Research ArticleDynamic Analysis of Two Policy Lags in a Kaldorian Model

Eiji Tsuzuki

Faculty of Economics Chiba Keizai University Chiba 263-0021 Japan

Correspondence should be addressed to Eiji Tsuzuki tsuzukie5gmailcom

Received 23 May 2015 Revised 11 August 2015 Accepted 13 August 2015

Academic Editor Mauro Sodini

Copyright copy 2015 Eiji Tsuzuki This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We examine the effects of policy lags on local economic stability using a Kaldorian model This study analyzes two cases the caseof 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 policylags examined in a previous study monetary policy lags have destabilizing effects on economic stability However in the case of theexistence of both fiscal and monetary policy lags there is a possibility that a monetary policy lag can stabilize an economy

1 Introduction

In this study we investigate the effects of policy lags (ie fiscaland monetary policy lags) on the stability of an economy Asthe analytical framework we employ a traditional Keynesianmacrodynamic model In general two types of time lagshave been studied distributed and fixed lags The presentstudy focuses on the fixed lag (the time-to-build modeldeveloped by Kalecki [1] is the origin of economic modelsthat consider a fixed lag Szydłowski [2 3] develops Kaleckirsquos[1] model to incorporate the factors of economic growtheconomic models that introduce a distributed lag includeFanti and Manfredirsquos [4] dynamic IS-LM model and Yoshidaand Asadarsquos [5] Keynes-Goodwin model Yoshida and Asadaalso examine the case of a fixed policy lag)

One of the representative traditional Keynesian macro-dynamic models is the dynamic IS-LM model developed bySchinasi [6] and Sasakura [7] In a previous study De Cesareand Sportelli [8] develop a dynamic IS-LM model with a taxcollection lag Furthermore Matsumoto and Szidarovszky[9] use the dynamic IS-LM model to compare the case of afixed lag with that of a distributed lag in tax collections anddemonstrate that a larger stable region can be established inthe case of a fixed lag compared with a distributed lag Bothof these studies evidently show that a policy lag restricts theability 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 representativemodel of the traditional Keynesian macrodynamic model

Asada and Yoshida [12] introduce a fiscal policy lag into thismodel and demonstrate that an increase in the responsivenessof a fiscal policy can lead to local instability as is the case withthe dynamic IS-LM model

Recently economic models that consider two time lagshave been developed For example Zhou and Li [13] proposea dynamic IS-LM model with two capital accumulationlags (Zhou and Lirsquos [13] model can be considered as anextension of Cairsquos [14] model which is a dynamic IS-LMmodel with only one capital accumulation lag) Sportelli etal [15] present a similar model with two time lags in thepublic sector government expenditure and tax collectionlags Moreover Matsumoto and Szidarovszky [16] proposea nonlinear multiplier-accelerator model with investmentand consumption lags These studies demonstrate that thesteady state fluctuates between stability and instability withan increase in a certain policy lag that is multiple stabilityswitches can occur

In the present study we first propose a Kaldorian modelthat only introduces the monetary policy lag which canbe considered as the monetary policy lag version of Asadaand Yoshidarsquos [12] model and perform a local stabilityanalysis Furthermore we propose a model that considersthe existence of both fiscal and monetary policy lags For theanalysis of this model we employ the mathematical methoddeveloped by Gu et al [17] Their method enables us torepresent a stability crossing curve (a curve that separatesstable and unstable regions on a lag parameter plane) as aperfectly parameterized form Before the advent of Gu et alrsquos

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 927138 12 pageshttpdxdoiorg1011552015927138

2 Discrete Dynamics in Nature and Society

[17] method one could only represent one lag as a function ofanother lag Using their method we can present a more exactfigure of a stability crossing curve Few studies have employedthis method for economic analysis

This study proceeds as follows Section 2 presents adynamic system that represents a model economy whileSection 3 examines the local stability in the case of amonetarypolicy with a lag Subsequently Section 4 examines the casewherein both fiscal and monetary policy lags exist FinallySection 5 presents our conclusion

2 The Model

21 Basic Equations The model economy comprises the fol-lowing equations

119884 (119905) = 120572 [119862 (119905) + 119868 (119905) + 119866 (119905) minus 119884 (119905)] 120572 gt 0 (1)

119862 (119905) = 119888 [119884 (119905) minus 119879 (119905)] + 119862 0 lt 119888 lt 1 119862 gt 0 (2)

119879 (119905) = 120591119884 (119905) minus 119879 0 lt 120591 lt 1 119879 ge 0 (3)

119868 (119905) = 119868 (119884 (119905) 119870 (119905) 119903 (119905))

119868

119884gt 0 119868

119870lt 0 119868

119903lt 0

(4)

119866 (119905) = 120573 [119884 minus 119884 (119905 minus 1205791)] + 119866

120573 gt 0 119884 gt 0 119866 gt 0

(5)

119870 (119905) = 119868 (119884 (119905) 119870 (119905) 119903 (119905)) (6)

119872(119905)

119875

= 119871 (119884 (119905) 119903 (119905)) 119875 gt 0 119871

119884gt 0 119871

119903lt 0 (7)

119872(119905) = 119875120574 [119884 minus 119884 (119905 minus 1205792)] + 119872 120574 gt 0 119872 gt 0 (8)

where 119884 = real national income 119862 = real private con-sumption 119868 = real private investment 119866 = real governmentexpenditure 119879 = real income tax 119870 = real capital stock119903 = nominal interest rate 119872 = nominal money supply120572 = adjustment speed of the goods market 119888 = marginalpropensity to consume 119862 = base consumption 120591 = marginaltax rate 119879 = real subsidy 120573 = activeness level of fiscal policy119884 = target level of real national income 119866 = target level ofreal government expenditure 119875 = price level 120574 = activenesslevel of monetary policy119872 = target level of nominal moneysupply 119905 = time 120579

1= fiscal policy lag and 120579

2= monetary

policy lagEquations (1) (2) (3) and (4) represent a disequilibrium

adjustment function of the goods market a consumptionfunction a tax collection function and an investment func-tion respectively Equation (5) represents a fiscal policyreaction function with a government expenditure lag while(6) signifies a capital accumulation function For simplic-ity we assume that capital does not depreciate Equation(7) represents the monetary market equilibrium conditionwhere the left-hand side denotes real money balance and theright-hand side indicates real money demand (in this studywe assume that the price level 119875 is constant by assuming

119875(119905) = 119875(119884(119905)) (where 119875119884gt 0) we can allow for variations

in the price level however this change does not affect thenature of our argument) Here we ensure that the adjustmentof the monetary market is rapid and therefore the balanceof demand and supply of this market is always maintainedFinally (8) represents a monetary policy reaction functionwith a money supply lag

If real money balance is constant (ie 120574 = 0) then thesystem compounded from (1)ndash(8) essentially becomes similarto that of Asada and Yoshida [12] However the assumptionsof 120574 gt 0 and 120579

2gt 0 significantly complicate the dynamic

property of the system thereby resulting in a major changein the economic implication of time lags

22 Summarization We summarize (1)ndash(8) in a two-dimen-sional dynamic system First by substituting (8) into (7) andsolving for 119903 we obtain

119903 (119905) = 119903 (119884 (119905) 119884 (119905 minus 1205792)) (9)

where 119903119884(119905)

= minus119871

119884119871

119903gt 0 and 119903

119884(119905minus1205792)= minus120574119871

119903gt 0

Furthermore by substituting (3) into (2) and substituting(9) into (4) we obtain

119862 (119905) = 119888 (1 minus 120591) 119884 (119905) + 119888119879 + 119862(10)

119868 (119905) = 119868 (119884 (119905) 119870 (119905) 119903 (119884 (119905) 119884 (119905 minus 1205792))) (11)

Finally by substituting (5) (10) and (11) into (1) andsubstituting (11) into (6) we obtain the following system ofdifferential equations with two time delays

119884 (119905) = 120572 [119868 (119884 (119905) 119870 (119905) 119903 (119884 (119905) 119884 (119905 minus 1205792)))

minus 1 minus 119888 (1 minus 120591) 119884 (119905) minus 120573119884 (119905 minus 1205791) + 119888119879 + 119862 + 120573119884

+ 119866]

119870 (119905) = 119868 (119884 (119905) 119870 (119905) 119903 (119884 (119905) 119884 (119905 minus 1205792)))

(12)

23 Linearization We assume that steady-state solutions119884

lowastgt 0 and 119870lowast gt 0 that satisfy

119884(119905) =

119870(119905) = 0 exist andthese solutions are continuous with respect to all parametersIn order to analyze the local stability around the steady-statepoint (119884lowast 119870lowast) we linearize the system and obtain

119884 (119905) = 120572 [1198601minus 1 + 119888 (1 minus 120591)

119884 (119905) minus 120573

119884 (119905 minus 120579

1)

+ 119860

2

119884 (119905 minus 120579

2) + 119868

119870

119870 (119905)]

119870 (119905) = 1198601

119884 (119905) + 1198602

119884 (119905 minus 120579

2) + 119868

119870

119870 (119905)

(13)

where 119884(119905) = 119884(119905) minus 119884lowast 119870(119905) = 119870(119905) minus 119870lowast 1198601= 119868

119884+ 119868

119903119903

119884(119905)

and 1198602= 119868

119903119903

119884(119905minus1205792)

Assuming the exponential functions 119884(119905) = 119862

1119890

120582119905 and

119870(119905) = 119862

2119890

120582119905 where 1198621and 119862

2are arbitrary constants and

Discrete Dynamics in Nature and Society 3

120582 denotes the eigenvalue as the solutions of system (13) andsubstituting these into the system we obtain

[

120582 minus 120572 119860

1minus 1 + 119888 (1 minus 120591) + 120572120573119890

minus1205791120582minus 120572119860

2119890

minus1205792120582

minus120572119868

119870

minus119860

1minus 119860

2119890

minus1205792120582

120582 minus 119868

119870

][

119884 (119905)

119870 (119905)

] = [

0

0

] (14)

For nontrivial solutions to exist the determinant of theleft-hand sidematrix denoted byΔ(120582) must be equal to zerothat is

Δ (120582) = 120582

2minus [119868

119870+ 120572 119860

1minus 1 + 119888 (1 minus 120591)] 120582

minus 120572 1 minus 119888 (1 minus 120591) 119868119870+ 120572120573 (120582 minus 119868

119870) 119890

minus1205791120582

minus 120572119860

2120582119890

minus1205792120582= 0

= 119901

0 (120582) + 1199011 (

120582) 119890

minus1205791120582+ 119901

2 (120582) 119890

minus1205792120582= 0

(15)

where

119901

0 (120582) = 120582

2+ 119887

1120582 + 119887

2

119887

1= minus [119868

119870+ 120572 119860

1minus 1 + 119888 (1 minus 120591)]

119887

2= minus120572 1 minus 119888 (1 minus 120591) 119868119870

gt 0

119901

1 (120582) = 120572120573 (120582 minus 119868119870

)

119901

2 (120582) = minus1205721198602

120582

(16)

Equation (15) is a characteristic equation of system (13) Thesignificant feature of this equation is found in the existence ofthe exponential terms 119890minus1205791120582 and 119890minus1205792120582

First we examine the case of no policy lags By substitut-ing 1205791= 120579

2= 0 into (15) we obtain

120582

2+ [119887

1+ 120572 (120573 minus 119860

2)] 120582 + 119887

2minus 120572120573119868

119870= 0 (17)

which is an ordinary quadratic equation of 120582Thus if 119887

1+120572(120573minus119860

2) gt 0 (ie the coefficient of120582 from (17)

is positive) the real parts of the roots of (17) are negative (seeChapter 18 in Gandolfo [18] for details regarding the relationbetween the roots and the coefficients of a quadric equation)In contrast if 119887

1+ 120572(120573 minus 119860

2) lt 0 the real parts of the roots

are positive Thus if 1198871+ 120572(120573 minus 119860

2) gt 0 then the steady state

is locally stable and if 1198871+ 120572(120573 minus 119860

2) lt 0 then it is unstable

The larger the values of 120573 and 120574 are the more likely thestability condition holds (note that119860

2= minus120574119868

119903119871

119903)Therefore

if policy lags do not exist then traditional fiscal andmonetarypolicies could function effectively for stabilization

In the following discussion we assume that if both fiscaland monetary policy lags do not exist then an economy isstable that is we have the following

Assumption 1 We have

120573 +

119868

119903

119871

119903

120574 +

119887

1

120572

gt 0 (18)

Under this assumption we analyze the effects of the policylags (120579

1 120579

2) on stability

Incidentally if [1198871+ 120572(120573 minus 119860

2)]

2lt 4(119887

2minus 120572120573119868

119870) then

both roots of (17) are conjugate complex Furthermore bydefining 120573lowast equiv 119860

2minus 119887

1120572 the conditions for Hopf bifurcation

are satisfied at 120573 = 120573

lowast that is (i) (17) includes a pair of pureimaginary roots and (ii) 119889Re120582119889120573 = 0 Therefore a cycleexists for a certain range of 120573 in the neighborhood 120573lowast Wecan develop a similar argument using the monetary policyparameter 120574

3 Case of Monetary Policy with a Lag

Asada and Yoshida [12] examine the situation where a timelag exists in fiscal policy responses which is equivalent to thecase wherein 120579

1gt 0 and 120579

2= 0 in system (12) In this section

we analyze the situation where a time lag exists in monetarypolicy responses by assuming that 120579

1= 0 and 120579

2gt 0 in system

(12)In this case the characteristic equation in (15) can be

rewritten as follows

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870) minus 120572119860

2120582119890

minus1205792120582= 0

(19)

The following analysis is based on the study ofMatsumotoand Szidarovszky [19] (Bellman and Cooke [20] provide ahelpful introductory textbook of delay differential equations)The procedures of the analysis are given as follows

(1) We characterize the points (if any) at which thedynamics around the steady state can change that isthe points at which a zero real root or pure imaginaryroots appear These points are referred to as ldquocrossingpointsrdquo

(2) We reveal the directions of the changes in the signs ofthe real parts that occur when 120579

2crosses the crossing

points

31 Crossing Points First we examine whether the points atwhich a zero real root appears can exist By substituting 120582 = 0into (19) we obtain 119887

2minus 120572120573119868

119870= 0 However since 119887

2gt 0 and

119868

119870lt 0 this equality cannot hold Accordingly 120582 = 0 cannot

be a rootNext we examine whether the points at which the pure

imaginary roots appear can exist By substituting 120582 = 119894120596where 120596 = imaginary part gt 0 (pure imaginary roots arealways conjugated therefore we can assume 120596 gt 0 withoutloss of generality) and 119894 = radicminus1 into (19) we obtain

minus120596

2+ 119894120596 (119887

1+ 120572120573) + (119887

2minus 120572120573119868

119870) minus 119894120596120572119860

2119890

minus1198941205961205792= 0

(20)

4 Discrete Dynamics in Nature and Society

Application of Eulerrsquos formula (forall119909 isin R 119890

plusmn119894119909= cos119909 plusmn 119894 sin119909)

to this equation yields

minus 120596

2+ (119887

2minus 120572120573119868

119870) minus 120596120572119860

2sin120596120579

2

+ 119894 [(119887

1+ 120572120573)120596 minus 120596120572119860

2cos120596120579

2] = 0

(21)

This equality holds only when both real and imaginary partsof the left-hand side equal zero that is

minus120596

2+ (119887

2minus 120572120573119868

119870) = 120596120572119860

2sin120596120579

2

(119887

1+ 120572120573)120596 = 120596120572119860

2cos120596120579

2

(22)

Or equivalently

cos1205961205792=

119887

1+ 120572120573

120572119860

2

(23)

sin1205961205792=

minus120596

2+ 119887

2minus 120572120573119868

119870

120572119860

2120596

(24)

For values of 1205961205792that satisfy these equations (19) includes

pure imaginary roots where 2120587ℎ lt 120596120579

2lt 2120587(1 + ℎ) ℎ =

0 1 2 3 Furthermore the sum of the squares of (22) yields the

following

120596

4minus 2 (119887

2minus 120572120573119868

119870) minus (119887

1+ 120572120573)

2+ (120572119860

2)

2 120596

2

+ (119887

2minus 120572120573119868

119870)

2= 0

(25)

Solving for 1205962 we obtain

120596

2

plusmn=

2 (119887

2minus 120572120573119868

119870) minus (119887

1+ 120572120573)

2+ (120572119860

2)

2plusmnradic119873

2

(26)

where119873 equiv 2(119887

2minus120572120573119868

119870)minus(119887

1+120572120573)

2+(120572119860

2)

2

2minus4(119887

2minus120572120573119868

119870)

2We denote real and positive values of 120596 by 120596

lowast Bysubstituting 120596lowast into (23) and solving for 120579

2 we obtain

120579

2ℎ=

cosminus1 ((1198871+ 120572120573) 120572119860

2) + 2120587ℎ

120596

lowast

ℎ = 0 1 2 3

(27)

Thus there exists an infinite number of values of 1205792(ie

120579

20 120579

21 120579

22 ) that generate pure imaginary roots (ie

crossing points)Finally if a real and positive value of120596 does not exist then

changes in the signs of the real parts of the roots do not occurTherefore in this case a policy lag does not affect stability

Direction of Crossing Here we reveal how the signs of thecomplex roots change when 120579

2crosses 120579

2ℎThis is determined

by the sign of 119889Re1205821198891205792|

120582=119894120596lowast If 119889Re120582119889120579

2|

120582=119894120596lowast gt 0

then the signs of the real parts of the roots change fromnegative to positive with an increase in 120579

2(which indicates

destabilization) In contrast if 119889Re1205821198891205792|

120582=119894120596lowast lt 0 then the

signs of the real parts of the roots change from positive tonegative with an increase in 120579

2(which indicates stabilization)

Table 1 Baseline parameters

120572 119888 120591 119862 + 119888119879 + 119866 119884

09 0625 02 200 400

For convenience of calculation we will observe the sign ofRe(119889120582119889120579

2)

minus1|

120582=119894120596lowast instead of that of 119889Re120582119889120579

2|

120582=119894120596lowast

Using (19) we can demonstrate that Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt

0 holds for any value of ℎ (see the appendix) Thus with anycrossing the signs of the real parts of the roots change fromnegative to positive

If a policy lag does not present (1205792= 0) then roots with

positive real parts do not exist under Assumption 1 Withincreases in 120579

2 the number of roots with positive real parts

increases by two Accordingly we can state that 2ℎ roots withpositive real parts exist in the region of 120579

2isin (120579

2(ℎminus1) 120579

2ℎ)

(where 1205792(minus1)

= 0) Therefore we can offer the followingproposition

Proposition 2 For 1205792isin (0 120579

20) the steady state is stable and

for 1205792isin (120579

20infin) it is unstable

This proposition suggests that even if a monetary policyis sufficiently active enough to satisfy Assumption 1 aneconomy can become unstable due to the existence of a policylag A similar result has already been shown by Asada andYoshida [12] concerning fiscal policies

32 Numerical Simulations Let us consider a numericalexample Following the study of Asada and Yoshida [12] weset baseline parameters as shown in Table 1

Furthermore Asada and Yoshida [12] employ the invest-ment function that is highly nonlinear with respect to 119884(119905)which is formulated as follows (in their model 120574 = 0)

