (13). The significant feature of this equation is found in the existence of the exponential terms e-θ1λ and e-θ2λ.First, we examine the case of no policy lags. By substitutingθ1=θ2=0 into (15), we obtain(17)λ2+b1+αβ-A2λ+b2-αβIK=0,which is an ordinary quadratic equation of λ.Thus, ifb1+α(β-A2)>0 (i.e., the coefficient of λ from (17) is positive), the real parts of the roots of (17) are negative (see Chapter 18 in Gandolfo [18] for details regarding the relation between the roots and the coefficients of a quadric equation). In contrast, if b1+α(β-A2)<0, the real parts of the roots are positive. Thus, if b1+α(β-A2)>0, then the steady state is locally stable, and if b1+α(β-A2)<0, then it is unstable.The larger the values ofβ and γ are, the more likely the stability condition holds (note that A2=-γIr/Lr). Therefore, if policy lags do not exist, then traditional fiscal and monetary policies could function effectively for stabilization.In the following discussion, we assume that if both fiscal and monetary policy lags do not exist, then an economy is stable; that is, we have the following.Assumption 1.
We have(18)β+IrLrγ+b1α>0.Under this assumption, we analyze the effects of the policy lags(θ1,θ2) on stability.Incidentally, if[b1+α(β-A2)]2<4(b2-αβIK), then both roots of (17) are conjugate complex. Furthermore, by defining β∗≡A2-b1/α, the conditions for Hopf bifurcation are satisfied at β=β∗; that is, (i) (17) includes a pair of pure imaginary roots and (ii) dReλ/dβ≠0. Therefore, a cycle exists for a certain range of β in the neighborhood β∗. We can develop a similar argument using the monetary policy parameter γ.


## 2.1. Basic Equations

The model economy comprises the following equations:(1)Y˙t=αCt+It+Gt-Yt;α>0,(2)Ct=cYt-Tt+C¯;0<c<1;C¯>0,(3)Tt=τYt-T¯;0<τ<1;T¯≥0,(4)It=IYt,Kt,rt;IY>0;IK<0;Ir<0,(5)Gt=βY¯-Yt-θ1+G¯;β>0;Y¯>0;G¯>0,(6)K˙t=IYt,Kt,rt,(7)MtP¯=LYt,rt;P¯>0;LY>0;Lr<0,(8)Mt=P¯γY¯-Yt-θ2+M¯;γ>0;M¯>0,where Y = real national income; C = real private consumption; I = real private investment; G = real government expenditure; T = real income tax; K = real capital stock; r = nominal interest rate; M = nominal money supply; α = adjustment speed of the goods market; c = marginal propensity to consume; C¯ = base consumption; τ = marginal tax rate; T¯ = real subsidy; β = activeness level of fiscal policy; Y¯ = target level of real national income; G¯ = target level of real government expenditure; P¯ = price level; γ = activeness level of monetary policy; M¯ = target level of nominal money supply; t = time; θ1 = fiscal policy lag; and θ2 = monetary policy lag.Equations (1), (2), (3), and (4) represent a disequilibrium adjustment function of the goods market, a consumption function, a tax collection function, and an investment function, respectively. Equation (5) represents a fiscal policy reaction function with a government expenditure lag, while (6) signifies a capital accumulation function. For simplicity, we assume that capital does not depreciate. Equation (7) represents the monetary market equilibrium condition, where the left-hand side denotes real money balance and the right-hand side indicates real money demand (in this study, we assume that the price level P is constant; by assuming P(t)=P(Y(t)) (where PY>0), we can allow for variations in the price level; however, this change does not affect the nature of our argument). Here, we ensure that the adjustment of the monetary market is rapid, and therefore the balance of demand and supply of this market is always maintained. Finally, (8) represents a monetary policy reaction function with a money supply lag.If real money balance is constant (i.e.,γ=0), then the system compounded from (1)–(8) essentially becomes similar to that of Asada and Yoshida [12]. However, the assumptions of γ>0 and θ2>0 significantly complicate the dynamic property of the system, thereby resulting in a major change in the economic implication of time lags.


## 2.2. Summarization

We summarize (1)–(8) in a two-dimensional dynamic system. First, by substituting (8) into (7) and solving for r, we obtain(9)rt=rYt,Yt-θ2,where rY(t)=-LY/Lr>0 and rY(t-θ2)=-γ/Lr>0.Furthermore, by substituting (3) into (2) and substituting (9) into (4), we obtain(10)Ct=c1-τYt+cT¯+C¯,(11)It=IYt,Kt,rYt,Yt-θ2.Finally, by substituting (5), (10), and (11) into (1) and substituting (11) into (6), we obtain the following system of differential equations with two time delays:(12)Y˙t=αIYt,Kt,rYt,Yt-θ2-1-c1-τYt-βYt-θ1+cT¯+C¯+βY¯+G¯,K˙t=IYt,Kt,rYt,Yt-θ2.