119868 =

119884

1 + 12119890

minus01(119884(119905)minus119884)minus 119903 minus 05119870 (119905) minus

119884

13

+ 02 + 30

119903 = minus005120574 (119884 minus 119884 (119905 minus 120579

2)) + 001

radic119884 (119905)

(28)

Based on these assumptions the steady-state values areprovided by (119884lowast 119870lowast) = (400 60)

We also set the baseline values of 120573 and 120574 as 25 and60 respectively (generally it seems that the responsivenessof monetary policies to economic fluctuations is often largerthan that of fiscal policies therefore we assume 120574 gt 120573)When 120574 is fixed at the baseline value 120596lowast exists only for 120573 isin

(149 208) Outside this range a policy lag does not affectstability In addition Assumption 1 is satisfied for 120573 gt 149Accordingly irrespective of the value of a monetary policylag the steady state is unstable for 0 lt 120573 lt 149 and it isstable for 120573 gt 208

Likewise when 120573 is fixed at the baseline value 120596lowast existsfor 120574 gt 1432 In addition Assumption 1 is satisfied for 120574 gtminus1432 Hence the steady state is stable for 0 lt 120574 lt 1432irrespective of the value of a lag

Figure 1 describes 1205792ℎ

(ℎ = 0 1 2) that corresponds to120596

lowast as a function of 120573 and 120574 These curves are referred to as

Discrete Dynamics in Nature and Society 5

16 18 20

5

10

15

16 18 20 220

2

4

6

8

10

12

14

120573 120574

1205792

1205792

12057922

12057921

12057920

12057922

12057921

12057920

Figure 1 Crossing curves

14 145 15 155 16 165 17 175 18250

300

350

400

450

500

Y

120574

Figure 2 Bifurcation diagram (case with (28))

ldquocrossing curvesrdquo Here the stability condition is expressedas 1205792lt 120579

20 Therefore we can state that an increase in 120573 has a

stabilizing effect whereas that in 120574 has a destabilizing effectThus an increase in the activeness level of a monetary

policymay cause instability due to the existence of amonetarypolicy lag A similar result concerning a fiscal policy hasalready been shown by Asada and Yoshida [12]

Incidentally conditions for Hopf bifurcation are satisfiedat 1205792= 120579

20 that is (i) (19) includes a pair of pure imaginary

roots and (ii) 119889Re1205821198891205792

= 0 Therefore when 120579

2crosses

120579

20 a cycle emerges A similar argument can be developed

using 120573 and 120574 We now suppose that 1205792= 20 In this case

a bifurcation diagram can be described for 120574 isin [14 18] asshown in Figure 2 (the vertical axis represents the maximal

15 155 16 165 17250

300

350

400

450

500

Y

120574

Figure 3 Bifurcation diagram (case with (29))

and minimal values of 119884(119905)) This figure shows that not onlyone period but also multiple period cycles can emerge Theemergence ofmultiple period cycles is attributed to the strongnonlinearity of the investment function with respect to 119884(119905)When we use the investment function that has relativelysimpler form with respect to 119884(119905) as below (we set thecoefficient of radic119884(119905) as 1136 so that the value of 119868

119884equals

that of (28)) the bifurcation diagram becomes as shown inFigure 3 In this case only one period cycle exists Hence

119868 = 1136radic119884 (119905) minus 119903 minus 05119870 (119905) minus 1136

radic

119884 + 02

+ 30

(29)

6 Discrete Dynamics in Nature and Society

Although Asada and Yoshida [12] do not clearly specifythe time dimension similar models that include a disequi-librium adjustment function of the goods market (120572 is setaround unity) specify the measure of 119905 as a year (refer to egYoshida and Asada [5]) Therefore we will consider the timedimension as a year For example when 120573 = 18 the steadystate is unstable for 120579

2gt 14 years

4 Case of Policies with Fiscal andMonetary Policy Lags

In this section we investigate the case wherein both fiscal andmonetary policy lags exist (ie 120579

1gt 0 and 120579

2gt 0) In order to

examine the signs of the roots of (15) we use a mathematicalmethod developed by Gu et al [17]

41 Preconditions First to apply the technique of Gu et al[17] some preconditions should be examined According totheir study (15) should satisfy the following conditions

(I) deg(1199010(120582)) ge maxdeg(119901

1(120582)) deg(119901

2(120582))

(II) Δ(0) = 0(III) A solution common to all three polynomials 119901

0(120582) =

0 1199011(120582) = 0 and 119901

2(120582) = 0 does not exist

(IV) lim120582rarrinfin

(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) lt 1

Condition (I) is satisfied by 2 gt max1 1 while Condi-tion (II) is also satisfied by Δ(0) = minus120572119868

1198701 minus 119888(1 minus 120591) + 120573 gt 0

Condition (III) is obviously satisfied Finally Condition (IV)is satisfied by lim

120582rarrinfin(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) = 0

Now we examine the effects of lags (1205791 120579

2) on the stability

of the steady state The analysis proceeds as follows

(1) We characterize the sets (1205791 120579

2) (ie crossing sets) at

which the pure imaginary roots appear (it is ensuredfrom precondition (II) that zero cannot be a root)

(2) We depict the crossing sets on the (120579

1 120579

2) plane

(which we refer to as the crossing curves) via numer-ical simulation

(3) We reveal the directions of changes in the signs ofthe real parts that occur when lags (120579

1 120579

2) cross the

crossing curves

42 Crossing Sets Dividing (15) by 1199010(120582) we obtain

1 + 119886

1 (120582) 119890

minus1205791120582+ 119886

2 (120582) 119890

minus1205792120582= 0

(30)

where

119886

1 (120582) =

119901

1 (120582)

119901

0 (120582)

=

120572120573 (120582 minus 119868

119870)

120582

2+ 119887

1120582 + 119887

2

119886

2 (120582) =

119901

2 (120582)

119901

0 (120582)

= minus

120572119860

2120582

120582

2+ 119887

1120582 + 119887

2

(31)

Denoting a pure imaginary root by 120582 = 119894120596 the values of120596that satisfy (30) can be characterized by the following lemma

0

Im

1

a2(i120596)eminusi1205961205792

1205752 1205751

Re

a1(i120596)eminusi 1120596120579

Figure 4 Triangle formed by 1 |1198861(119894120596)| and |119886

2(119894120596)| on the complex

plane

Lemma 3 (see [17 Proposition 31]) For each 120596 satisfying119901

0(119894120596) = 0 120582 = 119894120596 is a solution of Δ(120582) = 0 for some

(120579

1 120579

2) isin R2+if and only if

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

ge 1

minus1 le

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

le 1

(32)

We denote the set of 120596 gt 0 that satisfy conditions (32)as Ω which is termed the ldquocrossing frequencyrdquo For all 120596 isin

Ω the sets (1205791 120579

2) satisfying (30) (ie crossing sets) should

satisfy the following conditions (Figure 4)

∓120575

1= arg (119886

1 (119894120596) 119890

minus1198941205961205791) + 2119898120587 119898 = 0 1 2

plusmn120575

2= arg (119886

2 (119894120596) 119890

minus1198941205961205792) + 2119899120587 119899 = 0 1 2

(33)

where 1205751 120575

2isin [0 120587]

Incidentally on the complex plane multiplication ofamplitudes becomes a sum of the parts therefore we obtain

arg (1198861 (119894120596) 119890

minus1198941205961205791) = arg (119886

1 (119894120596)) minus 1205961205791

arg (1198862 (119894120596) 119890

minus1198941205961205792) = arg (119886

2 (119894120596)) minus 1205961205792

(34)

In addition Figure 4 demonstrates that the followingrelations hold

arg (1198861 (119894120596)) = tanminus1 (

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

)

arg (1198862 (119894120596)) = tanminus1 (

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

)

(35)

Moreover after some manipulation (31) derive the fol-lowing expressions

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

=

119887

1120596119868

119870+ 120596 (119887

2minus 120596

2)

119887

1120596

2minus 119868

119870(119887

2minus 120596

2)

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

=

119887

2minus 120596

2

119887

1120596

(36)

Discrete Dynamics in Nature and Society 7

Thus using (34) to (36) (33) can be rewritten as follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) plusmn 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) ∓ 120575

2+ 2119899120587

120596

(37)

where the interior angles of the triangle denoted by 1205751and

120575

2are given by the cosine theorem as follows

120575

1= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572120573119868

119870)

2+ (120572120573120596)

2minus (120572119860

2120596)

2

2

radic

(120572120573119868

119870)

2+ (120572120573120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

120575

2= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572119860

2120596)

2minus (120572120573119868

119870)

2minus (120572120573120596)

2

2

radic

(120572119860

2120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

(38)

Equations (37) characterize the crossing sets (1205791 120579

2) isin

R2+ Depending on the signs of 120575

1and 120575

2 we can define two

types of crossing sets denoted by 1198711(119898 119899) and 119871

2(119898 119899) as

follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) + 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) minus 120575

2+ 2119899120587

120596

119871

1(119898 119899)

120579

1=

tanminus1 ((1198871120596119868

119870+ V (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) minus 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) + 120575

2+ 2119899120587

120596

119871

2(119898 119899)

As in the previous section we set parameter values andthe functional form of the investment function as shownin Table 1 (28) or (29) 120573 = 25 and 120574 = 60 In thiscase crossing frequency Ω is provided by 120596 isin (150 213)

(Figure 5) (according to Gu et alrsquos [17] classification Figure 5exhibits Type 13) For 120596 isin Ω describing the crossing sets119871

1(119898 119899) and 119871

2(119898 119899) on the 120579

1-1205792plane we obtain Figure 6

where119898 = 0 1 2 and 119899 = 0 1 2 The dotted curves represent119871

1(119898 119899) and the solid curves represent 119871

2(119898 119899) The points

marked by a dot indicate the starting points of both curvesthat is the points corresponding to 120596 = 150

We establish how the real parts of the roots change whenlags cross the crossing curves with an increase in 120579

1 For 120596 isin

Ω if Re(1198891205821198891205791)

minus1gt 0 then the real parts of the roots change

from negative to positive this indicates destabilization Incontrast if Re(119889120582119889120579

1)

minus1lt 0 then the real parts of the roots

change from positive to negative this indicates stabilizationDifferentiating (30) with respect to 120579

1 we obtain

[119886

1015840

1(120582) 119890

minus1205791120582minus 119886

1 (120582) 119890

minus1205791120582120579

1+ 119886

1015840

2(120582) 119890

minus1205792120582

minus 119886

2 (120582) 119890

minus1205792120579

2]

119889120582

119889120579

1

= 119886

1 (120582) 119890

minus1205791120582120582

(39)

8 Discrete Dynamics in Nature and Society

0 2 4 6 8 10

0

1

2

3

4

5

6

minus1

|a1(i120596)| + |a2(i120596)|

120596

Ω|a1(i120596)| minus |a2(i120596)|

Figure 5 Crossing frequency Ω

or equivalently

(

119889120582

119889120579

1

)

minus1

=

119886

1015840

1(120582) 119890

minus1205791120582+ 119886

1015840

2(120582) 119890

minus1205792120582minus 119886

2 (120582) 119890

minus1205792120582120579

2

119886

1 (120582) 119890

minus1205791120582120582

minus

120579

1

120582

(40)

where

119886

1015840

1(120582) =

120572120573119901

0 (120582) minus 120572120573 (120582 minus 119868119870

) (2120582 + 119887

1)

119901

0 (120582)

2

119886

1015840

2(120582) = minus

120572119860

2119901

0 (120582) minus 1205721198602

120582 (2120582 + 119887

1)

119901

0 (120582)

2

(41)

Describing the real parts of (40) as a function of 120596 isin Ωwe obtain Figure 7 where the dotted curves are the functionsevaluated on curve 119871

1and the solid curves are the functions

evaluated on curve 1198712 As for curve 119871

1 stabilization only

occurs on the intermediate region of the segments 1198711(119898 2)

destabilization occurs on the other segments of 1198711 As for

curve 1198712 destabilization occurs on all segments Here we

define a curve formed by connecting curves 119871119895(0 119899) (where

119895 = 1 2 and 119899 = 0 1 2) as 1198980 (Figure 8) We also definea curve formed by connecting the points on curve 1198980 thatminimize the value of 120579

1for a given 120579

2as curve 1198720 Then

we can express the stability condition as follows set (1205791 120579

2)

exists in the left-hand side region of curve1198720On curve 1198720 conditions for Hopf bifurcation are sat-

isfied and therefore a cycle emerges In addition as isobvious from Figure 6 an increase in the fiscal policy lag 120579

1

necessarily destabilizes the steady state However an increasein the monetary policy lag 120579

2has the potential to stabilize

the steady state which depends on the value of 1205791 When

120579

1isin (022 036) we can see that the steady state fluctuates

between stability and instability with an increase in 1205792 Such

a phenomenon cannot be observed in models with only onetime lag Moreover Figure 9 presents a bifurcation diagram

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

L2(0 2)

L1(0 2)

L2(0 1)

L1(0 1)

L2(0 0)

L1(0 0)

L2(1 0)

L1(1 0) L1(2 0)

L2(2 0)

L2(2 1)

L1(2 1)

L2(2 2)

L1(2 2)

L2(1 2)

L1(1 2)

L2(1 1)

L1(1 1)

1205792

1205791

Figure 6 Crossing curves

wherein 1205791= 025 and (28) is used as the investment function

This figure evidently indicates that alternations of stabilityand instability emerge with an increase in 120579

2

In particular when 1205791= 025 stability can be established

by setting 1205792isin (0 106) 120579

2isin (253 433) or 120579

2isin (671 761)

that is for equilibrium stability monetary policy lag shouldbe less than one year between 3 and 45 years or between 68and 75 years when a fiscal policy lag is approximately threemonths

Next we examine how 120573 and 120574 affect the stabilitycondition Both 120573 and 120574 affect 119871

1and 119871

2through 120596 isin Ω It is

difficult to analytically show how 119871

1and 119871

2change Hence

we numerically examine how curve 1198980 (which comprises119871

1(0 119899) and 119871

2(0 119899)) shifts when 120573 and 120574 increase from each

baseline value (120573 = 25 and 120574 = 60)Figures 10 and 11 describe curve 1198980 in cases with 120573 = 8

and 120573 = 15 respectively We can observe that an increasein 120573 shifts curve 1198980 to left Accordingly the stability region119878 reduces Likewise Figures 12 and 13 describe curve 1198980 incaseswith 120574 = 12 and 120574 = 22 respectivelyThe stability regionis also reduced by an increase in 120574 These results suggestthat increases in the activeness levels of fiscal and monetarypolicies increase economic instability as is the case with onlyone policy lag

43 Investment and Stability In general the formulation ofthe investment function has great significance to stabilityin macrodynamic models that is the larger the marginalpropensity to investment is the more likely it becomesunstable In this subsection we examine the effects of changesin the partial derivatives of the investment function 119868

119884and

119868

119870 on stability We compare three cases the case of no policy

lags the case of the existence of onlymonetary policy lag andthe case of the existence of both fiscal and monetary policylags

In the casewith no policy lags 119868119884and 119868119870affect the stability

condition given by Assumption 1 through 1198871 The smaller the

values of 119868119884and 119868119870are (the larger the absolute value of 119868

119870is)

Discrete Dynamics in Nature and Society 9

L1(0 0) and L2(0 0) L1(0 1) and L2(0 1) L1(0 2) and L2(0 2)

L1(1 0) and L2(1 0) L1(1 1) and L2(1 1) L1(1 2) and L2(1 2)

L1(2 0) and L2(2 0) L1(2 1) and L2(2 1) L1(2 2) and L2(2 2)

0 1 2 30

02

04Re

(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

Figure 7 Direction of crossing

0 01 02 03 04 050

1

2

3

4

5

6

7

8

9

S1205792

1205791

Figure 8 Stability region 119878

the more likely the stability condition is satisfied These arefairly typical results

In the case of a monetary policy with a lag 119868119884and 119868119870

affect the value of 120596lowast through 119887

2(119868119870

also directly affects120596

lowast) Accordingly the value of (27) changes In this case itis difficult to analytically show the quality and quantity of

1 2 3 4 5 6 7 8 9350

360

370

380

390

400

410

420

430

440

450

Y

1205792

Figure 9 Bifurcation diagram (1205791= 025)

these effects Hence we resort to a numerical method Basedon the parameter values proposed in Section 32 crossingcurves 120579

20can be described within each range of 119868

119884and

119868

119870that ensure the existence of 120596lowast as shown in Figure 14

The region located below these curves indicates the stabilityregions Similar to the case with no policy lags an increase

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

[17] method one could only represent one lag as a function ofanother lag Using their method we can present a more exactfigure of a stability crossing curve Few studies have employedthis method for economic analysis

This study proceeds as follows Section 2 presents adynamic system that represents a model economy whileSection 3 examines the local stability in the case of amonetarypolicy with a lag Subsequently Section 4 examines the casewherein both fiscal and monetary policy lags exist FinallySection 5 presents our conclusion

2 The Model

21 Basic Equations The model economy comprises the fol-lowing equations

119884 (119905) = 120572 [119862 (119905) + 119868 (119905) + 119866 (119905) minus 119884 (119905)] 120572 gt 0 (1)

119862 (119905) = 119888 [119884 (119905) minus 119879 (119905)] + 119862 0 lt 119888 lt 1 119862 gt 0 (2)

119879 (119905) = 120591119884 (119905) minus 119879 0 lt 120591 lt 1 119879 ge 0 (3)

119868 (119905) = 119868 (119884 (119905) 119870 (119905) 119903 (119905))

119868

119884gt 0 119868

119870lt 0 119868

119903lt 0

(4)

119866 (119905) = 120573 [119884 minus 119884 (119905 minus 1205791)] + 119866

120573 gt 0 119884 gt 0 119866 gt 0

(5)

119870 (119905) = 119868 (119884 (119905) 119870 (119905) 119903 (119905)) (6)

119872(119905)

119875

= 119871 (119884 (119905) 119903 (119905)) 119875 gt 0 119871

119884gt 0 119871

119903lt 0 (7)

119872(119905) = 119875120574 [119884 minus 119884 (119905 minus 1205792)] + 119872 120574 gt 0 119872 gt 0 (8)

where 119884 = real national income 119862 = real private con-sumption 119868 = real private investment 119866 = real governmentexpenditure 119879 = real income tax 119870 = real capital stock119903 = nominal interest rate 119872 = nominal money supply120572 = adjustment speed of the goods market 119888 = marginalpropensity to consume 119862 = base consumption 120591 = marginaltax rate 119879 = real subsidy 120573 = activeness level of fiscal policy119884 = target level of real national income 119866 = target level ofreal government expenditure 119875 = price level 120574 = activenesslevel of monetary policy119872 = target level of nominal moneysupply 119905 = time 120579

1= fiscal policy lag and 120579

2= monetary

policy lagEquations (1) (2) (3) and (4) represent a disequilibrium

adjustment function of the goods market a consumptionfunction a tax collection function and an investment func-tion respectively Equation (5) represents a fiscal policyreaction function with a government expenditure lag while(6) signifies a capital accumulation function For simplic-ity we assume that capital does not depreciate Equation(7) represents the monetary market equilibrium conditionwhere the left-hand side denotes real money balance and theright-hand side indicates real money demand (in this studywe assume that the price level 119875 is constant by assuming

119875(119905) = 119875(119884(119905)) (where 119875119884gt 0) we can allow for variations

in the price level however this change does not affect thenature of our argument) Here we ensure that the adjustmentof the monetary market is rapid and therefore the balanceof demand and supply of this market is always maintainedFinally (8) represents a monetary policy reaction functionwith a money supply lag

If real money balance is constant (ie 120574 = 0) then thesystem compounded from (1)ndash(8) essentially becomes similarto that of Asada and Yoshida [12] However the assumptionsof 120574 gt 0 and 120579

2gt 0 significantly complicate the dynamic

property of the system thereby resulting in a major changein the economic implication of time lags

22 Summarization We summarize (1)ndash(8) in a two-dimen-sional dynamic system First by substituting (8) into (7) andsolving for 119903 we obtain

119903 (119905) = 119903 (119884 (119905) 119884 (119905 minus 1205792)) (9)

where 119903119884(119905)

= minus119871

119884119871

119903gt 0 and 119903

119884(119905minus1205792)= minus120574119871

119903gt 0

Furthermore by substituting (3) into (2) and substituting(9) into (4) we obtain

119862 (119905) = 119888 (1 minus 120591) 119884 (119905) + 119888119879 + 119862(10)

119868 (119905) = 119868 (119884 (119905) 119870 (119905) 119903 (119884 (119905) 119884 (119905 minus 1205792))) (11)

Finally by substituting (5) (10) and (11) into (1) andsubstituting (11) into (6) we obtain the following system ofdifferential equations with two time delays

119884 (119905) = 120572 [119868 (119884 (119905) 119870 (119905) 119903 (119884 (119905) 119884 (119905 minus 1205792)))

minus 1 minus 119888 (1 minus 120591) 119884 (119905) minus 120573119884 (119905 minus 1205791) + 119888119879 + 119862 + 120573119884

+ 119866]

119870 (119905) = 119868 (119884 (119905) 119870 (119905) 119903 (119884 (119905) 119884 (119905 minus 1205792)))

(12)

23 Linearization We assume that steady-state solutions119884

lowastgt 0 and 119870lowast gt 0 that satisfy

119884(119905) =

119870(119905) = 0 exist andthese solutions are continuous with respect to all parametersIn order to analyze the local stability around the steady-statepoint (119884lowast 119870lowast) we linearize the system and obtain

119884 (119905) = 120572 [1198601minus 1 + 119888 (1 minus 120591)

119884 (119905) minus 120573

119884 (119905 minus 120579

1)

+ 119860

2

119884 (119905 minus 120579

2) + 119868

119870

119870 (119905)]

119870 (119905) = 1198601

119884 (119905) + 1198602

119884 (119905 minus 120579

2) + 119868

119870

119870 (119905)

(13)

where 119884(119905) = 119884(119905) minus 119884lowast 119870(119905) = 119870(119905) minus 119870lowast 1198601= 119868

119884+ 119868

119903119903

119884(119905)

and 1198602= 119868

119903119903

119884(119905minus1205792)

Assuming the exponential functions 119884(119905) = 119862

1119890

120582119905 and

119870(119905) = 119862

2119890

120582119905 where 1198621and 119862

2are arbitrary constants and

Discrete Dynamics in Nature and Society 3

120582 denotes the eigenvalue as the solutions of system (13) andsubstituting these into the system we obtain

[

120582 minus 120572 119860

1minus 1 + 119888 (1 minus 120591) + 120572120573119890

minus1205791120582minus 120572119860

2119890

minus1205792120582

minus120572119868

119870

minus119860

1minus 119860

2119890

minus1205792120582

120582 minus 119868

119870

][

119884 (119905)

119870 (119905)

] = [

0

0

] (14)

For nontrivial solutions to exist the determinant of theleft-hand sidematrix denoted byΔ(120582) must be equal to zerothat is

Δ (120582) = 120582

2minus [119868

119870+ 120572 119860

1minus 1 + 119888 (1 minus 120591)] 120582

minus 120572 1 minus 119888 (1 minus 120591) 119868119870+ 120572120573 (120582 minus 119868

119870) 119890

minus1205791120582

minus 120572119860

2120582119890

minus1205792120582= 0

= 119901

0 (120582) + 1199011 (

120582) 119890

minus1205791120582+ 119901

2 (120582) 119890

minus1205792120582= 0

(15)

where

119901

0 (120582) = 120582

2+ 119887

1120582 + 119887

2

119887

1= minus [119868

119870+ 120572 119860

1minus 1 + 119888 (1 minus 120591)]

119887

2= minus120572 1 minus 119888 (1 minus 120591) 119868119870

gt 0

119901

1 (120582) = 120572120573 (120582 minus 119868119870

)

119901

2 (120582) = minus1205721198602

120582

(16)

Equation (15) is a characteristic equation of system (13) Thesignificant feature of this equation is found in the existence ofthe exponential terms 119890minus1205791120582 and 119890minus1205792120582

First we examine the case of no policy lags By substitut-ing 1205791= 120579

2= 0 into (15) we obtain

120582

2+ [119887

1+ 120572 (120573 minus 119860

2)] 120582 + 119887

2minus 120572120573119868

119870= 0 (17)

which is an ordinary quadratic equation of 120582Thus if 119887

1+120572(120573minus119860

2) gt 0 (ie the coefficient of120582 from (17)

is positive) the real parts of the roots of (17) are negative (seeChapter 18 in Gandolfo [18] for details regarding the relationbetween the roots and the coefficients of a quadric equation)In contrast if 119887

1+ 120572(120573 minus 119860

2) lt 0 the real parts of the roots

are positive Thus if 1198871+ 120572(120573 minus 119860

2) gt 0 then the steady state

is locally stable and if 1198871+ 120572(120573 minus 119860

2) lt 0 then it is unstable

The larger the values of 120573 and 120574 are the more likely thestability condition holds (note that119860

2= minus120574119868

119903119871

119903)Therefore

if policy lags do not exist then traditional fiscal andmonetarypolicies could function effectively for stabilization

In the following discussion we assume that if both fiscaland monetary policy lags do not exist then an economy isstable that is we have the following

Assumption 1 We have

120573 +

119868

119903

119871

119903

120574 +

119887

1

120572

gt 0 (18)

Under this assumption we analyze the effects of the policylags (120579

1 120579

2) on stability

Incidentally if [1198871+ 120572(120573 minus 119860

2)]

2lt 4(119887

2minus 120572120573119868

119870) then

both roots of (17) are conjugate complex Furthermore bydefining 120573lowast equiv 119860

2minus 119887

1120572 the conditions for Hopf bifurcation

are satisfied at 120573 = 120573

lowast that is (i) (17) includes a pair of pureimaginary roots and (ii) 119889Re120582119889120573 = 0 Therefore a cycleexists for a certain range of 120573 in the neighborhood 120573lowast Wecan develop a similar argument using the monetary policyparameter 120574

3 Case of Monetary Policy with a Lag

Asada and Yoshida [12] examine the situation where a timelag exists in fiscal policy responses which is equivalent to thecase wherein 120579

1gt 0 and 120579

2= 0 in system (12) In this section

we analyze the situation where a time lag exists in monetarypolicy responses by assuming that 120579

1= 0 and 120579

2gt 0 in system

(12)In this case the characteristic equation in (15) can be

rewritten as follows

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870) minus 120572119860

2120582119890

minus1205792120582= 0

(19)

The following analysis is based on the study ofMatsumotoand Szidarovszky [19] (Bellman and Cooke [20] provide ahelpful introductory textbook of delay differential equations)The procedures of the analysis are given as follows

(1) We characterize the points (if any) at which thedynamics around the steady state can change that isthe points at which a zero real root or pure imaginaryroots appear These points are referred to as ldquocrossingpointsrdquo

(2) We reveal the directions of the changes in the signs ofthe real parts that occur when 120579

2crosses the crossing

points

31 Crossing Points First we examine whether the points atwhich a zero real root appears can exist By substituting 120582 = 0into (19) we obtain 119887

2minus 120572120573119868

119870= 0 However since 119887

2gt 0 and

119868

119870lt 0 this equality cannot hold Accordingly 120582 = 0 cannot

be a rootNext we examine whether the points at which the pure

imaginary roots appear can exist By substituting 120582 = 119894120596where 120596 = imaginary part gt 0 (pure imaginary roots arealways conjugated therefore we can assume 120596 gt 0 withoutloss of generality) and 119894 = radicminus1 into (19) we obtain

minus120596

2+ 119894120596 (119887

1+ 120572120573) + (119887

2minus 120572120573119868

119870) minus 119894120596120572119860

2119890

minus1198941205961205792= 0

(20)

4 Discrete Dynamics in Nature and Society

Application of Eulerrsquos formula (forall119909 isin R 119890

plusmn119894119909= cos119909 plusmn 119894 sin119909)

to this equation yields

minus 120596

2+ (119887

2minus 120572120573119868

119870) minus 120596120572119860

2sin120596120579

2

+ 119894 [(119887

1+ 120572120573)120596 minus 120596120572119860

2cos120596120579

2] = 0

(21)

This equality holds only when both real and imaginary partsof the left-hand side equal zero that is

minus120596

2+ (119887

2minus 120572120573119868

119870) = 120596120572119860

2sin120596120579

2

(119887

1+ 120572120573)120596 = 120596120572119860

2cos120596120579

2

(22)

Or equivalently

cos1205961205792=

119887

1+ 120572120573

120572119860

2

(23)

sin1205961205792=

minus120596

2+ 119887

2minus 120572120573119868

119870

120572119860

2120596

(24)

For values of 1205961205792that satisfy these equations (19) includes

pure imaginary roots where 2120587ℎ lt 120596120579

2lt 2120587(1 + ℎ) ℎ =

0 1 2 3 Furthermore the sum of the squares of (22) yields the

following

120596

4minus 2 (119887

2minus 120572120573119868

119870) minus (119887

1+ 120572120573)

2+ (120572119860

2)

2 120596

2

+ (119887

2minus 120572120573119868

119870)

2= 0

(25)

Solving for 1205962 we obtain

120596

2

plusmn=

2 (119887

2minus 120572120573119868

119870) minus (119887

1+ 120572120573)

2+ (120572119860

2)

2plusmnradic119873

2

(26)

where119873 equiv 2(119887

2minus120572120573119868

119870)minus(119887

1+120572120573)

2+(120572119860

2)

2

2minus4(119887

2minus120572120573119868

119870)

2We denote real and positive values of 120596 by 120596

lowast Bysubstituting 120596lowast into (23) and solving for 120579

2 we obtain

120579

2ℎ=

cosminus1 ((1198871+ 120572120573) 120572119860

2) + 2120587ℎ

120596

lowast

ℎ = 0 1 2 3

(27)

Thus there exists an infinite number of values of 1205792(ie

120579

20 120579

21 120579

22 ) that generate pure imaginary roots (ie

crossing points)Finally if a real and positive value of120596 does not exist then

changes in the signs of the real parts of the roots do not occurTherefore in this case a policy lag does not affect stability

Direction of Crossing Here we reveal how the signs of thecomplex roots change when 120579

2crosses 120579

2ℎThis is determined

by the sign of 119889Re1205821198891205792|

120582=119894120596lowast If 119889Re120582119889120579

2|

120582=119894120596lowast gt 0

then the signs of the real parts of the roots change fromnegative to positive with an increase in 120579

2(which indicates

destabilization) In contrast if 119889Re1205821198891205792|

120582=119894120596lowast lt 0 then the

signs of the real parts of the roots change from positive tonegative with an increase in 120579

2(which indicates stabilization)

Table 1 Baseline parameters

120572 119888 120591 119862 + 119888119879 + 119866 119884

09 0625 02 200 400

For convenience of calculation we will observe the sign ofRe(119889120582119889120579

2)

minus1|

120582=119894120596lowast instead of that of 119889Re120582119889120579

2|

120582=119894120596lowast

Using (19) we can demonstrate that Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt

0 holds for any value of ℎ (see the appendix) Thus with anycrossing the signs of the real parts of the roots change fromnegative to positive

If a policy lag does not present (1205792= 0) then roots with

positive real parts do not exist under Assumption 1 Withincreases in 120579

2 the number of roots with positive real parts

increases by two Accordingly we can state that 2ℎ roots withpositive real parts exist in the region of 120579

2isin (120579

2(ℎminus1) 120579

2ℎ)

(where 1205792(minus1)

= 0) Therefore we can offer the followingproposition

Proposition 2 For 1205792isin (0 120579

20) the steady state is stable and

for 1205792isin (120579

20infin) it is unstable

This proposition suggests that even if a monetary policyis sufficiently active enough to satisfy Assumption 1 aneconomy can become unstable due to the existence of a policylag A similar result has already been shown by Asada andYoshida [12] concerning fiscal policies

32 Numerical Simulations Let us consider a numericalexample Following the study of Asada and Yoshida [12] weset baseline parameters as shown in Table 1

Furthermore Asada and Yoshida [12] employ the invest-ment function that is highly nonlinear with respect to 119884(119905)which is formulated as follows (in their model 120574 = 0)

119868 =

119884

1 + 12119890

minus01(119884(119905)minus119884)minus 119903 minus 05119870 (119905) minus

119884

13

+ 02 + 30

119903 = minus005120574 (119884 minus 119884 (119905 minus 120579

2)) + 001

radic119884 (119905)

(28)

Based on these assumptions the steady-state values areprovided by (119884lowast 119870lowast) = (400 60)

We also set the baseline values of 120573 and 120574 as 25 and60 respectively (generally it seems that the responsivenessof monetary policies to economic fluctuations is often largerthan that of fiscal policies therefore we assume 120574 gt 120573)When 120574 is fixed at the baseline value 120596lowast exists only for 120573 isin

(149 208) Outside this range a policy lag does not affectstability In addition Assumption 1 is satisfied for 120573 gt 149Accordingly irrespective of the value of a monetary policylag the steady state is unstable for 0 lt 120573 lt 149 and it isstable for 120573 gt 208

Likewise when 120573 is fixed at the baseline value 120596lowast existsfor 120574 gt 1432 In addition Assumption 1 is satisfied for 120574 gtminus1432 Hence the steady state is stable for 0 lt 120574 lt 1432irrespective of the value of a lag

Figure 1 describes 1205792ℎ

(ℎ = 0 1 2) that corresponds to120596

lowast as a function of 120573 and 120574 These curves are referred to as

Discrete Dynamics in Nature and Society 5

16 18 20

5

10

15

16 18 20 220

2

4

6

8

10

12

14

120573 120574

1205792

1205792

12057922

12057921

12057920

12057922

12057921

12057920

Figure 1 Crossing curves

14 145 15 155 16 165 17 175 18250

300

350

400

450

500

Y

120574

Figure 2 Bifurcation diagram (case with (28))

ldquocrossing curvesrdquo Here the stability condition is expressedas 1205792lt 120579

20 Therefore we can state that an increase in 120573 has a

stabilizing effect whereas that in 120574 has a destabilizing effectThus an increase in the activeness level of a monetary

policymay cause instability due to the existence of amonetarypolicy lag A similar result concerning a fiscal policy hasalready been shown by Asada and Yoshida [12]

Incidentally conditions for Hopf bifurcation are satisfiedat 1205792= 120579

20 that is (i) (19) includes a pair of pure imaginary

roots and (ii) 119889Re1205821198891205792

= 0 Therefore when 120579

2crosses

120579

20 a cycle emerges A similar argument can be developed

using 120573 and 120574 We now suppose that 1205792= 20 In this case

a bifurcation diagram can be described for 120574 isin [14 18] asshown in Figure 2 (the vertical axis represents the maximal

15 155 16 165 17250

300

350

400

450

500

Y

120574

Figure 3 Bifurcation diagram (case with (29))

and minimal values of 119884(119905)) This figure shows that not onlyone period but also multiple period cycles can emerge Theemergence ofmultiple period cycles is attributed to the strongnonlinearity of the investment function with respect to 119884(119905)When we use the investment function that has relativelysimpler form with respect to 119884(119905) as below (we set thecoefficient of radic119884(119905) as 1136 so that the value of 119868

119884equals

that of (28)) the bifurcation diagram becomes as shown inFigure 3 In this case only one period cycle exists Hence

119868 = 1136radic119884 (119905) minus 119903 minus 05119870 (119905) minus 1136

radic

119884 + 02

+ 30

(29)

6 Discrete Dynamics in Nature and Society

Although Asada and Yoshida [12] do not clearly specifythe time dimension similar models that include a disequi-librium adjustment function of the goods market (120572 is setaround unity) specify the measure of 119905 as a year (refer to egYoshida and Asada [5]) Therefore we will consider the timedimension as a year For example when 120573 = 18 the steadystate is unstable for 120579

2gt 14 years

4 Case of Policies with Fiscal andMonetary Policy Lags

In this section we investigate the case wherein both fiscal andmonetary policy lags exist (ie 120579

1gt 0 and 120579

2gt 0) In order to

examine the signs of the roots of (15) we use a mathematicalmethod developed by Gu et al [17]

41 Preconditions First to apply the technique of Gu et al[17] some preconditions should be examined According totheir study (15) should satisfy the following conditions

(I) deg(1199010(120582)) ge maxdeg(119901

1(120582)) deg(119901

2(120582))

(II) Δ(0) = 0(III) A solution common to all three polynomials 119901

0(120582) =

0 1199011(120582) = 0 and 119901

2(120582) = 0 does not exist

(IV) lim120582rarrinfin

(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) lt 1

Condition (I) is satisfied by 2 gt max1 1 while Condi-tion (II) is also satisfied by Δ(0) = minus120572119868

1198701 minus 119888(1 minus 120591) + 120573 gt 0

Condition (III) is obviously satisfied Finally Condition (IV)is satisfied by lim

120582rarrinfin(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) = 0

Now we examine the effects of lags (1205791 120579

2) on the stability

of the steady state The analysis proceeds as follows

(1) We characterize the sets (1205791 120579

2) (ie crossing sets) at

which the pure imaginary roots appear (it is ensuredfrom precondition (II) that zero cannot be a root)

(2) We depict the crossing sets on the (120579

1 120579

2) plane

(which we refer to as the crossing curves) via numer-ical simulation

(3) We reveal the directions of changes in the signs ofthe real parts that occur when lags (120579

1 120579

2) cross the

crossing curves

42 Crossing Sets Dividing (15) by 1199010(120582) we obtain

1 + 119886

1 (120582) 119890

minus1205791120582+ 119886

2 (120582) 119890

minus1205792120582= 0

(30)

where

119886

1 (120582) =

119901

1 (120582)

119901

0 (120582)

=

120572120573 (120582 minus 119868

119870)

120582

2+ 119887

1120582 + 119887

2

119886

2 (120582) =

119901

2 (120582)

119901

0 (120582)

= minus

120572119860

2120582

120582

2+ 119887

1120582 + 119887

2

(31)

Denoting a pure imaginary root by 120582 = 119894120596 the values of120596that satisfy (30) can be characterized by the following lemma

0

Im

1

a2(i120596)eminusi1205961205792

1205752 1205751

Re

a1(i120596)eminusi 1120596120579

Figure 4 Triangle formed by 1 |1198861(119894120596)| and |119886

2(119894120596)| on the complex

plane

Lemma 3 (see [17 Proposition 31]) For each 120596 satisfying119901

0(119894120596) = 0 120582 = 119894120596 is a solution of Δ(120582) = 0 for some