## 2.3. Linearization

We assume that steady-state solutionsY∗>0 and K∗>0 that satisfy Y˙(t)=K˙(t)=0 exist and these solutions are continuous with respect to all parameters. In order to analyze the local stability around the steady-state point (Y∗,K∗), we linearize the system and obtain(13)Y^˙t=αA1-1+c1-τY^t-βY^t-θ1+A2Y^t-θ2+IKK^t,K^˙t=A1Y^t+A2Y^t-θ2+IKK^t,where Y^(t)=Y(t)-Y∗, K^(t)=K(t)-K∗, A1=IY+IrrY(t), and A2=IrrY(t-θ2).Assuming the exponential functionsY^(t)=C1eλt and K^(t)=C2eλt, where C1 and C2 are arbitrary constants and λ denotes the eigenvalue, as the solutions of system (13) and substituting these into the system, we obtain (14)λ-αA1-1+c1-τ+αβe-θ1λ-αA2e-θ2λ-αIK-A1-A2e-θ2λλ-IKY^tK^t=00.For nontrivial solutions to exist, the determinant of the left-hand side matrix, denoted byΔ(λ), must be equal to zero; that is,(15)Δλ=λ2-IK+αA1-1+c1-τλ-α1-c1-τIK+αβλ-IKe-θ1λ-αA2λe-θ2λ=0=p0λ+p1λe-θ1λ+p2λe-θ2λ=0,where(16)p0λ=λ2+b1λ+b2,b1=-IK+αA1-1+c1-τ,b2=-α1-c1-τIK>0,p1λ=αβλ-IK,p2λ=-αA2λ.Equation (15) is a characteristic equation of system (13). The significant feature of this equation is found in the existence of the exponential terms e-θ1λ and e-θ2λ.First, we examine the case of no policy lags. By substitutingθ1=θ2=0 into (15), we obtain(17)λ2+b1+αβ-A2λ+b2-αβIK=0,which is an ordinary quadratic equation of λ.Thus, ifb1+α(β-A2)>0 (i.e., the coefficient of λ from (17) is positive), the real parts of the roots of (17) are negative (see Chapter 18 in Gandolfo [18] for details regarding the relation between the roots and the coefficients of a quadric equation). In contrast, if b1+α(β-A2)<0, the real parts of the roots are positive. Thus, if b1+α(β-A2)>0, then the steady state is locally stable, and if b1+α(β-A2)<0, then it is unstable.The larger the values ofβ and γ are, the more likely the stability condition holds (note that A2=-γIr/Lr). Therefore, if policy lags do not exist, then traditional fiscal and monetary policies could function effectively for stabilization.In the following discussion, we assume that if both fiscal and monetary policy lags do not exist, then an economy is stable; that is, we have the following.Assumption 1.
We have(18)β+IrLrγ+b1α>0.Under this assumption, we analyze the effects of the policy lags(θ1,θ2) on stability.Incidentally, if[b1+α(β-A2)]2<4(b2-αβIK), then both roots of (17) are conjugate complex. Furthermore, by defining β∗≡A2-b1/α, the conditions for Hopf bifurcation are satisfied at β=β∗; that is, (i) (17) includes a pair of pure imaginary roots and (ii) dReλ/dβ≠0. Therefore, a cycle exists for a certain range of β in the neighborhood β∗. We can develop a similar argument using the monetary policy parameter γ.


## 3. Case of Monetary Policy with a Lag

Asada and Yoshida [12] examine the situation where a time lag exists in fiscal policy responses, which is equivalent to the case wherein θ1>0 and θ2=0 in system (12). In this section, we analyze the situation where a time lag exists in monetary policy responses by assuming that θ1=0 and θ2>0 in system (12).In this case, the characteristic equation in (15) can be rewritten as follows:(19)λ2+b1+αβλ+b2-αβIK-αA2λe-θ2λ=0.The following analysis is based on the study of Matsumoto and Szidarovszky [19] (Bellman and Cooke [20] provide a helpful introductory textbook of delay differential equations). The procedures of the analysis are given as follows:(1)
We characterize the points (if any) at which the dynamics around the steady state can change, that is, the points at which a zero real root or pure imaginary roots appear. These points are referred to as “crossing points.”(2)
We reveal the directions of the changes in the signs of the real parts that occur whenθ2 crosses the crossing points.


### 3.1. Crossing Points

First, we examine whether the points at which a zero real root appears can exist. By substitutingλ=0 into (19), we obtain b2-αβIK=0. However, since b2>0 and IK<0, this equality cannot hold. Accordingly, λ=0 cannot be a root.Next, we examine whether the points at which the pure imaginary roots appear can exist. By substitutingλ=iω, where ω = imaginary part >0 (pure imaginary roots are always conjugated; therefore, we can assume ω>0 without loss of generality) and i=-1, into (19), we obtain (20)-ω2+iωb1+αβ+b2-αβIK-iωαA2e-iωθ2=0.Application of Euler’s formula (∀x∈Re±ix=cos⁡x±isin⁡x) to this equation yields (21)-ω2+b2-αβIK-ωαA2sin⁡ωθ2+ib1+αβω-ωαA2cos⁡ωθ2=0.This equality holds only when both real and imaginary parts of the left-hand side equal zero; that is,(22)-ω2+b2-αβIK=ωαA2sin⁡ωθ2,b1+αβω=ωαA2cos⁡ωθ2.Or, equivalently,(23)cos⁡ωθ2=b1+αβαA2,(24)sin⁡ωθ2=-ω2+b2-αβIKαA2ω.For values of ωθ2 that satisfy these equations, (19) includes pure imaginary roots, where 2πh<ωθ2<2π(1+h), h=0,1,2,3,….Furthermore, the sum of the squares of (22) yields the following: (25)ω4-2b2-αβIK-b1+αβ2+αA22ω2+b2-αβIK2=0.Solving for ω2, we obtain(26)ω±2=2b2-αβIK-b1+αβ2+αA22±N2,where N≡{2(b2-αβIK)-(b1+αβ)2+(αA2)2}2-4(b2-αβIK)2.We denote real and positive values ofω by ω∗. By substituting ω∗ into (23) and solving for θ2, we obtain(27)θ2h=cos-1⁡b1+αβ/αA2+2πhω∗,h=0,1,2,3,….Thus, there exists an infinite number of values of θ2 (i.e., θ20,θ21,θ22,…) that generate pure imaginary roots (i.e., crossing points).Finally, if a real and positive value ofω does not exist, then changes in the signs of the real parts of the roots do not occur. Therefore, in this case, a policy lag does not affect stability.Direction of Crossing. Here, we reveal how the signs of the complex roots change when θ2 crosses θ2h. This is determined by the sign of dReλ/dθ2|λ=iω∗. If dReλ/dθ2|λ=iω∗>0, then the signs of the real parts of the roots change from negative to positive with an increase in θ2 (which indicates destabilization). In contrast, if dReλ/dθ2|λ=iω∗<0, then the signs of the real parts of the roots change from positive to negative with an increase in θ2 (which indicates stabilization). For convenience of calculation, we will observe the sign of Re(dλ/dθ2)-1|λ=iω∗ instead of that of dReλ/dθ2|λ=iω∗.Using (19), we can demonstrate that Re(dλ/dθ2)-1|λ=iω∗>0 holds for any value of h (see the appendix). Thus, with any crossing, the signs of the real parts of the roots change from negative to positive.If a policy lag does not present (θ2=0), then roots with positive real parts do not exist under Assumption 1. With increases in θ2, the number of roots with positive real parts increases by two. Accordingly, we can state that 2h roots with positive real parts exist in the region of θ2∈(θ2(h-1),θ2h) (where θ2(-1)=0). Therefore, we can offer the following proposition.Proposition 2.
Forθ2∈(0,θ20), the steady state is stable, and, for θ2∈(θ20,∞), it is unstable.This proposition suggests that even if a monetary policy is sufficiently active enough to satisfy Assumption1, an economy can become unstable due to the existence of a policy lag. A similar result has already been shown by Asada and Yoshida [12] concerning fiscal policies.


### 3.2. Numerical Simulations

Let us consider a numerical example. Following the study of Asada and Yoshida [12], we set baseline parameters, as shown in Table 1.Table 1
Baseline parameters.
α
c
τ
C
-
+
c
T
-
+
G
-
Y
-
0.9
0.625
0.2
200
400Furthermore, Asada and Yoshida [12] employ the investment function that is highly nonlinear with respect to Y(t), which is formulated as follows (in their model, γ=0):(28)I=Y¯1+12e-0.1Yt-Y¯-r-0.5Kt-Y¯13+0.2+30,r=-0.05γY¯-Yt-θ2+0.01Yt.Based on these assumptions, the steady-state values are provided by (Y∗,K∗)=(400,60).We also set the baseline values ofβ and γ as 2.5 and 6.0, respectively (generally, it seems that the responsiveness of monetary policies to economic fluctuations is often larger than that of fiscal policies; therefore, we assume γ>β). When γ is fixed at the baseline value, ω∗ exists only for β∈(1.49,2.08). Outside this range, a policy lag does not affect stability. In addition, Assumption 1 is satisfied for β>1.49. Accordingly, irrespective of the value of a monetary policy lag, the steady state is unstable for 0<β<1.49, and it is stable for β>2.08.Likewise, whenβ is fixed at the baseline value, ω∗ exists for γ>14.32. In addition, Assumption 1 is satisfied for γ>-14.32. Hence, the steady state is stable for 0<γ<14.32, irrespective of the value of a lag.Figure1 describes θ2h (h=0,1,2) that corresponds to ω∗ as a function of β and γ. These curves are referred to as “crossing curves.” Here, the stability condition is expressed as θ2<θ20. Therefore, we can state that an increase in β has a stabilizing effect, whereas that in γ has a destabilizing effect.Figure 1
Crossing curves.Thus, an increase in the activeness level of a monetary policy may cause instability due to the existence of a monetary policy lag. A similar result concerning a fiscal policy has already been shown by Asada and Yoshida [12].Incidentally, conditions for Hopf bifurcation are satisfied atθ2=θ20; that is, (i) (19) includes a pair of pure imaginary roots and (ii) dReλ/dθ2≠0. Therefore, when θ2 crosses θ20, a cycle emerges. A similar argument can be developed using β and γ. We now suppose that θ2=2.0. In this case, a bifurcation diagram can be described for γ∈[14,18], as shown in Figure 2 (the vertical axis represents the maximal and minimal values of Y(t)). This figure shows that not only one period but also multiple period cycles can emerge. The emergence of multiple period cycles is attributed to the strong nonlinearity of the investment function with respect to Y(t). When we use the investment function that has relatively simpler form with respect to Y(t) as below (we set the coefficient of Y(t) as 113.6 so that the value of IY equals that of (28)), the bifurcation diagram becomes as shown in Figure 3. In this case, only one period cycle exists. Hence,(29)I=113.6Yt-r-0.5Kt-113.6Y-+0.2+30.Figure 2
Bifurcation diagram (case with (28)).Figure 3
Bifurcation diagram (case with (29)).Although Asada and Yoshida [12] do not clearly specify the time dimension, similar models that include a disequilibrium adjustment function of the goods market (α is set around unity) specify the measure of t as a year (refer to, e.g., Yoshida and Asada [5]). Therefore, we will consider the time dimension as a year. For example, when β=1.8, the steady state is unstable for θ2>1.4 years.