(120579

1 120579

2) isin R2+if and only if

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

ge 1

minus1 le

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

le 1

(32)

We denote the set of 120596 gt 0 that satisfy conditions (32)as Ω which is termed the ldquocrossing frequencyrdquo For all 120596 isin

Ω the sets (1205791 120579

2) satisfying (30) (ie crossing sets) should

satisfy the following conditions (Figure 4)

∓120575

1= arg (119886

1 (119894120596) 119890

minus1198941205961205791) + 2119898120587 119898 = 0 1 2

plusmn120575

2= arg (119886

2 (119894120596) 119890

minus1198941205961205792) + 2119899120587 119899 = 0 1 2

(33)

where 1205751 120575

2isin [0 120587]

Incidentally on the complex plane multiplication ofamplitudes becomes a sum of the parts therefore we obtain

arg (1198861 (119894120596) 119890

minus1198941205961205791) = arg (119886

1 (119894120596)) minus 1205961205791

arg (1198862 (119894120596) 119890

minus1198941205961205792) = arg (119886

2 (119894120596)) minus 1205961205792

(34)

In addition Figure 4 demonstrates that the followingrelations hold

arg (1198861 (119894120596)) = tanminus1 (

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

)

arg (1198862 (119894120596)) = tanminus1 (

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

)

(35)

Moreover after some manipulation (31) derive the fol-lowing expressions

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

=

119887

1120596119868

119870+ 120596 (119887

2minus 120596

2)

119887

1120596

2minus 119868

119870(119887

2minus 120596

2)

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

=

119887

2minus 120596

2

119887

1120596

(36)

Discrete Dynamics in Nature and Society 7

Thus using (34) to (36) (33) can be rewritten as follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) plusmn 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) ∓ 120575

2+ 2119899120587

120596

(37)

where the interior angles of the triangle denoted by 1205751and

120575

2are given by the cosine theorem as follows

120575

1= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572120573119868

119870)

2+ (120572120573120596)

2minus (120572119860

2120596)

2

2

radic

(120572120573119868

119870)

2+ (120572120573120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

120575

2= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572119860

2120596)

2minus (120572120573119868

119870)

2minus (120572120573120596)

2

2

radic

(120572119860

2120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

(38)

Equations (37) characterize the crossing sets (1205791 120579

2) isin

R2+ Depending on the signs of 120575

1and 120575

2 we can define two

types of crossing sets denoted by 1198711(119898 119899) and 119871

2(119898 119899) as

follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) + 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) minus 120575

2+ 2119899120587

120596

119871

1(119898 119899)

120579

1=

tanminus1 ((1198871120596119868

119870+ V (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) minus 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) + 120575

2+ 2119899120587

120596

119871

2(119898 119899)

As in the previous section we set parameter values andthe functional form of the investment function as shownin Table 1 (28) or (29) 120573 = 25 and 120574 = 60 In thiscase crossing frequency Ω is provided by 120596 isin (150 213)

(Figure 5) (according to Gu et alrsquos [17] classification Figure 5exhibits Type 13) For 120596 isin Ω describing the crossing sets119871

1(119898 119899) and 119871

2(119898 119899) on the 120579

1-1205792plane we obtain Figure 6

where119898 = 0 1 2 and 119899 = 0 1 2 The dotted curves represent119871

1(119898 119899) and the solid curves represent 119871

2(119898 119899) The points

marked by a dot indicate the starting points of both curvesthat is the points corresponding to 120596 = 150

We establish how the real parts of the roots change whenlags cross the crossing curves with an increase in 120579

1 For 120596 isin

Ω if Re(1198891205821198891205791)

minus1gt 0 then the real parts of the roots change

from negative to positive this indicates destabilization Incontrast if Re(119889120582119889120579

1)

minus1lt 0 then the real parts of the roots

change from positive to negative this indicates stabilizationDifferentiating (30) with respect to 120579

1 we obtain

[119886

1015840

1(120582) 119890

minus1205791120582minus 119886

1 (120582) 119890

minus1205791120582120579

1+ 119886

1015840

2(120582) 119890

minus1205792120582

minus 119886

2 (120582) 119890

minus1205792120579

2]

119889120582

119889120579

1

= 119886

1 (120582) 119890

minus1205791120582120582

(39)

8 Discrete Dynamics in Nature and Society

0 2 4 6 8 10

0

1

2

3

4

5

6

minus1

|a1(i120596)| + |a2(i120596)|

120596

Ω|a1(i120596)| minus |a2(i120596)|

Figure 5 Crossing frequency Ω

or equivalently

(

119889120582

119889120579

1

)

minus1

=

119886

1015840

1(120582) 119890

minus1205791120582+ 119886

1015840

2(120582) 119890

minus1205792120582minus 119886

2 (120582) 119890

minus1205792120582120579

2

119886

1 (120582) 119890

minus1205791120582120582

minus

120579

1

120582

(40)

where

119886

1015840

1(120582) =

120572120573119901

0 (120582) minus 120572120573 (120582 minus 119868119870

) (2120582 + 119887

1)

119901

0 (120582)

2

119886

1015840

2(120582) = minus

120572119860

2119901

0 (120582) minus 1205721198602

120582 (2120582 + 119887

1)

119901

0 (120582)

2

(41)

Describing the real parts of (40) as a function of 120596 isin Ωwe obtain Figure 7 where the dotted curves are the functionsevaluated on curve 119871

1and the solid curves are the functions

evaluated on curve 1198712 As for curve 119871

1 stabilization only

occurs on the intermediate region of the segments 1198711(119898 2)

destabilization occurs on the other segments of 1198711 As for

curve 1198712 destabilization occurs on all segments Here we

define a curve formed by connecting curves 119871119895(0 119899) (where

119895 = 1 2 and 119899 = 0 1 2) as 1198980 (Figure 8) We also definea curve formed by connecting the points on curve 1198980 thatminimize the value of 120579

1for a given 120579

2as curve 1198720 Then

we can express the stability condition as follows set (1205791 120579

2)

exists in the left-hand side region of curve1198720On curve 1198720 conditions for Hopf bifurcation are sat-

isfied and therefore a cycle emerges In addition as isobvious from Figure 6 an increase in the fiscal policy lag 120579

1

necessarily destabilizes the steady state However an increasein the monetary policy lag 120579

2has the potential to stabilize

the steady state which depends on the value of 1205791 When

120579

1isin (022 036) we can see that the steady state fluctuates

between stability and instability with an increase in 1205792 Such

a phenomenon cannot be observed in models with only onetime lag Moreover Figure 9 presents a bifurcation diagram

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

L2(0 2)

L1(0 2)

L2(0 1)

L1(0 1)

L2(0 0)

L1(0 0)

L2(1 0)

L1(1 0) L1(2 0)

L2(2 0)

L2(2 1)

L1(2 1)

L2(2 2)

L1(2 2)

L2(1 2)

L1(1 2)

L2(1 1)

L1(1 1)

1205792

1205791

Figure 6 Crossing curves

wherein 1205791= 025 and (28) is used as the investment function

This figure evidently indicates that alternations of stabilityand instability emerge with an increase in 120579

2

In particular when 1205791= 025 stability can be established

by setting 1205792isin (0 106) 120579

2isin (253 433) or 120579

2isin (671 761)

that is for equilibrium stability monetary policy lag shouldbe less than one year between 3 and 45 years or between 68and 75 years when a fiscal policy lag is approximately threemonths

Next we examine how 120573 and 120574 affect the stabilitycondition Both 120573 and 120574 affect 119871

1and 119871

2through 120596 isin Ω It is

difficult to analytically show how 119871

1and 119871

2change Hence

we numerically examine how curve 1198980 (which comprises119871

1(0 119899) and 119871

2(0 119899)) shifts when 120573 and 120574 increase from each

baseline value (120573 = 25 and 120574 = 60)Figures 10 and 11 describe curve 1198980 in cases with 120573 = 8

and 120573 = 15 respectively We can observe that an increasein 120573 shifts curve 1198980 to left Accordingly the stability region119878 reduces Likewise Figures 12 and 13 describe curve 1198980 incaseswith 120574 = 12 and 120574 = 22 respectivelyThe stability regionis also reduced by an increase in 120574 These results suggestthat increases in the activeness levels of fiscal and monetarypolicies increase economic instability as is the case with onlyone policy lag

43 Investment and Stability In general the formulation ofthe investment function has great significance to stabilityin macrodynamic models that is the larger the marginalpropensity to investment is the more likely it becomesunstable In this subsection we examine the effects of changesin the partial derivatives of the investment function 119868

119884and

119868

119870 on stability We compare three cases the case of no policy

lags the case of the existence of onlymonetary policy lag andthe case of the existence of both fiscal and monetary policylags

In the casewith no policy lags 119868119884and 119868119870affect the stability

condition given by Assumption 1 through 1198871 The smaller the

values of 119868119884and 119868119870are (the larger the absolute value of 119868

119870is)

Discrete Dynamics in Nature and Society 9

L1(0 0) and L2(0 0) L1(0 1) and L2(0 1) L1(0 2) and L2(0 2)

L1(1 0) and L2(1 0) L1(1 1) and L2(1 1) L1(1 2) and L2(1 2)

L1(2 0) and L2(2 0) L1(2 1) and L2(2 1) L1(2 2) and L2(2 2)

0 1 2 30

02

04Re

(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

Figure 7 Direction of crossing

0 01 02 03 04 050

1

2

3

4

5

6

7

8

9

S1205792

1205791

Figure 8 Stability region 119878

the more likely the stability condition is satisfied These arefairly typical results

In the case of a monetary policy with a lag 119868119884and 119868119870

affect the value of 120596lowast through 119887

2(119868119870

also directly affects120596

lowast) Accordingly the value of (27) changes In this case itis difficult to analytically show the quality and quantity of

1 2 3 4 5 6 7 8 9350

360

370

380

390

400

410

420

430

440

450

Y

1205792

Figure 9 Bifurcation diagram (1205791= 025)

these effects Hence we resort to a numerical method Basedon the parameter values proposed in Section 32 crossingcurves 120579

20can be described within each range of 119868

119884and

119868

119870that ensure the existence of 120596lowast as shown in Figure 14

The region located below these curves indicates the stabilityregions Similar to the case with no policy lags an increase

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

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Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

120582 denotes the eigenvalue as the solutions of system (13) andsubstituting these into the system we obtain

[

120582 minus 120572 119860

1minus 1 + 119888 (1 minus 120591) + 120572120573119890

minus1205791120582minus 120572119860

2119890

minus1205792120582

minus120572119868

119870

minus119860

1minus 119860

2119890

minus1205792120582

120582 minus 119868

119870

][

119884 (119905)

119870 (119905)

] = [

0

0

] (14)

For nontrivial solutions to exist the determinant of theleft-hand sidematrix denoted byΔ(120582) must be equal to zerothat is

Δ (120582) = 120582

2minus [119868

119870+ 120572 119860

1minus 1 + 119888 (1 minus 120591)] 120582

minus 120572 1 minus 119888 (1 minus 120591) 119868119870+ 120572120573 (120582 minus 119868

119870) 119890

minus1205791120582

minus 120572119860

2120582119890

minus1205792120582= 0

= 119901

0 (120582) + 1199011 (

120582) 119890

minus1205791120582+ 119901

2 (120582) 119890

minus1205792120582= 0

(15)

where

119901

0 (120582) = 120582

2+ 119887

1120582 + 119887

2

119887

1= minus [119868

119870+ 120572 119860

1minus 1 + 119888 (1 minus 120591)]

119887

2= minus120572 1 minus 119888 (1 minus 120591) 119868119870

gt 0

119901

1 (120582) = 120572120573 (120582 minus 119868119870

)

119901

2 (120582) = minus1205721198602

120582

(16)

Equation (15) is a characteristic equation of system (13) Thesignificant feature of this equation is found in the existence ofthe exponential terms 119890minus1205791120582 and 119890minus1205792120582

First we examine the case of no policy lags By substitut-ing 1205791= 120579

2= 0 into (15) we obtain

120582

2+ [119887

1+ 120572 (120573 minus 119860

2)] 120582 + 119887

2minus 120572120573119868

119870= 0 (17)

which is an ordinary quadratic equation of 120582Thus if 119887

1+120572(120573minus119860

2) gt 0 (ie the coefficient of120582 from (17)

is positive) the real parts of the roots of (17) are negative (seeChapter 18 in Gandolfo [18] for details regarding the relationbetween the roots and the coefficients of a quadric equation)In contrast if 119887

1+ 120572(120573 minus 119860

2) lt 0 the real parts of the roots

are positive Thus if 1198871+ 120572(120573 minus 119860

2) gt 0 then the steady state

is locally stable and if 1198871+ 120572(120573 minus 119860

2) lt 0 then it is unstable

The larger the values of 120573 and 120574 are the more likely thestability condition holds (note that119860

2= minus120574119868

119903119871

119903)Therefore

if policy lags do not exist then traditional fiscal andmonetarypolicies could function effectively for stabilization

In the following discussion we assume that if both fiscaland monetary policy lags do not exist then an economy isstable that is we have the following

Assumption 1 We have

120573 +

119868

119903

119871

119903

120574 +

119887

1

120572

gt 0 (18)

Under this assumption we analyze the effects of the policylags (120579

1 120579

2) on stability

Incidentally if [1198871+ 120572(120573 minus 119860

2)]

2lt 4(119887

2minus 120572120573119868

119870) then

both roots of (17) are conjugate complex Furthermore bydefining 120573lowast equiv 119860

2minus 119887

1120572 the conditions for Hopf bifurcation

are satisfied at 120573 = 120573

lowast that is (i) (17) includes a pair of pureimaginary roots and (ii) 119889Re120582119889120573 = 0 Therefore a cycleexists for a certain range of 120573 in the neighborhood 120573lowast Wecan develop a similar argument using the monetary policyparameter 120574

3 Case of Monetary Policy with a Lag

Asada and Yoshida [12] examine the situation where a timelag exists in fiscal policy responses which is equivalent to thecase wherein 120579

1gt 0 and 120579

2= 0 in system (12) In this section

we analyze the situation where a time lag exists in monetarypolicy responses by assuming that 120579

1= 0 and 120579

2gt 0 in system

(12)In this case the characteristic equation in (15) can be

rewritten as follows

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870) minus 120572119860

2120582119890

minus1205792120582= 0

(19)

The following analysis is based on the study ofMatsumotoand Szidarovszky [19] (Bellman and Cooke [20] provide ahelpful introductory textbook of delay differential equations)The procedures of the analysis are given as follows

(1) We characterize the points (if any) at which thedynamics around the steady state can change that isthe points at which a zero real root or pure imaginaryroots appear These points are referred to as ldquocrossingpointsrdquo

(2) We reveal the directions of the changes in the signs ofthe real parts that occur when 120579

2crosses the crossing

points

31 Crossing Points First we examine whether the points atwhich a zero real root appears can exist By substituting 120582 = 0into (19) we obtain 119887

2minus 120572120573119868

119870= 0 However since 119887

2gt 0 and

119868

119870lt 0 this equality cannot hold Accordingly 120582 = 0 cannot

be a rootNext we examine whether the points at which the pure

imaginary roots appear can exist By substituting 120582 = 119894120596where 120596 = imaginary part gt 0 (pure imaginary roots arealways conjugated therefore we can assume 120596 gt 0 withoutloss of generality) and 119894 = radicminus1 into (19) we obtain

minus120596

2+ 119894120596 (119887

1+ 120572120573) + (119887

2minus 120572120573119868

119870) minus 119894120596120572119860

2119890

minus1198941205961205792= 0

(20)

4 Discrete Dynamics in Nature and Society

Application of Eulerrsquos formula (forall119909 isin R 119890

plusmn119894119909= cos119909 plusmn 119894 sin119909)

to this equation yields

minus 120596

2+ (119887

2minus 120572120573119868

119870) minus 120596120572119860

2sin120596120579

2

+ 119894 [(119887

1+ 120572120573)120596 minus 120596120572119860

2cos120596120579

2] = 0

(21)

This equality holds only when both real and imaginary partsof the left-hand side equal zero that is

minus120596

2+ (119887

2minus 120572120573119868

119870) = 120596120572119860

2sin120596120579

2

(119887

1+ 120572120573)120596 = 120596120572119860

2cos120596120579

2

(22)

Or equivalently

cos1205961205792=

119887

1+ 120572120573

120572119860

2

(23)

sin1205961205792=

minus120596

2+ 119887

2minus 120572120573119868

119870

120572119860

2120596

(24)

For values of 1205961205792that satisfy these equations (19) includes

pure imaginary roots where 2120587ℎ lt 120596120579

2lt 2120587(1 + ℎ) ℎ =

0 1 2 3 Furthermore the sum of the squares of (22) yields the

following

120596

4minus 2 (119887

2minus 120572120573119868

119870) minus (119887

1+ 120572120573)

2+ (120572119860

2)

2 120596

2

+ (119887

2minus 120572120573119868

119870)

2= 0

(25)

Solving for 1205962 we obtain

120596

2

plusmn=

2 (119887

2minus 120572120573119868

119870) minus (119887

1+ 120572120573)

2+ (120572119860

2)

2plusmnradic119873

2

(26)

where119873 equiv 2(119887

2minus120572120573119868

119870)minus(119887

1+120572120573)

2+(120572119860

2)

2

2minus4(119887

2minus120572120573119868

119870)

2We denote real and positive values of 120596 by 120596

lowast Bysubstituting 120596lowast into (23) and solving for 120579

2 we obtain

120579

2ℎ=

cosminus1 ((1198871+ 120572120573) 120572119860

2) + 2120587ℎ

120596

lowast

ℎ = 0 1 2 3

(27)

Thus there exists an infinite number of values of 1205792(ie

120579

20 120579

21 120579

22 ) that generate pure imaginary roots (ie

crossing points)Finally if a real and positive value of120596 does not exist then

changes in the signs of the real parts of the roots do not occurTherefore in this case a policy lag does not affect stability

Direction of Crossing Here we reveal how the signs of thecomplex roots change when 120579

2crosses 120579

2ℎThis is determined

by the sign of 119889Re1205821198891205792|

120582=119894120596lowast If 119889Re120582119889120579

2|

120582=119894120596lowast gt 0

then the signs of the real parts of the roots change fromnegative to positive with an increase in 120579

2(which indicates

destabilization) In contrast if 119889Re1205821198891205792|

120582=119894120596lowast lt 0 then the

signs of the real parts of the roots change from positive tonegative with an increase in 120579

2(which indicates stabilization)

Table 1 Baseline parameters

120572 119888 120591 119862 + 119888119879 + 119866 119884

09 0625 02 200 400

For convenience of calculation we will observe the sign ofRe(119889120582119889120579

2)

minus1|

120582=119894120596lowast instead of that of 119889Re120582119889120579

2|

120582=119894120596lowast

Using (19) we can demonstrate that Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt

0 holds for any value of ℎ (see the appendix) Thus with anycrossing the signs of the real parts of the roots change fromnegative to positive

If a policy lag does not present (1205792= 0) then roots with

positive real parts do not exist under Assumption 1 Withincreases in 120579

2 the number of roots with positive real parts

increases by two Accordingly we can state that 2ℎ roots withpositive real parts exist in the region of 120579

2isin (120579

2(ℎminus1) 120579

2ℎ)