## 3.1. Crossing Points

First, we examine whether the points at which a zero real root appears can exist. By substitutingλ=0 into (19), we obtain b2-αβIK=0. However, since b2>0 and IK<0, this equality cannot hold. Accordingly, λ=0 cannot be a root.Next, we examine whether the points at which the pure imaginary roots appear can exist. By substitutingλ=iω, where ω = imaginary part >0 (pure imaginary roots are always conjugated; therefore, we can assume ω>0 without loss of generality) and i=-1, into (19), we obtain (20)-ω2+iωb1+αβ+b2-αβIK-iωαA2e-iωθ2=0.Application of Euler’s formula (∀x∈Re±ix=cos⁡x±isin⁡x) to this equation yields (21)-ω2+b2-αβIK-ωαA2sin⁡ωθ2+ib1+αβω-ωαA2cos⁡ωθ2=0.This equality holds only when both real and imaginary parts of the left-hand side equal zero; that is,(22)-ω2+b2-αβIK=ωαA2sin⁡ωθ2,b1+αβω=ωαA2cos⁡ωθ2.Or, equivalently,(23)cos⁡ωθ2=b1+αβαA2,(24)sin⁡ωθ2=-ω2+b2-αβIKαA2ω.For values of ωθ2 that satisfy these equations, (19) includes pure imaginary roots, where 2πh<ωθ2<2π(1+h), h=0,1,2,3,….Furthermore, the sum of the squares of (22) yields the following: (25)ω4-2b2-αβIK-b1+αβ2+αA22ω2+b2-αβIK2=0.Solving for ω2, we obtain(26)ω±2=2b2-αβIK-b1+αβ2+αA22±N2,where N≡{2(b2-αβIK)-(b1+αβ)2+(αA2)2}2-4(b2-αβIK)2.We denote real and positive values ofω by ω∗. By substituting ω∗ into (23) and solving for θ2, we obtain(27)θ2h=cos-1⁡b1+αβ/αA2+2πhω∗,h=0,1,2,3,….Thus, there exists an infinite number of values of θ2 (i.e., θ20,θ21,θ22,…) that generate pure imaginary roots (i.e., crossing points).Finally, if a real and positive value ofω does not exist, then changes in the signs of the real parts of the roots do not occur. Therefore, in this case, a policy lag does not affect stability.Direction of Crossing. Here, we reveal how the signs of the complex roots change when θ2 crosses θ2h. This is determined by the sign of dReλ/dθ2|λ=iω∗. If dReλ/dθ2|λ=iω∗>0, then the signs of the real parts of the roots change from negative to positive with an increase in θ2 (which indicates destabilization). In contrast, if dReλ/dθ2|λ=iω∗<0, then the signs of the real parts of the roots change from positive to negative with an increase in θ2 (which indicates stabilization). For convenience of calculation, we will observe the sign of Re(dλ/dθ2)-1|λ=iω∗ instead of that of dReλ/dθ2|λ=iω∗.Using (19), we can demonstrate that Re(dλ/dθ2)-1|λ=iω∗>0 holds for any value of h (see the appendix). Thus, with any crossing, the signs of the real parts of the roots change from negative to positive.If a policy lag does not present (θ2=0), then roots with positive real parts do not exist under Assumption 1. With increases in θ2, the number of roots with positive real parts increases by two. Accordingly, we can state that 2h roots with positive real parts exist in the region of θ2∈(θ2(h-1),θ2h) (where θ2(-1)=0). Therefore, we can offer the following proposition.Proposition 2.
Forθ2∈(0,θ20), the steady state is stable, and, for θ2∈(θ20,∞), it is unstable.This proposition suggests that even if a monetary policy is sufficiently active enough to satisfy Assumption1, an economy can become unstable due to the existence of a policy lag. A similar result has already been shown by Asada and Yoshida [12] concerning fiscal policies.


## 3.2. Numerical Simulations

Let us consider a numerical example. Following the study of Asada and Yoshida [12], we set baseline parameters, as shown in Table 1.Table 1
Baseline parameters.
α
c
τ
C
-
+
c
T
-
+
G
-
Y
-
0.9
0.625
0.2
200
400Furthermore, Asada and Yoshida [12] employ the investment function that is highly nonlinear with respect to Y(t), which is formulated as follows (in their model, γ=0):(28)I=Y¯1+12e-0.1Yt-Y¯-r-0.5Kt-Y¯13+0.2+30,r=-0.05γY¯-Yt-θ2+0.01Yt.Based on these assumptions, the steady-state values are provided by (Y∗,K∗)=(400,60).We also set the baseline values ofβ and γ as 2.5 and 6.0, respectively (generally, it seems that the responsiveness of monetary policies to economic fluctuations is often larger than that of fiscal policies; therefore, we assume γ>β). When γ is fixed at the baseline value, ω∗ exists only for β∈(1.49,2.08). Outside this range, a policy lag does not affect stability. In addition, Assumption 1 is satisfied for β>1.49. Accordingly, irrespective of the value of a monetary policy lag, the steady state is unstable for 0<β<1.49, and it is stable for β>2.08.Likewise, whenβ is fixed at the baseline value, ω∗ exists for γ>14.32. In addition, Assumption 1 is satisfied for γ>-14.32. Hence, the steady state is stable for 0<γ<14.32, irrespective of the value of a lag.Figure1 describes θ2h (h=0,1,2) that corresponds to ω∗ as a function of β and γ. These curves are referred to as “crossing curves.” Here, the stability condition is expressed as θ2<θ20. Therefore, we can state that an increase in β has a stabilizing effect, whereas that in γ has a destabilizing effect.Figure 1
Crossing curves.Thus, an increase in the activeness level of a monetary policy may cause instability due to the existence of a monetary policy lag. A similar result concerning a fiscal policy has already been shown by Asada and Yoshida [12].Incidentally, conditions for Hopf bifurcation are satisfied atθ2=θ20; that is, (i) (19) includes a pair of pure imaginary roots and (ii) dReλ/dθ2≠0. Therefore, when θ2 crosses θ20, a cycle emerges. A similar argument can be developed using β and γ. We now suppose that θ2=2.0. In this case, a bifurcation diagram can be described for γ∈[14,18], as shown in Figure 2 (the vertical axis represents the maximal and minimal values of Y(t)). This figure shows that not only one period but also multiple period cycles can emerge. The emergence of multiple period cycles is attributed to the strong nonlinearity of the investment function with respect to Y(t). When we use the investment function that has relatively simpler form with respect to Y(t) as below (we set the coefficient of Y(t) as 113.6 so that the value of IY equals that of (28)), the bifurcation diagram becomes as shown in Figure 3. In this case, only one period cycle exists. Hence,(29)I=113.6Yt-r-0.5Kt-113.6Y-+0.2+30.Figure 2
Bifurcation diagram (case with (28)).Figure 3
Bifurcation diagram (case with (29)).Although Asada and Yoshida [12] do not clearly specify the time dimension, similar models that include a disequilibrium adjustment function of the goods market (α is set around unity) specify the measure of t as a year (refer to, e.g., Yoshida and Asada [5]). Therefore, we will consider the time dimension as a year. For example, when β=1.8, the steady state is unstable for θ2>1.4 years.