(where 1205792(minus1)

= 0) Therefore we can offer the followingproposition

Proposition 2 For 1205792isin (0 120579

20) the steady state is stable and

for 1205792isin (120579

20infin) it is unstable

This proposition suggests that even if a monetary policyis sufficiently active enough to satisfy Assumption 1 aneconomy can become unstable due to the existence of a policylag A similar result has already been shown by Asada andYoshida [12] concerning fiscal policies

32 Numerical Simulations Let us consider a numericalexample Following the study of Asada and Yoshida [12] weset baseline parameters as shown in Table 1

Furthermore Asada and Yoshida [12] employ the invest-ment function that is highly nonlinear with respect to 119884(119905)which is formulated as follows (in their model 120574 = 0)

119868 =

119884

1 + 12119890

minus01(119884(119905)minus119884)minus 119903 minus 05119870 (119905) minus

119884

13

+ 02 + 30

119903 = minus005120574 (119884 minus 119884 (119905 minus 120579

2)) + 001

radic119884 (119905)

(28)

Based on these assumptions the steady-state values areprovided by (119884lowast 119870lowast) = (400 60)

We also set the baseline values of 120573 and 120574 as 25 and60 respectively (generally it seems that the responsivenessof monetary policies to economic fluctuations is often largerthan that of fiscal policies therefore we assume 120574 gt 120573)When 120574 is fixed at the baseline value 120596lowast exists only for 120573 isin

(149 208) Outside this range a policy lag does not affectstability In addition Assumption 1 is satisfied for 120573 gt 149Accordingly irrespective of the value of a monetary policylag the steady state is unstable for 0 lt 120573 lt 149 and it isstable for 120573 gt 208

Likewise when 120573 is fixed at the baseline value 120596lowast existsfor 120574 gt 1432 In addition Assumption 1 is satisfied for 120574 gtminus1432 Hence the steady state is stable for 0 lt 120574 lt 1432irrespective of the value of a lag

Figure 1 describes 1205792ℎ

(ℎ = 0 1 2) that corresponds to120596

lowast as a function of 120573 and 120574 These curves are referred to as

Discrete Dynamics in Nature and Society 5

16 18 20

5

10

15

16 18 20 220

2

4

6

8

10

12

14

120573 120574

1205792

1205792

12057922

12057921

12057920

12057922

12057921

12057920

Figure 1 Crossing curves

14 145 15 155 16 165 17 175 18250

300

350

400

450

500

Y

120574

Figure 2 Bifurcation diagram (case with (28))

ldquocrossing curvesrdquo Here the stability condition is expressedas 1205792lt 120579

20 Therefore we can state that an increase in 120573 has a

stabilizing effect whereas that in 120574 has a destabilizing effectThus an increase in the activeness level of a monetary

policymay cause instability due to the existence of amonetarypolicy lag A similar result concerning a fiscal policy hasalready been shown by Asada and Yoshida [12]

Incidentally conditions for Hopf bifurcation are satisfiedat 1205792= 120579

20 that is (i) (19) includes a pair of pure imaginary

roots and (ii) 119889Re1205821198891205792

= 0 Therefore when 120579

2crosses

120579

20 a cycle emerges A similar argument can be developed

using 120573 and 120574 We now suppose that 1205792= 20 In this case

a bifurcation diagram can be described for 120574 isin [14 18] asshown in Figure 2 (the vertical axis represents the maximal

15 155 16 165 17250

300

350

400

450

500

Y

120574

Figure 3 Bifurcation diagram (case with (29))

and minimal values of 119884(119905)) This figure shows that not onlyone period but also multiple period cycles can emerge Theemergence ofmultiple period cycles is attributed to the strongnonlinearity of the investment function with respect to 119884(119905)When we use the investment function that has relativelysimpler form with respect to 119884(119905) as below (we set thecoefficient of radic119884(119905) as 1136 so that the value of 119868

119884equals

that of (28)) the bifurcation diagram becomes as shown inFigure 3 In this case only one period cycle exists Hence

119868 = 1136radic119884 (119905) minus 119903 minus 05119870 (119905) minus 1136

radic

119884 + 02

+ 30

(29)

6 Discrete Dynamics in Nature and Society

Although Asada and Yoshida [12] do not clearly specifythe time dimension similar models that include a disequi-librium adjustment function of the goods market (120572 is setaround unity) specify the measure of 119905 as a year (refer to egYoshida and Asada [5]) Therefore we will consider the timedimension as a year For example when 120573 = 18 the steadystate is unstable for 120579

2gt 14 years

4 Case of Policies with Fiscal andMonetary Policy Lags

In this section we investigate the case wherein both fiscal andmonetary policy lags exist (ie 120579

1gt 0 and 120579

2gt 0) In order to

examine the signs of the roots of (15) we use a mathematicalmethod developed by Gu et al [17]

41 Preconditions First to apply the technique of Gu et al[17] some preconditions should be examined According totheir study (15) should satisfy the following conditions

(I) deg(1199010(120582)) ge maxdeg(119901

1(120582)) deg(119901

2(120582))

(II) Δ(0) = 0(III) A solution common to all three polynomials 119901

0(120582) =

0 1199011(120582) = 0 and 119901

2(120582) = 0 does not exist

(IV) lim120582rarrinfin

(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) lt 1

Condition (I) is satisfied by 2 gt max1 1 while Condi-tion (II) is also satisfied by Δ(0) = minus120572119868

1198701 minus 119888(1 minus 120591) + 120573 gt 0

Condition (III) is obviously satisfied Finally Condition (IV)is satisfied by lim

120582rarrinfin(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) = 0

Now we examine the effects of lags (1205791 120579

2) on the stability

of the steady state The analysis proceeds as follows

(1) We characterize the sets (1205791 120579

2) (ie crossing sets) at

which the pure imaginary roots appear (it is ensuredfrom precondition (II) that zero cannot be a root)

(2) We depict the crossing sets on the (120579

1 120579

2) plane

(which we refer to as the crossing curves) via numer-ical simulation

(3) We reveal the directions of changes in the signs ofthe real parts that occur when lags (120579

1 120579

2) cross the

crossing curves

42 Crossing Sets Dividing (15) by 1199010(120582) we obtain

1 + 119886

1 (120582) 119890

minus1205791120582+ 119886

2 (120582) 119890

minus1205792120582= 0

(30)

where

119886

1 (120582) =

119901

1 (120582)

119901

0 (120582)

=

120572120573 (120582 minus 119868

119870)

120582

2+ 119887

1120582 + 119887

2

119886

2 (120582) =

119901

2 (120582)

119901

0 (120582)

= minus

120572119860

2120582

120582

2+ 119887

1120582 + 119887

2

(31)

Denoting a pure imaginary root by 120582 = 119894120596 the values of120596that satisfy (30) can be characterized by the following lemma

0

Im

1

a2(i120596)eminusi1205961205792

1205752 1205751

Re

a1(i120596)eminusi 1120596120579

Figure 4 Triangle formed by 1 |1198861(119894120596)| and |119886

2(119894120596)| on the complex

plane

Lemma 3 (see [17 Proposition 31]) For each 120596 satisfying119901

0(119894120596) = 0 120582 = 119894120596 is a solution of Δ(120582) = 0 for some

(120579

1 120579

2) isin R2+if and only if

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

ge 1

minus1 le

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

le 1

(32)

We denote the set of 120596 gt 0 that satisfy conditions (32)as Ω which is termed the ldquocrossing frequencyrdquo For all 120596 isin

Ω the sets (1205791 120579

2) satisfying (30) (ie crossing sets) should

satisfy the following conditions (Figure 4)

∓120575

1= arg (119886

1 (119894120596) 119890

minus1198941205961205791) + 2119898120587 119898 = 0 1 2

plusmn120575

2= arg (119886

2 (119894120596) 119890

minus1198941205961205792) + 2119899120587 119899 = 0 1 2

(33)

where 1205751 120575

2isin [0 120587]

Incidentally on the complex plane multiplication ofamplitudes becomes a sum of the parts therefore we obtain

arg (1198861 (119894120596) 119890

minus1198941205961205791) = arg (119886

1 (119894120596)) minus 1205961205791

arg (1198862 (119894120596) 119890

minus1198941205961205792) = arg (119886

2 (119894120596)) minus 1205961205792

(34)

In addition Figure 4 demonstrates that the followingrelations hold

arg (1198861 (119894120596)) = tanminus1 (

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

)

arg (1198862 (119894120596)) = tanminus1 (

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

)

(35)

Moreover after some manipulation (31) derive the fol-lowing expressions

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

=

119887

1120596119868

119870+ 120596 (119887

2minus 120596

2)

119887

1120596

2minus 119868

119870(119887

2minus 120596

2)

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

=

119887

2minus 120596

2

119887

1120596

(36)

Discrete Dynamics in Nature and Society 7

Thus using (34) to (36) (33) can be rewritten as follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) plusmn 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) ∓ 120575

2+ 2119899120587

120596

(37)

where the interior angles of the triangle denoted by 1205751and

120575

2are given by the cosine theorem as follows

120575

1= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572120573119868

119870)

2+ (120572120573120596)

2minus (120572119860

2120596)

2

2

radic

(120572120573119868

119870)

2+ (120572120573120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

120575

2= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572119860

2120596)

2minus (120572120573119868

119870)

2minus (120572120573120596)

2

2

radic

(120572119860

2120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

(38)

Equations (37) characterize the crossing sets (1205791 120579

2) isin

R2+ Depending on the signs of 120575

1and 120575

2 we can define two

types of crossing sets denoted by 1198711(119898 119899) and 119871

2(119898 119899) as

follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) + 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) minus 120575

2+ 2119899120587

120596

119871

1(119898 119899)

120579

1=

tanminus1 ((1198871120596119868

119870+ V (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) minus 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) + 120575

2+ 2119899120587

120596

119871

2(119898 119899)

As in the previous section we set parameter values andthe functional form of the investment function as shownin Table 1 (28) or (29) 120573 = 25 and 120574 = 60 In thiscase crossing frequency Ω is provided by 120596 isin (150 213)

(Figure 5) (according to Gu et alrsquos [17] classification Figure 5exhibits Type 13) For 120596 isin Ω describing the crossing sets119871

1(119898 119899) and 119871

2(119898 119899) on the 120579

1-1205792plane we obtain Figure 6

where119898 = 0 1 2 and 119899 = 0 1 2 The dotted curves represent119871

1(119898 119899) and the solid curves represent 119871

2(119898 119899) The points

marked by a dot indicate the starting points of both curvesthat is the points corresponding to 120596 = 150

We establish how the real parts of the roots change whenlags cross the crossing curves with an increase in 120579

1 For 120596 isin

Ω if Re(1198891205821198891205791)

minus1gt 0 then the real parts of the roots change

from negative to positive this indicates destabilization Incontrast if Re(119889120582119889120579

1)

minus1lt 0 then the real parts of the roots

change from positive to negative this indicates stabilizationDifferentiating (30) with respect to 120579

1 we obtain

[119886

1015840

1(120582) 119890

minus1205791120582minus 119886

1 (120582) 119890

minus1205791120582120579

1+ 119886

1015840

2(120582) 119890

minus1205792120582

minus 119886

2 (120582) 119890

minus1205792120579

2]

119889120582

119889120579

1

= 119886

1 (120582) 119890

minus1205791120582120582

(39)

8 Discrete Dynamics in Nature and Society

0 2 4 6 8 10

0

1

2

3

4

5

6

minus1

|a1(i120596)| + |a2(i120596)|

120596

Ω|a1(i120596)| minus |a2(i120596)|

Figure 5 Crossing frequency Ω

or equivalently

(

119889120582

119889120579

1

)

minus1

=

119886

1015840

1(120582) 119890

minus1205791120582+ 119886

1015840

2(120582) 119890

minus1205792120582minus 119886

2 (120582) 119890

minus1205792120582120579

2

119886

1 (120582) 119890

minus1205791120582120582

minus

120579

1

120582

(40)

where

119886

1015840

1(120582) =

120572120573119901

0 (120582) minus 120572120573 (120582 minus 119868119870

) (2120582 + 119887

1)

119901

0 (120582)

2

119886

1015840

2(120582) = minus

120572119860

2119901

0 (120582) minus 1205721198602

120582 (2120582 + 119887

1)

119901

0 (120582)

2

(41)

Describing the real parts of (40) as a function of 120596 isin Ωwe obtain Figure 7 where the dotted curves are the functionsevaluated on curve 119871

1and the solid curves are the functions

evaluated on curve 1198712 As for curve 119871

1 stabilization only

occurs on the intermediate region of the segments 1198711(119898 2)

destabilization occurs on the other segments of 1198711 As for

curve 1198712 destabilization occurs on all segments Here we

define a curve formed by connecting curves 119871119895(0 119899) (where

119895 = 1 2 and 119899 = 0 1 2) as 1198980 (Figure 8) We also definea curve formed by connecting the points on curve 1198980 thatminimize the value of 120579

1for a given 120579

2as curve 1198720 Then

we can express the stability condition as follows set (1205791 120579

2)

exists in the left-hand side region of curve1198720On curve 1198720 conditions for Hopf bifurcation are sat-

isfied and therefore a cycle emerges In addition as isobvious from Figure 6 an increase in the fiscal policy lag 120579

1

necessarily destabilizes the steady state However an increasein the monetary policy lag 120579

2has the potential to stabilize

the steady state which depends on the value of 1205791 When

120579

1isin (022 036) we can see that the steady state fluctuates

between stability and instability with an increase in 1205792 Such

a phenomenon cannot be observed in models with only onetime lag Moreover Figure 9 presents a bifurcation diagram

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

L2(0 2)

L1(0 2)

L2(0 1)

L1(0 1)

L2(0 0)

L1(0 0)

L2(1 0)

L1(1 0) L1(2 0)

L2(2 0)

L2(2 1)

L1(2 1)

L2(2 2)

L1(2 2)

L2(1 2)

L1(1 2)

L2(1 1)

L1(1 1)

1205792

1205791

Figure 6 Crossing curves

wherein 1205791= 025 and (28) is used as the investment function

This figure evidently indicates that alternations of stabilityand instability emerge with an increase in 120579

2

In particular when 1205791= 025 stability can be established

by setting 1205792isin (0 106) 120579

2isin (253 433) or 120579

2isin (671 761)

that is for equilibrium stability monetary policy lag shouldbe less than one year between 3 and 45 years or between 68and 75 years when a fiscal policy lag is approximately threemonths

Next we examine how 120573 and 120574 affect the stabilitycondition Both 120573 and 120574 affect 119871

1and 119871

2through 120596 isin Ω It is

difficult to analytically show how 119871

1and 119871

2change Hence

we numerically examine how curve 1198980 (which comprises119871

1(0 119899) and 119871

2(0 119899)) shifts when 120573 and 120574 increase from each

baseline value (120573 = 25 and 120574 = 60)Figures 10 and 11 describe curve 1198980 in cases with 120573 = 8

and 120573 = 15 respectively We can observe that an increasein 120573 shifts curve 1198980 to left Accordingly the stability region119878 reduces Likewise Figures 12 and 13 describe curve 1198980 incaseswith 120574 = 12 and 120574 = 22 respectivelyThe stability regionis also reduced by an increase in 120574 These results suggestthat increases in the activeness levels of fiscal and monetarypolicies increase economic instability as is the case with onlyone policy lag

43 Investment and Stability In general the formulation ofthe investment function has great significance to stabilityin macrodynamic models that is the larger the marginalpropensity to investment is the more likely it becomesunstable In this subsection we examine the effects of changesin the partial derivatives of the investment function 119868

119884and

119868

119870 on stability We compare three cases the case of no policy

lags the case of the existence of onlymonetary policy lag andthe case of the existence of both fiscal and monetary policylags

In the casewith no policy lags 119868119884and 119868119870affect the stability

condition given by Assumption 1 through 1198871 The smaller the

values of 119868119884and 119868119870are (the larger the absolute value of 119868

119870is)

Discrete Dynamics in Nature and Society 9

L1(0 0) and L2(0 0) L1(0 1) and L2(0 1) L1(0 2) and L2(0 2)

L1(1 0) and L2(1 0) L1(1 1) and L2(1 1) L1(1 2) and L2(1 2)

L1(2 0) and L2(2 0) L1(2 1) and L2(2 1) L1(2 2) and L2(2 2)

0 1 2 30

02

04Re

(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

Figure 7 Direction of crossing

0 01 02 03 04 050

1

2

3

4

5

6

7

8

9

S1205792

1205791

Figure 8 Stability region 119878

the more likely the stability condition is satisfied These arefairly typical results

In the case of a monetary policy with a lag 119868119884and 119868119870

affect the value of 120596lowast through 119887

2(119868119870

also directly affects120596

lowast) Accordingly the value of (27) changes In this case itis difficult to analytically show the quality and quantity of

1 2 3 4 5 6 7 8 9350

360

370

380

390

400

410

420

430

440

450

Y

1205792

Figure 9 Bifurcation diagram (1205791= 025)

these effects Hence we resort to a numerical method Basedon the parameter values proposed in Section 32 crossingcurves 120579

20can be described within each range of 119868

119884and

119868

119870that ensure the existence of 120596lowast as shown in Figure 14

The region located below these curves indicates the stabilityregions Similar to the case with no policy lags an increase

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Application of Eulerrsquos formula (forall119909 isin R 119890

plusmn119894119909= cos119909 plusmn 119894 sin119909)

to this equation yields

minus 120596

2+ (119887

2minus 120572120573119868

119870) minus 120596120572119860

2sin120596120579

2

+ 119894 [(119887

1+ 120572120573)120596 minus 120596120572119860

2cos120596120579

2] = 0

(21)

This equality holds only when both real and imaginary partsof the left-hand side equal zero that is

minus120596

2+ (119887

2minus 120572120573119868

119870) = 120596120572119860

2sin120596120579

2

(119887

1+ 120572120573)120596 = 120596120572119860

2cos120596120579

2

(22)

Or equivalently

cos1205961205792=

119887

1+ 120572120573

120572119860

2

(23)

sin1205961205792=

minus120596

2+ 119887

2minus 120572120573119868

119870

120572119860

2120596

(24)

For values of 1205961205792that satisfy these equations (19) includes

pure imaginary roots where 2120587ℎ lt 120596120579

2lt 2120587(1 + ℎ) ℎ =

0 1 2 3 Furthermore the sum of the squares of (22) yields the

following

120596

4minus 2 (119887

2minus 120572120573119868

119870) minus (119887

1+ 120572120573)

2+ (120572119860

2)

2 120596

2

+ (119887

2minus 120572120573119868

119870)

2= 0

(25)

Solving for 1205962 we obtain

120596

2

plusmn=

2 (119887

2minus 120572120573119868

119870) minus (119887

1+ 120572120573)

2+ (120572119860

2)

2plusmnradic119873

2

(26)

where119873 equiv 2(119887

2minus120572120573119868

119870)minus(119887

1+120572120573)

2+(120572119860

2)

2

2minus4(119887

2minus120572120573119868

119870)

2We denote real and positive values of 120596 by 120596

lowast Bysubstituting 120596lowast into (23) and solving for 120579

2 we obtain

120579

2ℎ=

cosminus1 ((1198871+ 120572120573) 120572119860

2) + 2120587ℎ

120596

lowast

ℎ = 0 1 2 3

(27)