## 4. Case of Policies with Fiscal and Monetary Policy Lags

In this section, we investigate the case wherein both fiscal and monetary policy lags exist (i.e.,θ1>0 and θ2>0). In order to examine the signs of the roots of (15), we use a mathematical method developed by Gu et al. [17].


### 4.1. Preconditions

First, to apply the technique of Gu et al. [17], some preconditions should be examined. According to their study, (15) should satisfy the following conditions:(I)
deg⁡(p0(λ))≥max⁡{deg⁡(p1(λ)),deg⁡(p2(λ))}.(II)
Δ(0)≠0.(III)
A solution common to all three polynomialsp0(λ)=0, p1(λ)=0, and p2(λ)=0 does not exist.(IV)
limλ→∞⁡(|p1(λ)/p0(λ)|+|p2(λ)/p0(λ)|)<1.Condition (I) is satisfied by2>max⁡{1,1}, while Condition (II) is also satisfied by Δ(0)=-αIK{1-c(1-τ)+β}>0. Condition (III) is obviously 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 sets(θ1,θ2) (i.e., crossing sets) at which the pure imaginary roots appear (it is ensured from precondition (II) that zero cannot be a root).(2)
We depict the crossing sets on the(θ1,θ2) plane (which we refer to as the crossing curves) via numerical simulation.(3)
We reveal the directions of changes in the signs of the real parts that occur when lags(θ1,θ2) cross the crossing curves.


### 4.2. Crossing Sets

Dividing (15) by p0(λ), we obtain(30)1+a1λe-θ1λ+a2λe-θ2λ=0,where(31)a1λ=p1λp0λ=αβλ-IKλ2+b1λ+b2,a2λ=p2λp0λ=-αA2λλ2+b1λ+b2.Denoting a pure imaginary root byλ=iω, the values of ω that satisfy (30) can be characterized by the following lemma.Lemma 3 (see [17, Proposition 3.1]).
For eachω satisfying p0(iω)≠0, λ=iω is a solution of Δ(λ)=0 for some (θ1,θ2)∈R+2 if and only if(32)a1iω+a2iω≥1,-1≤a1iω-a2iω≤1.We denote the set ofω>0 that satisfy conditions (32) as Ω, which is termed the “crossing frequency.” For all ω∈Ω, the sets (θ1,θ2) satisfying (30) (i.e., crossing sets) should satisfy the following conditions (Figure 4):(33)∓δ1=arg⁡a1iωe-iωθ1+2mπ;m=0,1,2,…,±δ2=arg⁡a2iωe-iωθ2+2nπ;n=0,1,2,…,where δ1,δ2∈[0,π].Figure 4
Triangle formed by1, |a1(iω)|, and |a2(iω)| on the complex plane.Incidentally, on the complex plane, multiplication of amplitudes becomes a sum of the parts; therefore, we obtain(34)arg⁡a1iωe-iωθ1=arg⁡a1iω-ωθ1,arg⁡a2iωe-iωθ2=arg⁡a2iω-ωθ2.In addition, Figure4 demonstrates that the following relations hold:(35)arg⁡a1iω=tan-1⁡Im⁡a1iωRe⁡a1iω,arg⁡a2iω=tan-1⁡Im⁡a2iωRe⁡a2iω.Moreover, after some manipulation, (31) derive the following expressions:(36)Im⁡a1iωRe⁡a1iω=b1ωIK+ωb2-ω2b1ω2-IKb2-ω2,Im⁡a2iωRe⁡a2iω=b2-ω2b1ω.Thus, using (34) to (36), (33) can be rewritten as follows:(37)θ1=tan-1⁡b1ωIK+ωb2-ω2/b1ω2-IKb2-ω2±δ1+2mπω,θ2=tan-1⁡b2-ω2/b1ω∓δ2+2nπω,where the interior angles of the triangle denoted by δ1 and δ2 are given by the cosine theorem as follows: (38)δ1=cos-1⁡1+a1iω2-a2iω22a1iω=cos-1⁡b2-ω22+b1ω2+αβIK2+αβω2-αA2ω22αβIK2+αβω2b2-ω22+b1ω2,δ2=cos-1⁡1+a2iω2-a1iω22a2iω=cos-1⁡b2-ω22+b1ω2+αA2ω2-αβIK2-αβω22αA2ω2b2-ω22+b1ω2.Equations (37) characterize the crossing sets (θ1,θ2)∈R+2. Depending on the signs of δ1 and δ2, we can define two types of crossing sets, denoted by L1m,n and L2m,n, as follows: L1m,nθ1=tan-1b1ωIK+ωb2-ω2/b1ω2-IKb2-ω2+δ1+2mπω,θ2=tan-1b2-ω2/b1ω-δ2+2nπω,L2m,nθ1=tan-1b1ωIK+vb2-ω2/b1ω2-IKb2-ω2-δ1+2mπω,θ2=tan-1b2-ω2/b1ω+δ2+2nπω.As in the previous section, we set parameter values and the functional form of the investment function, as shown in Table1, (28) or (29), β=2.5 and γ=6.0. In this case, crossing frequency Ω is provided by ω∈(1.50,2.13) (Figure 5) (according to Gu et al.’s [17] classification, Figure 5 exhibits Type 13). For ω∈Ω, describing the crossing sets L1m,n and L2m,n on the θ1-θ2 plane, we obtain Figure 6, where m=0,1,2 and n=0,1,2. The dotted curves represent L1m,n, and the solid curves represent L2m,n. The points marked by a dot indicate the starting points of both curves, that is, the points corresponding to ω=1.50.Figure 5
Crossing frequencyΩ.Figure 6
Crossing curves.We establish how the real parts of the roots change when lags cross the crossing curves with an increase inθ1. For ω∈Ω, if Re(dλ/dθ1)-1>0, then the real parts of the roots change from negative to positive; this indicates destabilization. In contrast, if Re(dλ/dθ1)-1<0, then the real parts of the roots change from positive to negative; this indicates stabilization.Differentiating (30) with respect to θ1, we obtain (39)a1′λe-θ1λ-a1λe-θ1λθ1+a2′λe-θ2λ-a2λe-θ2θ2dλdθ1=a1λe-θ1λλ,or, equivalently,(40)dλdθ1-1=a1′λe-θ1λ+a2′λe-θ2λ-a2λe-θ2λθ2a1λe-θ1λλ-θ1λ,where (41)a1′λ=αβp0λ-αβλ-IK2λ+b1p0λ2,a2′λ=-αA2p0λ-αA2λ2λ+b1p0λ2.Describing the real parts of (40) as a function of ω∈Ω, we obtain Figure 7, where the dotted curves are the functions evaluated on curve L1 and the solid curves are the functions evaluated on curve L2. As for curve L1, stabilization only occurs on the intermediate region of the segments L1(m,2); destabilization occurs on the other segments of L1. As for curve L2, destabilization occurs on all segments. Here, we define a curve formed by connecting curves Lj(0,n) (where j=1,2 and n=0,1,2) as m0 (Figure 8). We also define a curve formed by connecting the points on curve m0 that minimize the value of θ1 for a given θ2 as curve M0. Then, we can express the stability condition as follows: set (θ1,θ2) exists in the left-hand side region of curve M0.Figure 7
Direction of crossing.Figure 8
Stability regionS.On curveM0, conditions for Hopf bifurcation are satisfied, and therefore a cycle emerges. In addition, as is obvious from Figure 6, an increase in the fiscal policy lag θ1 necessarily destabilizes the steady state. However, an increase in the monetary policy lag θ2 has the potential to stabilize the steady state, which depends on the value of θ1. When θ1∈(0.22,0.36), we can see that the steady state fluctuates between stability and instability with an increase in θ2. Such a phenomenon cannot be observed in models with only one time lag. Moreover, Figure 9 presents a bifurcation diagram wherein θ1=0.25 and (28) is used as the investment function. This figure evidently indicates that alternations of stability and instability emerge with an increase in θ2.Figure 9
Bifurcation diagram (θ1=0.25).In particular, whenθ1=0.25, stability can be established by setting θ2∈(0,1.06), θ2∈(2.53,4.33), or θ2∈(6.71,7.61); that is, for equilibrium stability, monetary policy lag should be less than one year, between 3 and 4.5 years, or between 6.8 and 7.5 years when a fiscal policy lag is approximately three months.Next, we examine howβ and γ affect the stability condition. Both β and γ affect L1 and L2 through ω∈Ω. It is difficult to analytically show how L1 and L2 change. Hence, we numerically examine how curve m0 (which comprises L1(0,n) and L2(0,n)) shifts when β and γ increase from each baseline value (β=2.5 and γ=6.0).Figures10 and 11 describe curve m0 in cases with β=8 and β=15, respectively. We can observe that an increase in β shifts curve m0 to left. Accordingly, the stability region S reduces. Likewise, Figures 12 and 13 describe curve m0 in cases with γ=12 and γ=22, respectively. The stability region is also reduced by an increase in γ. These results suggest that increases in the activeness levels of fiscal and monetary policies increase economic instability, as is the case with only one policy lag.Figure 10
β
=
8.Figure 11
β
=
15.Figure 12
γ
=
12.Figure 13
γ
=
22.


### 4.3. Investment and Stability

In general, the formulation of the investment function has great significance to stability in macrodynamic models; that is, the larger the marginal propensity to investment is, the more likely it becomes unstable. In this subsection, we examine the effects of changes in the partial derivatives of the investment function,IY and IK, on stability. We compare three cases: the case of no policy lags, the case of the existence of only monetary policy lag, and the case of the existence of both fiscal and monetary policy lags.In the case with no policy lags,IY and IK affect the stability condition given by Assumption 1 through b1. The smaller the values of IY and IK are (the larger the absolute value of IK is), the more likely the stability condition is satisfied. These are fairly typical results.In the case of a monetary policy with a lag,IY and IK affect the value of ω∗ through b2 (IK also directly affects ω∗). Accordingly, the value of (27) changes. In this case, it is difficult to analytically show the quality and quantity of these effects. Hence, we resort to a numerical method. Based on the parameter values proposed in Section 3.2, crossing curves θ20 can be described within each range of IY and IK that ensure the existence of ω∗, as shown in Figure 14. The region located below these curves indicates the stability regions. Similar to the case with no policy lags, an increase in IY causes destabilization. However, due to the concavity of the crossing curve, there is a possibility that a decrease in IK causes destabilization. Thus, the existence of a lag complicates the effects of investment on stability.Figure 14
Crossing curveθ20.Finally, we investigate the case wherein both fiscal and monetary policy lags exist. In this case,IY and IK affect the stability condition (i.e., location of curve M0) through ω∈Ω, b1, and b2. According to numerical simulations, increases in the absolute values of IY and IK generally shift the curve M0 toward the left. Therefore, these changes decrease the stability region. However, if M0 completely shifts even to the negative region of θ1, then the curve that specifies the stability region is no longer M0 but is M1. For example, when IY=3.0, the stability region S is defined as the left-hand side region of the curve M0 in Figure 15. However, when IY=4.0, the stability region S is defined as the left-hand side region of the curve M1 in Figure 16. Thus, an increase in IY may increase the stability region.Figure 15
I
Y
=
3.0.Figure 16
I
Y
=
4.0.