Thus there exists an infinite number of values of 1205792(ie

120579

20 120579

21 120579

22 ) that generate pure imaginary roots (ie

crossing points)Finally if a real and positive value of120596 does not exist then

changes in the signs of the real parts of the roots do not occurTherefore in this case a policy lag does not affect stability

Direction of Crossing Here we reveal how the signs of thecomplex roots change when 120579

2crosses 120579

2ℎThis is determined

by the sign of 119889Re1205821198891205792|

120582=119894120596lowast If 119889Re120582119889120579

2|

120582=119894120596lowast gt 0

then the signs of the real parts of the roots change fromnegative to positive with an increase in 120579

2(which indicates

destabilization) In contrast if 119889Re1205821198891205792|

120582=119894120596lowast lt 0 then the

signs of the real parts of the roots change from positive tonegative with an increase in 120579

2(which indicates stabilization)

Table 1 Baseline parameters

120572 119888 120591 119862 + 119888119879 + 119866 119884

09 0625 02 200 400

For convenience of calculation we will observe the sign ofRe(119889120582119889120579

2)

minus1|

120582=119894120596lowast instead of that of 119889Re120582119889120579

2|

120582=119894120596lowast

Using (19) we can demonstrate that Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt

0 holds for any value of ℎ (see the appendix) Thus with anycrossing the signs of the real parts of the roots change fromnegative to positive

If a policy lag does not present (1205792= 0) then roots with

positive real parts do not exist under Assumption 1 Withincreases in 120579

2 the number of roots with positive real parts

increases by two Accordingly we can state that 2ℎ roots withpositive real parts exist in the region of 120579

2isin (120579

2(ℎminus1) 120579

2ℎ)

(where 1205792(minus1)

= 0) Therefore we can offer the followingproposition

Proposition 2 For 1205792isin (0 120579

20) the steady state is stable and

for 1205792isin (120579

20infin) it is unstable

This proposition suggests that even if a monetary policyis sufficiently active enough to satisfy Assumption 1 aneconomy can become unstable due to the existence of a policylag A similar result has already been shown by Asada andYoshida [12] concerning fiscal policies

32 Numerical Simulations Let us consider a numericalexample Following the study of Asada and Yoshida [12] weset baseline parameters as shown in Table 1

Furthermore Asada and Yoshida [12] employ the invest-ment function that is highly nonlinear with respect to 119884(119905)which is formulated as follows (in their model 120574 = 0)

119868 =

119884

1 + 12119890

minus01(119884(119905)minus119884)minus 119903 minus 05119870 (119905) minus

119884

13

+ 02 + 30

119903 = minus005120574 (119884 minus 119884 (119905 minus 120579

2)) + 001

radic119884 (119905)

(28)

Based on these assumptions the steady-state values areprovided by (119884lowast 119870lowast) = (400 60)

We also set the baseline values of 120573 and 120574 as 25 and60 respectively (generally it seems that the responsivenessof monetary policies to economic fluctuations is often largerthan that of fiscal policies therefore we assume 120574 gt 120573)When 120574 is fixed at the baseline value 120596lowast exists only for 120573 isin

(149 208) Outside this range a policy lag does not affectstability In addition Assumption 1 is satisfied for 120573 gt 149Accordingly irrespective of the value of a monetary policylag the steady state is unstable for 0 lt 120573 lt 149 and it isstable for 120573 gt 208

Likewise when 120573 is fixed at the baseline value 120596lowast existsfor 120574 gt 1432 In addition Assumption 1 is satisfied for 120574 gtminus1432 Hence the steady state is stable for 0 lt 120574 lt 1432irrespective of the value of a lag

Figure 1 describes 1205792ℎ

(ℎ = 0 1 2) that corresponds to120596

lowast as a function of 120573 and 120574 These curves are referred to as

Discrete Dynamics in Nature and Society 5

16 18 20

5

10

15

16 18 20 220

2

4

6

8

10

12

14

120573 120574

1205792

1205792

12057922

12057921

12057920

12057922

12057921

12057920

Figure 1 Crossing curves

14 145 15 155 16 165 17 175 18250

300

350

400

450

500

Y

120574

Figure 2 Bifurcation diagram (case with (28))

ldquocrossing curvesrdquo Here the stability condition is expressedas 1205792lt 120579

20 Therefore we can state that an increase in 120573 has a

stabilizing effect whereas that in 120574 has a destabilizing effectThus an increase in the activeness level of a monetary

policymay cause instability due to the existence of amonetarypolicy lag A similar result concerning a fiscal policy hasalready been shown by Asada and Yoshida [12]

Incidentally conditions for Hopf bifurcation are satisfiedat 1205792= 120579

20 that is (i) (19) includes a pair of pure imaginary

roots and (ii) 119889Re1205821198891205792

= 0 Therefore when 120579

2crosses

120579

20 a cycle emerges A similar argument can be developed

using 120573 and 120574 We now suppose that 1205792= 20 In this case

a bifurcation diagram can be described for 120574 isin [14 18] asshown in Figure 2 (the vertical axis represents the maximal

15 155 16 165 17250

300

350

400

450

500

Y

120574

Figure 3 Bifurcation diagram (case with (29))

and minimal values of 119884(119905)) This figure shows that not onlyone period but also multiple period cycles can emerge Theemergence ofmultiple period cycles is attributed to the strongnonlinearity of the investment function with respect to 119884(119905)When we use the investment function that has relativelysimpler form with respect to 119884(119905) as below (we set thecoefficient of radic119884(119905) as 1136 so that the value of 119868

119884equals

that of (28)) the bifurcation diagram becomes as shown inFigure 3 In this case only one period cycle exists Hence

119868 = 1136radic119884 (119905) minus 119903 minus 05119870 (119905) minus 1136

radic

119884 + 02

+ 30

(29)

6 Discrete Dynamics in Nature and Society

Although Asada and Yoshida [12] do not clearly specifythe time dimension similar models that include a disequi-librium adjustment function of the goods market (120572 is setaround unity) specify the measure of 119905 as a year (refer to egYoshida and Asada [5]) Therefore we will consider the timedimension as a year For example when 120573 = 18 the steadystate is unstable for 120579

2gt 14 years

4 Case of Policies with Fiscal andMonetary Policy Lags

In this section we investigate the case wherein both fiscal andmonetary policy lags exist (ie 120579

1gt 0 and 120579

2gt 0) In order to

examine the signs of the roots of (15) we use a mathematicalmethod developed by Gu et al [17]

41 Preconditions First to apply the technique of Gu et al[17] some preconditions should be examined According totheir study (15) should satisfy the following conditions

(I) deg(1199010(120582)) ge maxdeg(119901

1(120582)) deg(119901

2(120582))

(II) Δ(0) = 0(III) A solution common to all three polynomials 119901

0(120582) =

0 1199011(120582) = 0 and 119901

2(120582) = 0 does not exist

(IV) lim120582rarrinfin

(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) lt 1

Condition (I) is satisfied by 2 gt max1 1 while Condi-tion (II) is also satisfied by Δ(0) = minus120572119868

1198701 minus 119888(1 minus 120591) + 120573 gt 0

Condition (III) is obviously satisfied Finally Condition (IV)is satisfied by lim

120582rarrinfin(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) = 0

Now we examine the effects of lags (1205791 120579

2) on the stability

of the steady state The analysis proceeds as follows

(1) We characterize the sets (1205791 120579

2) (ie crossing sets) at

which the pure imaginary roots appear (it is ensuredfrom precondition (II) that zero cannot be a root)

(2) We depict the crossing sets on the (120579

1 120579

2) plane

(which we refer to as the crossing curves) via numer-ical simulation

(3) We reveal the directions of changes in the signs ofthe real parts that occur when lags (120579

1 120579

2) cross the

crossing curves

42 Crossing Sets Dividing (15) by 1199010(120582) we obtain

1 + 119886

1 (120582) 119890

minus1205791120582+ 119886

2 (120582) 119890

minus1205792120582= 0

(30)

where

119886

1 (120582) =

119901

1 (120582)

119901

0 (120582)

=

120572120573 (120582 minus 119868

119870)

120582

2+ 119887

1120582 + 119887

2

119886

2 (120582) =

119901

2 (120582)

119901

0 (120582)

= minus

120572119860

2120582

120582

2+ 119887

1120582 + 119887

2

(31)

Denoting a pure imaginary root by 120582 = 119894120596 the values of120596that satisfy (30) can be characterized by the following lemma

0

Im

1

a2(i120596)eminusi1205961205792

1205752 1205751

Re

a1(i120596)eminusi 1120596120579

Figure 4 Triangle formed by 1 |1198861(119894120596)| and |119886

2(119894120596)| on the complex

plane

Lemma 3 (see [17 Proposition 31]) For each 120596 satisfying119901

0(119894120596) = 0 120582 = 119894120596 is a solution of Δ(120582) = 0 for some

(120579

1 120579

2) isin R2+if and only if

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

ge 1

minus1 le

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

le 1

(32)

We denote the set of 120596 gt 0 that satisfy conditions (32)as Ω which is termed the ldquocrossing frequencyrdquo For all 120596 isin

Ω the sets (1205791 120579

2) satisfying (30) (ie crossing sets) should

satisfy the following conditions (Figure 4)

∓120575

1= arg (119886

1 (119894120596) 119890

minus1198941205961205791) + 2119898120587 119898 = 0 1 2

plusmn120575

2= arg (119886

2 (119894120596) 119890

minus1198941205961205792) + 2119899120587 119899 = 0 1 2

(33)

where 1205751 120575

2isin [0 120587]

Incidentally on the complex plane multiplication ofamplitudes becomes a sum of the parts therefore we obtain

arg (1198861 (119894120596) 119890

minus1198941205961205791) = arg (119886

1 (119894120596)) minus 1205961205791

arg (1198862 (119894120596) 119890

minus1198941205961205792) = arg (119886

2 (119894120596)) minus 1205961205792

(34)

In addition Figure 4 demonstrates that the followingrelations hold

arg (1198861 (119894120596)) = tanminus1 (

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

)

arg (1198862 (119894120596)) = tanminus1 (

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

)

(35)

Moreover after some manipulation (31) derive the fol-lowing expressions

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

=

119887

1120596119868

119870+ 120596 (119887

2minus 120596

2)

119887

1120596

2minus 119868

119870(119887

2minus 120596

2)

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

=

119887

2minus 120596

2

119887

1120596

(36)

Discrete Dynamics in Nature and Society 7

Thus using (34) to (36) (33) can be rewritten as follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) plusmn 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) ∓ 120575

2+ 2119899120587

120596

(37)

where the interior angles of the triangle denoted by 1205751and

120575

2are given by the cosine theorem as follows

120575

1= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572120573119868

119870)

2+ (120572120573120596)

2minus (120572119860

2120596)

2

2

radic

(120572120573119868

119870)

2+ (120572120573120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

120575

2= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572119860

2120596)

2minus (120572120573119868

119870)

2minus (120572120573120596)

2

2

radic

(120572119860

2120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

(38)

Equations (37) characterize the crossing sets (1205791 120579

2) isin

R2+ Depending on the signs of 120575

1and 120575

2 we can define two

types of crossing sets denoted by 1198711(119898 119899) and 119871

2(119898 119899) as

follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) + 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) minus 120575

2+ 2119899120587

120596

119871

1(119898 119899)

120579

1=

tanminus1 ((1198871120596119868

119870+ V (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) minus 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) + 120575

2+ 2119899120587

120596

119871

2(119898 119899)

As in the previous section we set parameter values andthe functional form of the investment function as shownin Table 1 (28) or (29) 120573 = 25 and 120574 = 60 In thiscase crossing frequency Ω is provided by 120596 isin (150 213)

(Figure 5) (according to Gu et alrsquos [17] classification Figure 5exhibits Type 13) For 120596 isin Ω describing the crossing sets119871

1(119898 119899) and 119871

2(119898 119899) on the 120579

1-1205792plane we obtain Figure 6

where119898 = 0 1 2 and 119899 = 0 1 2 The dotted curves represent119871

1(119898 119899) and the solid curves represent 119871

2(119898 119899) The points

marked by a dot indicate the starting points of both curvesthat is the points corresponding to 120596 = 150

We establish how the real parts of the roots change whenlags cross the crossing curves with an increase in 120579

1 For 120596 isin

Ω if Re(1198891205821198891205791)

minus1gt 0 then the real parts of the roots change

from negative to positive this indicates destabilization Incontrast if Re(119889120582119889120579

1)

minus1lt 0 then the real parts of the roots

change from positive to negative this indicates stabilizationDifferentiating (30) with respect to 120579

1 we obtain

[119886

1015840

1(120582) 119890

minus1205791120582minus 119886

1 (120582) 119890

minus1205791120582120579

1+ 119886

1015840

2(120582) 119890

minus1205792120582

minus 119886

2 (120582) 119890

minus1205792120579

2]

119889120582

119889120579

1

= 119886

1 (120582) 119890

minus1205791120582120582

(39)

8 Discrete Dynamics in Nature and Society

0 2 4 6 8 10

0

1

2

3

4

5

6

minus1

|a1(i120596)| + |a2(i120596)|

120596

Ω|a1(i120596)| minus |a2(i120596)|

Figure 5 Crossing frequency Ω

or equivalently

(

119889120582

119889120579

1

)

minus1

=

119886

1015840

1(120582) 119890

minus1205791120582+ 119886

1015840

2(120582) 119890

minus1205792120582minus 119886

2 (120582) 119890

minus1205792120582120579

2

119886

1 (120582) 119890

minus1205791120582120582

minus

120579

1

120582

(40)

where

119886

1015840

1(120582) =

120572120573119901

0 (120582) minus 120572120573 (120582 minus 119868119870

) (2120582 + 119887

1)

119901

0 (120582)

2

119886

1015840

2(120582) = minus

120572119860

2119901

0 (120582) minus 1205721198602

120582 (2120582 + 119887

1)

119901

0 (120582)

2

(41)

Describing the real parts of (40) as a function of 120596 isin Ωwe obtain Figure 7 where the dotted curves are the functionsevaluated on curve 119871

1and the solid curves are the functions

evaluated on curve 1198712 As for curve 119871

1 stabilization only

occurs on the intermediate region of the segments 1198711(119898 2)

destabilization occurs on the other segments of 1198711 As for

curve 1198712 destabilization occurs on all segments Here we

define a curve formed by connecting curves 119871119895(0 119899) (where

119895 = 1 2 and 119899 = 0 1 2) as 1198980 (Figure 8) We also definea curve formed by connecting the points on curve 1198980 thatminimize the value of 120579

1for a given 120579

2as curve 1198720 Then

we can express the stability condition as follows set (1205791 120579

2)

exists in the left-hand side region of curve1198720On curve 1198720 conditions for Hopf bifurcation are sat-

isfied and therefore a cycle emerges In addition as isobvious from Figure 6 an increase in the fiscal policy lag 120579

1

necessarily destabilizes the steady state However an increasein the monetary policy lag 120579

2has the potential to stabilize

the steady state which depends on the value of 1205791 When

120579

1isin (022 036) we can see that the steady state fluctuates

between stability and instability with an increase in 1205792 Such

a phenomenon cannot be observed in models with only onetime lag Moreover Figure 9 presents a bifurcation diagram

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

L2(0 2)

L1(0 2)

L2(0 1)

L1(0 1)

L2(0 0)

L1(0 0)

L2(1 0)

L1(1 0) L1(2 0)

L2(2 0)

L2(2 1)

L1(2 1)

L2(2 2)

L1(2 2)

L2(1 2)

L1(1 2)

L2(1 1)

L1(1 1)

1205792

1205791

Figure 6 Crossing curves

wherein 1205791= 025 and (28) is used as the investment function

This figure evidently indicates that alternations of stabilityand instability emerge with an increase in 120579

2

In particular when 1205791= 025 stability can be established

by setting 1205792isin (0 106) 120579

2isin (253 433) or 120579

2isin (671 761)

that is for equilibrium stability monetary policy lag shouldbe less than one year between 3 and 45 years or between 68and 75 years when a fiscal policy lag is approximately threemonths

Next we examine how 120573 and 120574 affect the stabilitycondition Both 120573 and 120574 affect 119871

1and 119871

2through 120596 isin Ω It is

difficult to analytically show how 119871

1and 119871

2change Hence

we numerically examine how curve 1198980 (which comprises119871

1(0 119899) and 119871

2(0 119899)) shifts when 120573 and 120574 increase from each

baseline value (120573 = 25 and 120574 = 60)Figures 10 and 11 describe curve 1198980 in cases with 120573 = 8

and 120573 = 15 respectively We can observe that an increasein 120573 shifts curve 1198980 to left Accordingly the stability region119878 reduces Likewise Figures 12 and 13 describe curve 1198980 incaseswith 120574 = 12 and 120574 = 22 respectivelyThe stability regionis also reduced by an increase in 120574 These results suggestthat increases in the activeness levels of fiscal and monetarypolicies increase economic instability as is the case with onlyone policy lag

43 Investment and Stability In general the formulation ofthe investment function has great significance to stabilityin macrodynamic models that is the larger the marginalpropensity to investment is the more likely it becomesunstable In this subsection we examine the effects of changesin the partial derivatives of the investment function 119868

119884and

119868

119870 on stability We compare three cases the case of no policy

lags the case of the existence of onlymonetary policy lag andthe case of the existence of both fiscal and monetary policylags

In the casewith no policy lags 119868119884and 119868119870affect the stability

condition given by Assumption 1 through 1198871 The smaller the

values of 119868119884and 119868119870are (the larger the absolute value of 119868

119870is)

Discrete Dynamics in Nature and Society 9

L1(0 0) and L2(0 0) L1(0 1) and L2(0 1) L1(0 2) and L2(0 2)

L1(1 0) and L2(1 0) L1(1 1) and L2(1 1) L1(1 2) and L2(1 2)

L1(2 0) and L2(2 0) L1(2 1) and L2(2 1) L1(2 2) and L2(2 2)

0 1 2 30

02

04Re

(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

Figure 7 Direction of crossing

0 01 02 03 04 050

1

2

3

4

5

6

7

8

9

S1205792

1205791

Figure 8 Stability region 119878

the more likely the stability condition is satisfied These arefairly typical results

In the case of a monetary policy with a lag 119868119884and 119868119870

affect the value of 120596lowast through 119887

2(119868119870

also directly affects120596

lowast) Accordingly the value of (27) changes In this case itis difficult to analytically show the quality and quantity of

1 2 3 4 5 6 7 8 9350

360

370

380

390

400

410

420

430

440

450

Y

1205792

Figure 9 Bifurcation diagram (1205791= 025)

these effects Hence we resort to a numerical method Basedon the parameter values proposed in Section 32 crossingcurves 120579

20can be described within each range of 119868

119884and

119868

119870that ensure the existence of 120596lowast as shown in Figure 14

The region located below these curves indicates the stabilityregions Similar to the case with no policy lags an increase

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

16 18 20

5

10

15

16 18 20 220

2

4

6

8

10

12

14

120573 120574

1205792

1205792

12057922

12057921

12057920

12057922

12057921

12057920

Figure 1 Crossing curves

14 145 15 155 16 165 17 175 18250

300

350

400

450

500

Y

120574

Figure 2 Bifurcation diagram (case with (28))

ldquocrossing curvesrdquo Here the stability condition is expressedas 1205792lt 120579