## 4.1. Preconditions

First, to apply the technique of Gu et al. [17], some preconditions should be examined. According to their study, (15) should satisfy the following conditions:(I)
deg⁡(p0(λ))≥max⁡{deg⁡(p1(λ)),deg⁡(p2(λ))}.(II)
Δ(0)≠0.(III)
A solution common to all three polynomialsp0(λ)=0, p1(λ)=0, and p2(λ)=0 does not exist.(IV)
limλ→∞⁡(|p1(λ)/p0(λ)|+|p2(λ)/p0(λ)|)<1.Condition (I) is satisfied by2>max⁡{1,1}, while Condition (II) is also satisfied by Δ(0)=-αIK{1-c(1-τ)+β}>0. Condition (III) is obviously 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 sets(θ1,θ2) (i.e., crossing sets) at which the pure imaginary roots appear (it is ensured from precondition (II) that zero cannot be a root).(2)
We depict the crossing sets on the(θ1,θ2) plane (which we refer to as the crossing curves) via numerical simulation.(3)
We reveal the directions of changes in the signs of the real parts that occur when lags(θ1,θ2) cross the crossing curves.


## 4.2. Crossing Sets

Dividing (15) by p0(λ), we obtain(30)1+a1λe-θ1λ+a2λe-θ2λ=0,where(31)a1λ=p1λp0λ=αβλ-IKλ2+b1λ+b2,a2λ=p2λp0λ=-αA2λλ2+b1λ+b2.Denoting a pure imaginary root byλ=iω, the values of ω that satisfy (30) can be characterized by the following lemma.Lemma 3 (see [17, Proposition 3.1]).
For eachω satisfying p0(iω)≠0, λ=iω is a solution of Δ(λ)=0 for some (θ1,θ2)∈R+2 if and only if(32)a1iω+a2iω≥1,-1≤a1iω-a2iω≤1.We denote the set ofω>0 that satisfy conditions (32) as Ω, which is termed the “crossing frequency.” For all ω∈Ω, the sets (θ1,θ2) satisfying (30) (i.e., crossing sets) should satisfy the following conditions (Figure 4):(33)∓δ1=arg⁡a1iωe-iωθ1+2mπ;m=0,1,2,…,±δ2=arg⁡a2iωe-iωθ2+2nπ;n=0,1,2,…,where δ1,δ2∈[0,π].Figure 4
Triangle formed by1, |a1(iω)|, and |a2(iω)| on the complex plane.Incidentally, on the complex plane, multiplication of amplitudes becomes a sum of the parts; therefore, we obtain(34)arg⁡a1iωe-iωθ1=arg⁡a1iω-ωθ1,arg⁡a2iωe-iωθ2=arg⁡a2iω-ωθ2.In addition, Figure4 demonstrates that the following relations hold:(35)arg⁡a1iω=tan-1⁡Im⁡a1iωRe⁡a1iω,arg⁡a2iω=tan-1⁡Im⁡a2iωRe⁡a2iω.Moreover, after some manipulation, (31) derive the following expressions:(36)Im⁡a1iωRe⁡a1iω=b1ωIK+ωb2-ω2b1ω2-IKb2-ω2,Im⁡a2iωRe⁡a2iω=b2-ω2b1ω.Thus, using (34) to (36), (33) can be rewritten as follows:(37)θ1=tan-1⁡b1ωIK+ωb2-ω2/b1ω2-IKb2-ω2±δ1+2mπω,θ2=tan-1⁡b2-ω2/b1ω∓δ2+2nπω,where the interior angles of the triangle denoted by δ1 and δ2 are given by the cosine theorem as follows: (38)δ1=cos-1⁡1+a1iω2-a2iω22a1iω=cos-1⁡b2-ω22+b1ω2+αβIK2+αβω2-αA2ω22αβIK2+αβω2b2-ω22+b1ω2,δ2=cos-1⁡1+a2iω2-a1iω22a2iω=cos-1⁡b2-ω22+b1ω2+αA2ω2-αβIK2-αβω22αA2ω2b2-ω22+b1ω2.Equations (37) characterize the crossing sets (θ1,θ2)∈R+2. Depending on the signs of δ1 and δ2, we can define two types of crossing sets, denoted by L1m,n and L2m,n, as follows: L1m,nθ1=tan-1b1ωIK+ωb2-ω2/b1ω2-IKb2-ω2+δ1+2mπω,θ2=tan-1b2-ω2/b1ω-δ2+2nπω,L2m,nθ1=tan-1b1ωIK+vb2-ω2/b1ω2-IKb2-ω2-δ1+2mπω,θ2=tan-1b2-ω2/b1ω+δ2+2nπω.As in the previous section, we set parameter values and the functional form of the investment function, as shown in Table1, (28) or (29), β=2.5 and γ=6.0. In this case, crossing frequency Ω is provided by ω∈(1.50,2.13) (Figure 5) (according to Gu et al.’s [17] classification, Figure 5 exhibits Type 13). For ω∈Ω, describing the crossing sets L1m,n and L2m,n on the θ1-θ2 plane, we obtain Figure 6, where m=0,1,2 and n=0,1,2. The dotted curves represent L1m,n, and the solid curves represent L2m,n. The points marked by a dot indicate the starting points of both curves, that is, the points corresponding to ω=1.50.Figure 5
Crossing frequencyΩ.Figure 6
Crossing curves.We establish how the real parts of the roots change when lags cross the crossing curves with an increase inθ1. For ω∈Ω, if Re(dλ/dθ1)-1>0, then the real parts of the roots change from negative to positive; this indicates destabilization. In contrast, if Re(dλ/dθ1)-1<0, then the real parts of the roots change from positive to negative; this indicates stabilization.Differentiating (30) with respect to θ1, we obtain (39)a1′λe-θ1λ-a1λe-θ1λθ1+a2′λe-θ2λ-a2λe-θ2θ2dλdθ1=a1λe-θ1λλ,or, equivalently,(40)dλdθ1-1=a1′λe-θ1λ+a2′λe-θ2λ-a2λe-θ2λθ2a1λe-θ1λλ-θ1λ,where (41)a1′λ=αβp0λ-αβλ-IK2λ+b1p0λ2,a2′λ=-αA2p0λ-αA2λ2λ+b1p0λ2.Describing the real parts of (40) as a function of ω∈Ω, we obtain Figure 7, where the dotted curves are the functions evaluated on curve L1 and the solid curves are the functions evaluated on curve L2. As for curve L1, stabilization only occurs on the intermediate region of the segments L1(m,2); destabilization occurs on the other segments of L1. As for curve L2, destabilization occurs on all segments. Here, we define a curve formed by connecting curves Lj(0,n) (where j=1,2 and n=0,1,2) as m0 (Figure 8). We also define a curve formed by connecting the points on curve m0 that minimize the value of θ1 for a given θ2 as curve M0. Then, we can express the stability condition as follows: set (θ1,θ2) exists in the left-hand side region of curve M0.Figure 7
Direction of crossing.Figure 8
Stability regionS.On curveM0, conditions for Hopf bifurcation are satisfied, and therefore a cycle emerges. In addition, as is obvious from Figure 6, an increase in the fiscal policy lag θ1 necessarily destabilizes the steady state. However, an increase in the monetary policy lag θ2 has the potential to stabilize the steady state, which depends on the value of θ1. When θ1∈(0.22,0.36), we can see that the steady state fluctuates between stability and instability with an increase in θ2. Such a phenomenon cannot be observed in models with only one time lag. Moreover, Figure 9 presents a bifurcation diagram wherein θ1=0.25 and (28) is used as the investment function. This figure evidently indicates that alternations of stability and instability emerge with an increase in θ2.Figure 9
Bifurcation diagram (θ1=0.25).In particular, whenθ1=0.25, stability can be established by setting θ2∈(0,1.06), θ2∈(2.53,4.33), or θ2∈(6.71,7.61); that is, for equilibrium stability, monetary policy lag should be less than one year, between 3 and 4.5 years, or between 6.8 and 7.5 years when a fiscal policy lag is approximately three months.Next, we examine howβ and γ affect the stability condition. Both β and γ affect L1 and L2 through ω∈Ω. It is difficult to analytically show how L1 and L2 change. Hence, we numerically examine how curve m0 (which comprises L1(0,n) and L2(0,n)) shifts when β and γ increase from each baseline value (β=2.5 and γ=6.0).Figures10 and 11 describe curve m0 in cases with β=8 and β=15, respectively. We can observe that an increase in β shifts curve m0 to left. Accordingly, the stability region S reduces. Likewise, Figures 12 and 13 describe curve m0 in cases with γ=12 and γ=22, respectively. The stability region is also reduced by an increase in γ. These results suggest that increases in the activeness levels of fiscal and monetary policies increase economic instability, as is the case with only one policy lag.Figure 10
β
=
8.Figure 11
β
=
15.Figure 12
γ
=
12.Figure 13
γ
=
22.