20 Therefore we can state that an increase in 120573 has a

stabilizing effect whereas that in 120574 has a destabilizing effectThus an increase in the activeness level of a monetary

policymay cause instability due to the existence of amonetarypolicy lag A similar result concerning a fiscal policy hasalready been shown by Asada and Yoshida [12]

Incidentally conditions for Hopf bifurcation are satisfiedat 1205792= 120579

20 that is (i) (19) includes a pair of pure imaginary

roots and (ii) 119889Re1205821198891205792

= 0 Therefore when 120579

2crosses

120579

20 a cycle emerges A similar argument can be developed

using 120573 and 120574 We now suppose that 1205792= 20 In this case

a bifurcation diagram can be described for 120574 isin [14 18] asshown in Figure 2 (the vertical axis represents the maximal

15 155 16 165 17250

300

350

400

450

500

Y

120574

Figure 3 Bifurcation diagram (case with (29))

and minimal values of 119884(119905)) This figure shows that not onlyone period but also multiple period cycles can emerge Theemergence ofmultiple period cycles is attributed to the strongnonlinearity of the investment function with respect to 119884(119905)When we use the investment function that has relativelysimpler form with respect to 119884(119905) as below (we set thecoefficient of radic119884(119905) as 1136 so that the value of 119868

119884equals

that of (28)) the bifurcation diagram becomes as shown inFigure 3 In this case only one period cycle exists Hence

119868 = 1136radic119884 (119905) minus 119903 minus 05119870 (119905) minus 1136

radic

119884 + 02

+ 30

(29)

6 Discrete Dynamics in Nature and Society

Although Asada and Yoshida [12] do not clearly specifythe time dimension similar models that include a disequi-librium adjustment function of the goods market (120572 is setaround unity) specify the measure of 119905 as a year (refer to egYoshida and Asada [5]) Therefore we will consider the timedimension as a year For example when 120573 = 18 the steadystate is unstable for 120579

2gt 14 years

4 Case of Policies with Fiscal andMonetary Policy Lags

In this section we investigate the case wherein both fiscal andmonetary policy lags exist (ie 120579

1gt 0 and 120579

2gt 0) In order to

examine the signs of the roots of (15) we use a mathematicalmethod developed by Gu et al [17]

41 Preconditions First to apply the technique of Gu et al[17] some preconditions should be examined According totheir study (15) should satisfy the following conditions

(I) deg(1199010(120582)) ge maxdeg(119901

1(120582)) deg(119901

2(120582))

(II) Δ(0) = 0(III) A solution common to all three polynomials 119901

0(120582) =

0 1199011(120582) = 0 and 119901

2(120582) = 0 does not exist

(IV) lim120582rarrinfin

(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) lt 1

Condition (I) is satisfied by 2 gt max1 1 while Condi-tion (II) is also satisfied by Δ(0) = minus120572119868

1198701 minus 119888(1 minus 120591) + 120573 gt 0

Condition (III) is obviously satisfied Finally Condition (IV)is satisfied by lim

120582rarrinfin(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) = 0

Now we examine the effects of lags (1205791 120579

2) on the stability

of the steady state The analysis proceeds as follows

(1) We characterize the sets (1205791 120579

2) (ie crossing sets) at

which the pure imaginary roots appear (it is ensuredfrom precondition (II) that zero cannot be a root)

(2) We depict the crossing sets on the (120579

1 120579

2) plane

(which we refer to as the crossing curves) via numer-ical simulation

(3) We reveal the directions of changes in the signs ofthe real parts that occur when lags (120579

1 120579

2) cross the

crossing curves

42 Crossing Sets Dividing (15) by 1199010(120582) we obtain

1 + 119886

1 (120582) 119890

minus1205791120582+ 119886

2 (120582) 119890

minus1205792120582= 0

(30)

where

119886

1 (120582) =

119901

1 (120582)

119901

0 (120582)

=

120572120573 (120582 minus 119868

119870)

120582

2+ 119887

1120582 + 119887

2

119886

2 (120582) =

119901

2 (120582)

119901

0 (120582)

= minus

120572119860

2120582

120582

2+ 119887

1120582 + 119887

2

(31)

Denoting a pure imaginary root by 120582 = 119894120596 the values of120596that satisfy (30) can be characterized by the following lemma

0

Im

1

a2(i120596)eminusi1205961205792

1205752 1205751

Re

a1(i120596)eminusi 1120596120579

Figure 4 Triangle formed by 1 |1198861(119894120596)| and |119886

2(119894120596)| on the complex

plane

Lemma 3 (see [17 Proposition 31]) For each 120596 satisfying119901

0(119894120596) = 0 120582 = 119894120596 is a solution of Δ(120582) = 0 for some

(120579

1 120579

2) isin R2+if and only if

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

ge 1

minus1 le

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

le 1

(32)

We denote the set of 120596 gt 0 that satisfy conditions (32)as Ω which is termed the ldquocrossing frequencyrdquo For all 120596 isin

Ω the sets (1205791 120579

2) satisfying (30) (ie crossing sets) should

satisfy the following conditions (Figure 4)

∓120575

1= arg (119886

1 (119894120596) 119890

minus1198941205961205791) + 2119898120587 119898 = 0 1 2

plusmn120575

2= arg (119886

2 (119894120596) 119890

minus1198941205961205792) + 2119899120587 119899 = 0 1 2

(33)

where 1205751 120575

2isin [0 120587]

Incidentally on the complex plane multiplication ofamplitudes becomes a sum of the parts therefore we obtain

arg (1198861 (119894120596) 119890

minus1198941205961205791) = arg (119886

1 (119894120596)) minus 1205961205791

arg (1198862 (119894120596) 119890

minus1198941205961205792) = arg (119886

2 (119894120596)) minus 1205961205792

(34)

In addition Figure 4 demonstrates that the followingrelations hold

arg (1198861 (119894120596)) = tanminus1 (

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

)

arg (1198862 (119894120596)) = tanminus1 (

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

)

(35)

Moreover after some manipulation (31) derive the fol-lowing expressions

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

=

119887

1120596119868

119870+ 120596 (119887

2minus 120596

2)

119887

1120596

2minus 119868

119870(119887

2minus 120596

2)

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

=

119887

2minus 120596

2

119887

1120596

(36)

Discrete Dynamics in Nature and Society 7

Thus using (34) to (36) (33) can be rewritten as follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) plusmn 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) ∓ 120575

2+ 2119899120587

120596

(37)

where the interior angles of the triangle denoted by 1205751and

120575

2are given by the cosine theorem as follows

120575

1= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572120573119868

119870)

2+ (120572120573120596)

2minus (120572119860

2120596)

2

2

radic

(120572120573119868

119870)

2+ (120572120573120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

120575

2= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572119860

2120596)

2minus (120572120573119868

119870)

2minus (120572120573120596)

2

2

radic

(120572119860

2120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

(38)

Equations (37) characterize the crossing sets (1205791 120579

2) isin

R2+ Depending on the signs of 120575

1and 120575

2 we can define two

types of crossing sets denoted by 1198711(119898 119899) and 119871

2(119898 119899) as

follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) + 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) minus 120575

2+ 2119899120587

120596

119871

1(119898 119899)

120579

1=

tanminus1 ((1198871120596119868

119870+ V (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) minus 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) + 120575

2+ 2119899120587

120596

119871

2(119898 119899)

As in the previous section we set parameter values andthe functional form of the investment function as shownin Table 1 (28) or (29) 120573 = 25 and 120574 = 60 In thiscase crossing frequency Ω is provided by 120596 isin (150 213)

(Figure 5) (according to Gu et alrsquos [17] classification Figure 5exhibits Type 13) For 120596 isin Ω describing the crossing sets119871

1(119898 119899) and 119871

2(119898 119899) on the 120579

1-1205792plane we obtain Figure 6

where119898 = 0 1 2 and 119899 = 0 1 2 The dotted curves represent119871

1(119898 119899) and the solid curves represent 119871

2(119898 119899) The points

marked by a dot indicate the starting points of both curvesthat is the points corresponding to 120596 = 150

We establish how the real parts of the roots change whenlags cross the crossing curves with an increase in 120579

1 For 120596 isin

Ω if Re(1198891205821198891205791)

minus1gt 0 then the real parts of the roots change

from negative to positive this indicates destabilization Incontrast if Re(119889120582119889120579

1)

minus1lt 0 then the real parts of the roots

change from positive to negative this indicates stabilizationDifferentiating (30) with respect to 120579

1 we obtain

[119886

1015840

1(120582) 119890

minus1205791120582minus 119886

1 (120582) 119890

minus1205791120582120579

1+ 119886

1015840

2(120582) 119890

minus1205792120582

minus 119886

2 (120582) 119890

minus1205792120579

2]

119889120582

119889120579

1

= 119886

1 (120582) 119890

minus1205791120582120582

(39)

8 Discrete Dynamics in Nature and Society

0 2 4 6 8 10

0

1

2

3

4

5

6

minus1

|a1(i120596)| + |a2(i120596)|

120596

Ω|a1(i120596)| minus |a2(i120596)|

Figure 5 Crossing frequency Ω

or equivalently

(

119889120582

119889120579

1

)

minus1

=

119886

1015840

1(120582) 119890

minus1205791120582+ 119886

1015840

2(120582) 119890

minus1205792120582minus 119886

2 (120582) 119890

minus1205792120582120579

2

119886

1 (120582) 119890

minus1205791120582120582

minus

120579

1

120582

(40)

where

119886

1015840

1(120582) =

120572120573119901

0 (120582) minus 120572120573 (120582 minus 119868119870

) (2120582 + 119887

1)

119901

0 (120582)

2

119886

1015840

2(120582) = minus

120572119860

2119901

0 (120582) minus 1205721198602

120582 (2120582 + 119887

1)

119901

0 (120582)

2

(41)

Describing the real parts of (40) as a function of 120596 isin Ωwe obtain Figure 7 where the dotted curves are the functionsevaluated on curve 119871

1and the solid curves are the functions

evaluated on curve 1198712 As for curve 119871

1 stabilization only

occurs on the intermediate region of the segments 1198711(119898 2)

destabilization occurs on the other segments of 1198711 As for

curve 1198712 destabilization occurs on all segments Here we

define a curve formed by connecting curves 119871119895(0 119899) (where

119895 = 1 2 and 119899 = 0 1 2) as 1198980 (Figure 8) We also definea curve formed by connecting the points on curve 1198980 thatminimize the value of 120579

1for a given 120579

2as curve 1198720 Then

we can express the stability condition as follows set (1205791 120579

2)

exists in the left-hand side region of curve1198720On curve 1198720 conditions for Hopf bifurcation are sat-

isfied and therefore a cycle emerges In addition as isobvious from Figure 6 an increase in the fiscal policy lag 120579

1

necessarily destabilizes the steady state However an increasein the monetary policy lag 120579

2has the potential to stabilize

the steady state which depends on the value of 1205791 When

120579

1isin (022 036) we can see that the steady state fluctuates

between stability and instability with an increase in 1205792 Such

a phenomenon cannot be observed in models with only onetime lag Moreover Figure 9 presents a bifurcation diagram

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

L2(0 2)

L1(0 2)

L2(0 1)

L1(0 1)

L2(0 0)

L1(0 0)

L2(1 0)

L1(1 0) L1(2 0)

L2(2 0)

L2(2 1)

L1(2 1)

L2(2 2)

L1(2 2)

L2(1 2)

L1(1 2)

L2(1 1)

L1(1 1)

1205792

1205791

Figure 6 Crossing curves

wherein 1205791= 025 and (28) is used as the investment function

This figure evidently indicates that alternations of stabilityand instability emerge with an increase in 120579

2

In particular when 1205791= 025 stability can be established

by setting 1205792isin (0 106) 120579

2isin (253 433) or 120579

2isin (671 761)

that is for equilibrium stability monetary policy lag shouldbe less than one year between 3 and 45 years or between 68and 75 years when a fiscal policy lag is approximately threemonths

Next we examine how 120573 and 120574 affect the stabilitycondition Both 120573 and 120574 affect 119871

1and 119871

2through 120596 isin Ω It is

difficult to analytically show how 119871

1and 119871

2change Hence

we numerically examine how curve 1198980 (which comprises119871

1(0 119899) and 119871

2(0 119899)) shifts when 120573 and 120574 increase from each

baseline value (120573 = 25 and 120574 = 60)Figures 10 and 11 describe curve 1198980 in cases with 120573 = 8

and 120573 = 15 respectively We can observe that an increasein 120573 shifts curve 1198980 to left Accordingly the stability region119878 reduces Likewise Figures 12 and 13 describe curve 1198980 incaseswith 120574 = 12 and 120574 = 22 respectivelyThe stability regionis also reduced by an increase in 120574 These results suggestthat increases in the activeness levels of fiscal and monetarypolicies increase economic instability as is the case with onlyone policy lag

43 Investment and Stability In general the formulation ofthe investment function has great significance to stabilityin macrodynamic models that is the larger the marginalpropensity to investment is the more likely it becomesunstable In this subsection we examine the effects of changesin the partial derivatives of the investment function 119868

119884and

119868

119870 on stability We compare three cases the case of no policy

lags the case of the existence of onlymonetary policy lag andthe case of the existence of both fiscal and monetary policylags

In the casewith no policy lags 119868119884and 119868119870affect the stability

condition given by Assumption 1 through 1198871 The smaller the

values of 119868119884and 119868119870are (the larger the absolute value of 119868

119870is)

Discrete Dynamics in Nature and Society 9

L1(0 0) and L2(0 0) L1(0 1) and L2(0 1) L1(0 2) and L2(0 2)

L1(1 0) and L2(1 0) L1(1 1) and L2(1 1) L1(1 2) and L2(1 2)

L1(2 0) and L2(2 0) L1(2 1) and L2(2 1) L1(2 2) and L2(2 2)

0 1 2 30

02

04Re

(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

Figure 7 Direction of crossing

0 01 02 03 04 050

1

2

3

4

5

6

7

8

9

S1205792

1205791

Figure 8 Stability region 119878

the more likely the stability condition is satisfied These arefairly typical results

In the case of a monetary policy with a lag 119868119884and 119868119870

affect the value of 120596lowast through 119887

2(119868119870

also directly affects120596

lowast) Accordingly the value of (27) changes In this case itis difficult to analytically show the quality and quantity of

1 2 3 4 5 6 7 8 9350

360

370

380

390

400

410

420

430

440

450

Y

1205792

Figure 9 Bifurcation diagram (1205791= 025)

these effects Hence we resort to a numerical method Basedon the parameter values proposed in Section 32 crossingcurves 120579

20can be described within each range of 119868

119884and

119868

119870that ensure the existence of 120596lowast as shown in Figure 14

The region located below these curves indicates the stabilityregions Similar to the case with no policy lags an increase

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Although Asada and Yoshida [12] do not clearly specifythe time dimension similar models that include a disequi-librium adjustment function of the goods market (120572 is setaround unity) specify the measure of 119905 as a year (refer to egYoshida and Asada [5]) Therefore we will consider the timedimension as a year For example when 120573 = 18 the steadystate is unstable for 120579

2gt 14 years

4 Case of Policies with Fiscal andMonetary Policy Lags

In this section we investigate the case wherein both fiscal andmonetary policy lags exist (ie 120579

1gt 0 and 120579

2gt 0) In order to

examine the signs of the roots of (15) we use a mathematicalmethod developed by Gu et al [17]

41 Preconditions First to apply the technique of Gu et al[17] some preconditions should be examined According totheir study (15) should satisfy the following conditions

(I) deg(1199010(120582)) ge maxdeg(119901

1(120582)) deg(119901

2(120582))

(II) Δ(0) = 0(III) A solution common to all three polynomials 119901

0(120582) =

0 1199011(120582) = 0 and 119901

2(120582) = 0 does not exist

(IV) lim120582rarrinfin

(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) lt 1

Condition (I) is satisfied by 2 gt max1 1 while Condi-tion (II) is also satisfied by Δ(0) = minus120572119868

1198701 minus 119888(1 minus 120591) + 120573 gt 0

Condition (III) is obviously satisfied Finally Condition (IV)is satisfied by lim

120582rarrinfin(|119901

1(120582)119901

0(120582)| + |119901

2(120582)119901

0(120582)|) = 0

Now we examine the effects of lags (1205791 120579

2) on the stability

of the steady state The analysis proceeds as follows

(1) We characterize the sets (1205791 120579

2) (ie crossing sets) at

which the pure imaginary roots appear (it is ensuredfrom precondition (II) that zero cannot be a root)

(2) We depict the crossing sets on the (120579

1 120579

2) plane

(which we refer to as the crossing curves) via numer-ical simulation

(3) We reveal the directions of changes in the signs ofthe real parts that occur when lags (120579

1 120579

2) cross the

crossing curves

42 Crossing Sets Dividing (15) by 1199010(120582) we obtain

1 + 119886

1 (120582) 119890

minus1205791120582+ 119886

2 (120582) 119890

minus1205792120582= 0

(30)

where

119886

1 (120582) =

119901

1 (120582)

119901

0 (120582)

=

120572120573 (120582 minus 119868

119870)

120582

2+ 119887

1120582 + 119887

2

119886

2 (120582) =

119901

2 (120582)

119901

0 (120582)

= minus

120572119860

2120582

120582

2+ 119887

1120582 + 119887

2

(31)

Denoting a pure imaginary root by 120582 = 119894120596 the values of120596that satisfy (30) can be characterized by the following lemma

0

Im

1

a2(i120596)eminusi1205961205792

1205752 1205751

Re

a1(i120596)eminusi 1120596120579

Figure 4 Triangle formed by 1 |1198861(119894120596)| and |119886

2(119894120596)| on the complex

plane

Lemma 3 (see [17 Proposition 31]) For each 120596 satisfying119901

0(119894120596) = 0 120582 = 119894120596 is a solution of Δ(120582) = 0 for some

(120579

1 120579

2) isin R2+if and only if

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

ge 1

minus1 le

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

le 1

(32)

We denote the set of 120596 gt 0 that satisfy conditions (32)as Ω which is termed the ldquocrossing frequencyrdquo For all 120596 isin

Ω the sets (1205791 120579

2) satisfying (30) (ie crossing sets) should

satisfy the following conditions (Figure 4)

∓120575

1= arg (119886

1 (119894120596) 119890

minus1198941205961205791) + 2119898120587 119898 = 0 1 2

plusmn120575

2= arg (119886

2 (119894120596) 119890

minus1198941205961205792) + 2119899120587 119899 = 0 1 2

(33)

where 1205751 120575

2isin [0 120587]

Incidentally on the complex plane multiplication ofamplitudes becomes a sum of the parts therefore we obtain

arg (1198861 (119894120596) 119890

minus1198941205961205791) = arg (119886

1 (119894120596)) minus 1205961205791

arg (1198862 (119894120596) 119890

minus1198941205961205792) = arg (119886

2 (119894120596)) minus 1205961205792

(34)

In addition Figure 4 demonstrates that the followingrelations hold

arg (1198861 (119894120596)) = tanminus1 (

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

)

arg (1198862 (119894120596)) = tanminus1 (

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

)

(35)

Moreover after some manipulation (31) derive the fol-lowing expressions

Im (119886

1 (119894120596))