## 4.3. Investment and Stability

In general, the formulation of the investment function has great significance to stability in macrodynamic models; that is, the larger the marginal propensity to investment is, the more likely it becomes unstable. In this subsection, we examine the effects of changes in the partial derivatives of the investment function,IY and IK, on stability. We compare three cases: the case of no policy lags, the case of the existence of only monetary policy lag, and the case of the existence of both fiscal and monetary policy lags.In the case with no policy lags,IY and IK affect the stability condition given by Assumption 1 through b1. The smaller the values of IY and IK are (the larger the absolute value of IK is), the more likely the stability condition is satisfied. These are fairly typical results.In the case of a monetary policy with a lag,IY and IK affect the value of ω∗ through b2 (IK also directly affects ω∗). Accordingly, the value of (27) changes. In this case, it is difficult to analytically show the quality and quantity of these effects. Hence, we resort to a numerical method. Based on the parameter values proposed in Section 3.2, crossing curves θ20 can be described within each range of IY and IK that ensure the existence of ω∗, as shown in Figure 14. The region located below these curves indicates the stability regions. Similar to the case with no policy lags, an increase in IY causes destabilization. However, due to the concavity of the crossing curve, there is a possibility that a decrease in IK causes destabilization. Thus, the existence of a lag complicates the effects of investment on stability.Figure 14
Crossing curveθ20.Finally, we investigate the case wherein both fiscal and monetary policy lags exist. In this case,IY and IK affect the stability condition (i.e., location of curve M0) through ω∈Ω, b1, and b2. According to numerical simulations, increases in the absolute values of IY and IK generally shift the curve M0 toward the left. Therefore, these changes decrease the stability region. However, if M0 completely shifts even to the negative region of θ1, then the curve that specifies the stability region is no longer M0 but is M1. For example, when IY=3.0, the stability region S is defined as the left-hand side region of the curve M0 in Figure 15. However, when IY=4.0, the stability region S is defined as the left-hand side region of the curve M1 in Figure 16. Thus, an increase in IY may increase the stability region.Figure 15
I
Y
=
3.0.Figure 16
I
Y
=
4.0.


## 5. Conclusion

In this study, we analyzed the effects of fiscal and monetary policy lags on economic stability. In Section3, we examined the case of a monetary policy with a lag. We confirmed that the results of Asada and Yoshida [12] that examined the case of a fiscal policy with a lag also basically hold in our model; that is, policy lags may cause economic instability and suppress the effect of expeditious policies aimed at economic stability (a model with a fiscal policy that does not consider a policy lag is proposed by Asada [21]).Moreover, in Section4, we examined the case wherein both fiscal and monetary policy lags exist and demonstrated that, in the neighborhood of a stability crossing curve, a monetary policy lag can have a stabilizing effect.According to Friedman [22], policy lags can generally be classified into three types: recognition, implementation, and diffusion lags. Unlike recognition and diffusion lags, implementation lag can be considered as purposefully 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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