Re (1198861 (119894120596))

=

119887

1120596119868

119870+ 120596 (119887

2minus 120596

2)

119887

1120596

2minus 119868

119870(119887

2minus 120596

2)

Im (119886

2 (119894120596))

Re (1198862 (119894120596))

=

119887

2minus 120596

2

119887

1120596

(36)

Discrete Dynamics in Nature and Society 7

Thus using (34) to (36) (33) can be rewritten as follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) plusmn 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) ∓ 120575

2+ 2119899120587

120596

(37)

where the interior angles of the triangle denoted by 1205751and

120575

2are given by the cosine theorem as follows

120575

1= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572120573119868

119870)

2+ (120572120573120596)

2minus (120572119860

2120596)

2

2

radic

(120572120573119868

119870)

2+ (120572120573120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

120575

2= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572119860

2120596)

2minus (120572120573119868

119870)

2minus (120572120573120596)

2

2

radic

(120572119860

2120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

(38)

Equations (37) characterize the crossing sets (1205791 120579

2) isin

R2+ Depending on the signs of 120575

1and 120575

2 we can define two

types of crossing sets denoted by 1198711(119898 119899) and 119871

2(119898 119899) as

follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) + 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) minus 120575

2+ 2119899120587

120596

119871

1(119898 119899)

120579

1=

tanminus1 ((1198871120596119868

119870+ V (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) minus 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) + 120575

2+ 2119899120587

120596

119871

2(119898 119899)

As in the previous section we set parameter values andthe functional form of the investment function as shownin Table 1 (28) or (29) 120573 = 25 and 120574 = 60 In thiscase crossing frequency Ω is provided by 120596 isin (150 213)

(Figure 5) (according to Gu et alrsquos [17] classification Figure 5exhibits Type 13) For 120596 isin Ω describing the crossing sets119871

1(119898 119899) and 119871

2(119898 119899) on the 120579

1-1205792plane we obtain Figure 6

where119898 = 0 1 2 and 119899 = 0 1 2 The dotted curves represent119871

1(119898 119899) and the solid curves represent 119871

2(119898 119899) The points

marked by a dot indicate the starting points of both curvesthat is the points corresponding to 120596 = 150

We establish how the real parts of the roots change whenlags cross the crossing curves with an increase in 120579

1 For 120596 isin

Ω if Re(1198891205821198891205791)

minus1gt 0 then the real parts of the roots change

from negative to positive this indicates destabilization Incontrast if Re(119889120582119889120579

1)

minus1lt 0 then the real parts of the roots

change from positive to negative this indicates stabilizationDifferentiating (30) with respect to 120579

1 we obtain

[119886

1015840

1(120582) 119890

minus1205791120582minus 119886

1 (120582) 119890

minus1205791120582120579

1+ 119886

1015840

2(120582) 119890

minus1205792120582

minus 119886

2 (120582) 119890

minus1205792120579

2]

119889120582

119889120579

1

= 119886

1 (120582) 119890

minus1205791120582120582

(39)

8 Discrete Dynamics in Nature and Society

0 2 4 6 8 10

0

1

2

3

4

5

6

minus1

|a1(i120596)| + |a2(i120596)|

120596

Ω|a1(i120596)| minus |a2(i120596)|

Figure 5 Crossing frequency Ω

or equivalently

(

119889120582

119889120579

1

)

minus1

=

119886

1015840

1(120582) 119890

minus1205791120582+ 119886

1015840

2(120582) 119890

minus1205792120582minus 119886

2 (120582) 119890

minus1205792120582120579

2

119886

1 (120582) 119890

minus1205791120582120582

minus

120579

1

120582

(40)

where

119886

1015840

1(120582) =

120572120573119901

0 (120582) minus 120572120573 (120582 minus 119868119870

) (2120582 + 119887

1)

119901

0 (120582)

2

119886

1015840

2(120582) = minus

120572119860

2119901

0 (120582) minus 1205721198602

120582 (2120582 + 119887

1)

119901

0 (120582)

2

(41)

Describing the real parts of (40) as a function of 120596 isin Ωwe obtain Figure 7 where the dotted curves are the functionsevaluated on curve 119871

1and the solid curves are the functions

evaluated on curve 1198712 As for curve 119871

1 stabilization only

occurs on the intermediate region of the segments 1198711(119898 2)

destabilization occurs on the other segments of 1198711 As for

curve 1198712 destabilization occurs on all segments Here we

define a curve formed by connecting curves 119871119895(0 119899) (where

119895 = 1 2 and 119899 = 0 1 2) as 1198980 (Figure 8) We also definea curve formed by connecting the points on curve 1198980 thatminimize the value of 120579

1for a given 120579

2as curve 1198720 Then

we can express the stability condition as follows set (1205791 120579

2)

exists in the left-hand side region of curve1198720On curve 1198720 conditions for Hopf bifurcation are sat-

isfied and therefore a cycle emerges In addition as isobvious from Figure 6 an increase in the fiscal policy lag 120579

1

necessarily destabilizes the steady state However an increasein the monetary policy lag 120579

2has the potential to stabilize

the steady state which depends on the value of 1205791 When

120579

1isin (022 036) we can see that the steady state fluctuates

between stability and instability with an increase in 1205792 Such

a phenomenon cannot be observed in models with only onetime lag Moreover Figure 9 presents a bifurcation diagram

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

L2(0 2)

L1(0 2)

L2(0 1)

L1(0 1)

L2(0 0)

L1(0 0)

L2(1 0)

L1(1 0) L1(2 0)

L2(2 0)

L2(2 1)

L1(2 1)

L2(2 2)

L1(2 2)

L2(1 2)

L1(1 2)

L2(1 1)

L1(1 1)

1205792

1205791

Figure 6 Crossing curves

wherein 1205791= 025 and (28) is used as the investment function

This figure evidently indicates that alternations of stabilityand instability emerge with an increase in 120579

2

In particular when 1205791= 025 stability can be established

by setting 1205792isin (0 106) 120579

2isin (253 433) or 120579

2isin (671 761)

that is for equilibrium stability monetary policy lag shouldbe less than one year between 3 and 45 years or between 68and 75 years when a fiscal policy lag is approximately threemonths

Next we examine how 120573 and 120574 affect the stabilitycondition Both 120573 and 120574 affect 119871

1and 119871

2through 120596 isin Ω It is

difficult to analytically show how 119871

1and 119871

2change Hence

we numerically examine how curve 1198980 (which comprises119871

1(0 119899) and 119871

2(0 119899)) shifts when 120573 and 120574 increase from each

baseline value (120573 = 25 and 120574 = 60)Figures 10 and 11 describe curve 1198980 in cases with 120573 = 8

and 120573 = 15 respectively We can observe that an increasein 120573 shifts curve 1198980 to left Accordingly the stability region119878 reduces Likewise Figures 12 and 13 describe curve 1198980 incaseswith 120574 = 12 and 120574 = 22 respectivelyThe stability regionis also reduced by an increase in 120574 These results suggestthat increases in the activeness levels of fiscal and monetarypolicies increase economic instability as is the case with onlyone policy lag

43 Investment and Stability In general the formulation ofthe investment function has great significance to stabilityin macrodynamic models that is the larger the marginalpropensity to investment is the more likely it becomesunstable In this subsection we examine the effects of changesin the partial derivatives of the investment function 119868

119884and

119868

119870 on stability We compare three cases the case of no policy

lags the case of the existence of onlymonetary policy lag andthe case of the existence of both fiscal and monetary policylags

In the casewith no policy lags 119868119884and 119868119870affect the stability

condition given by Assumption 1 through 1198871 The smaller the

values of 119868119884and 119868119870are (the larger the absolute value of 119868

119870is)

Discrete Dynamics in Nature and Society 9

L1(0 0) and L2(0 0) L1(0 1) and L2(0 1) L1(0 2) and L2(0 2)

L1(1 0) and L2(1 0) L1(1 1) and L2(1 1) L1(1 2) and L2(1 2)

L1(2 0) and L2(2 0) L1(2 1) and L2(2 1) L1(2 2) and L2(2 2)

0 1 2 30

02

04Re

(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

Figure 7 Direction of crossing

0 01 02 03 04 050

1

2

3

4

5

6

7

8

9

S1205792

1205791

Figure 8 Stability region 119878

the more likely the stability condition is satisfied These arefairly typical results

In the case of a monetary policy with a lag 119868119884and 119868119870

affect the value of 120596lowast through 119887

2(119868119870

also directly affects120596

lowast) Accordingly the value of (27) changes In this case itis difficult to analytically show the quality and quantity of

1 2 3 4 5 6 7 8 9350

360

370

380

390

400

410

420

430

440

450

Y

1205792

Figure 9 Bifurcation diagram (1205791= 025)

these effects Hence we resort to a numerical method Basedon the parameter values proposed in Section 32 crossingcurves 120579

20can be described within each range of 119868

119884and

119868

119870that ensure the existence of 120596lowast as shown in Figure 14

The region located below these curves indicates the stabilityregions Similar to the case with no policy lags an increase

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

Thus using (34) to (36) (33) can be rewritten as follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) plusmn 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) ∓ 120575

2+ 2119899120587

120596

(37)

where the interior angles of the triangle denoted by 1205751and

120575

2are given by the cosine theorem as follows

120575

1= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572120573119868

119870)

2+ (120572120573120596)

2minus (120572119860

2120596)

2

2

radic

(120572120573119868

119870)

2+ (120572120573120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

120575

2= cosminus1(

1 +

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

2minus

1003816

1003816

1003816

1003816

119886

1 (119894120596)

1003816

1003816

1003816

1003816

2

2

1003816

1003816

1003816

1003816

119886

2 (119894120596)

1003816

1003816

1003816

1003816

) = cosminus1((119887

2minus 120596

2)

2

+ (119887

1120596)

2+ (120572119860

2120596)

2minus (120572120573119868

119870)

2minus (120572120573120596)

2

2

radic

(120572119860

2120596)

2radic

(119887

2minus 120596

2)

2+ (119887

1120596)

2

)

(38)

Equations (37) characterize the crossing sets (1205791 120579

2) isin

R2+ Depending on the signs of 120575

1and 120575

2 we can define two

types of crossing sets denoted by 1198711(119898 119899) and 119871

2(119898 119899) as

follows

120579

1=

tanminus1 ((1198871120596119868

119870+ 120596 (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) + 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) minus 120575

2+ 2119899120587

120596

119871

1(119898 119899)

120579

1=

tanminus1 ((1198871120596119868

119870+ V (119887

2minus 120596

2)) (119887

1120596

2minus 119868

119870(119887

2minus 120596

2))) minus 120575

1+ 2119898120587

120596

120579

2=

tanminus1 ((1198872minus 120596

2) 119887

1120596) + 120575

2+ 2119899120587

120596

119871

2(119898 119899)

As in the previous section we set parameter values andthe functional form of the investment function as shownin Table 1 (28) or (29) 120573 = 25 and 120574 = 60 In thiscase crossing frequency Ω is provided by 120596 isin (150 213)

(Figure 5) (according to Gu et alrsquos [17] classification Figure 5exhibits Type 13) For 120596 isin Ω describing the crossing sets119871

1(119898 119899) and 119871

2(119898 119899) on the 120579

1-1205792plane we obtain Figure 6

where119898 = 0 1 2 and 119899 = 0 1 2 The dotted curves represent119871

1(119898 119899) and the solid curves represent 119871

2(119898 119899) The points

marked by a dot indicate the starting points of both curvesthat is the points corresponding to 120596 = 150

We establish how the real parts of the roots change whenlags cross the crossing curves with an increase in 120579

1 For 120596 isin

Ω if Re(1198891205821198891205791)

minus1gt 0 then the real parts of the roots change

from negative to positive this indicates destabilization Incontrast if Re(119889120582119889120579

1)

minus1lt 0 then the real parts of the roots

change from positive to negative this indicates stabilizationDifferentiating (30) with respect to 120579

1 we obtain

[119886

1015840

1(120582) 119890

minus1205791120582minus 119886

1 (120582) 119890

minus1205791120582120579

1+ 119886

1015840

2(120582) 119890

minus1205792120582

minus 119886

2 (120582) 119890

minus1205792120579

2]

119889120582

119889120579

1

= 119886

1 (120582) 119890

minus1205791120582120582

(39)

8 Discrete Dynamics in Nature and Society

0 2 4 6 8 10

0

1

2

3

4

5

6

minus1

|a1(i120596)| + |a2(i120596)|

120596

Ω|a1(i120596)| minus |a2(i120596)|

Figure 5 Crossing frequency Ω

or equivalently

(

119889120582

119889120579

1

)

minus1

=

119886

1015840

1(120582) 119890

minus1205791120582+ 119886

1015840

2(120582) 119890

minus1205792120582minus 119886

2 (120582) 119890

minus1205792120582120579

2

119886

1 (120582) 119890

minus1205791120582120582

minus

120579

1

120582

(40)

where

119886

1015840

1(120582) =

120572120573119901

0 (120582) minus 120572120573 (120582 minus 119868119870

) (2120582 + 119887

1)

119901

0 (120582)

2

119886

1015840

2(120582) = minus

120572119860

2119901

0 (120582) minus 1205721198602

120582 (2120582 + 119887

1)

119901

0 (120582)

2

(41)

Describing the real parts of (40) as a function of 120596 isin Ωwe obtain Figure 7 where the dotted curves are the functionsevaluated on curve 119871

1and the solid curves are the functions

evaluated on curve 1198712 As for curve 119871

1 stabilization only

occurs on the intermediate region of the segments 1198711(119898 2)

destabilization occurs on the other segments of 1198711 As for

curve 1198712 destabilization occurs on all segments Here we

define a curve formed by connecting curves 119871119895(0 119899) (where

119895 = 1 2 and 119899 = 0 1 2) as 1198980 (Figure 8) We also definea curve formed by connecting the points on curve 1198980 thatminimize the value of 120579

1for a given 120579

2as curve 1198720 Then

we can express the stability condition as follows set (1205791 120579

2)

exists in the left-hand side region of curve1198720On curve 1198720 conditions for Hopf bifurcation are sat-

isfied and therefore a cycle emerges In addition as isobvious from Figure 6 an increase in the fiscal policy lag 120579

1

necessarily destabilizes the steady state However an increasein the monetary policy lag 120579

2has the potential to stabilize

the steady state which depends on the value of 1205791 When

120579

1isin (022 036) we can see that the steady state fluctuates

between stability and instability with an increase in 1205792 Such

a phenomenon cannot be observed in models with only onetime lag Moreover Figure 9 presents a bifurcation diagram

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

L2(0 2)

L1(0 2)

L2(0 1)

L1(0 1)

L2(0 0)

L1(0 0)

L2(1 0)

L1(1 0) L1(2 0)

L2(2 0)

L2(2 1)

L1(2 1)

L2(2 2)

L1(2 2)

L2(1 2)

L1(1 2)

L2(1 1)

L1(1 1)

1205792

1205791

Figure 6 Crossing curves

wherein 1205791= 025 and (28) is used as the investment function

This figure evidently indicates that alternations of stabilityand instability emerge with an increase in 120579

2

In particular when 1205791= 025 stability can be established

by setting 1205792isin (0 106) 120579

2isin (253 433) or 120579

2isin (671 761)

that is for equilibrium stability monetary policy lag shouldbe less than one year between 3 and 45 years or between 68and 75 years when a fiscal policy lag is approximately threemonths

Next we examine how 120573 and 120574 affect the stabilitycondition Both 120573 and 120574 affect 119871

1and 119871

2through 120596 isin Ω It is

difficult to analytically show how 119871

1and 119871

2change Hence

we numerically examine how curve 1198980 (which comprises119871

1(0 119899) and 119871

2(0 119899)) shifts when 120573 and 120574 increase from each

baseline value (120573 = 25 and 120574 = 60)Figures 10 and 11 describe curve 1198980 in cases with 120573 = 8

and 120573 = 15 respectively We can observe that an increasein 120573 shifts curve 1198980 to left Accordingly the stability region119878 reduces Likewise Figures 12 and 13 describe curve 1198980 incaseswith 120574 = 12 and 120574 = 22 respectivelyThe stability regionis also reduced by an increase in 120574 These results suggestthat increases in the activeness levels of fiscal and monetarypolicies increase economic instability as is the case with onlyone policy lag

43 Investment and Stability In general the formulation ofthe investment function has great significance to stabilityin macrodynamic models that is the larger the marginalpropensity to investment is the more likely it becomesunstable In this subsection we examine the effects of changesin the partial derivatives of the investment function 119868

119884and

119868

119870 on stability We compare three cases the case of no policy

lags the case of the existence of onlymonetary policy lag andthe case of the existence of both fiscal and monetary policylags

In the casewith no policy lags 119868119884and 119868119870affect the stability

condition given by Assumption 1 through 1198871 The smaller the

values of 119868119884and 119868119870are (the larger the absolute value of 119868

119870is)

Discrete Dynamics in Nature and Society 9

L1(0 0) and L2(0 0) L1(0 1) and L2(0 1) L1(0 2) and L2(0 2)

L1(1 0) and L2(1 0) L1(1 1) and L2(1 1) L1(1 2) and L2(1 2)

L1(2 0) and L2(2 0) L1(2 1) and L2(2 1) L1(2 2) and L2(2 2)

0 1 2 30

02

04Re

(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

Figure 7 Direction of crossing

0 01 02 03 04 050

1

2

3

4

5

6

7

8

9

S1205792

1205791

Figure 8 Stability region 119878

the more likely the stability condition is satisfied These arefairly typical results

In the case of a monetary policy with a lag 119868119884and 119868119870

affect the value of 120596lowast through 119887

2(119868119870

also directly affects120596

lowast) Accordingly the value of (27) changes In this case itis difficult to analytically show the quality and quantity of

1 2 3 4 5 6 7 8 9350

360

370

380

390

400

410

420

430

440

450

Y

1205792

Figure 9 Bifurcation diagram (1205791= 025)

these effects Hence we resort to a numerical method Basedon the parameter values proposed in Section 32 crossingcurves 120579

20can be described within each range of 119868

119884and

119868

119870that ensure the existence of 120596lowast as shown in Figure 14

The region located below these curves indicates the stabilityregions Similar to the case with no policy lags an increase

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

0 2 4 6 8 10

0

1

2

3

4

5

6

minus1

|a1(i120596)| + |a2(i120596)|

120596

Ω|a1(i120596)| minus |a2(i120596)|

Figure 5 Crossing frequency Ω

or equivalently

(

119889120582

119889120579

1

)

minus1

=

119886

1015840

1(120582) 119890

minus1205791120582+ 119886

1015840

2(120582) 119890

minus1205792120582minus 119886

2 (120582) 119890

minus1205792120582120579

2

119886

1 (120582) 119890

minus1205791120582120582

minus

120579

1

120582

(40)

where

119886

1015840

1(120582) =

120572120573119901

0 (120582) minus 120572120573 (120582 minus 119868119870

) (2120582 + 119887

1)

119901

0 (120582)

2

119886

1015840

2(120582) = minus

120572119860

2119901

0 (120582) minus 1205721198602

120582 (2120582 + 119887

1)

119901

0 (120582)

2

(41)

Describing the real parts of (40) as a function of 120596 isin Ωwe obtain Figure 7 where the dotted curves are the functionsevaluated on curve 119871

1and the solid curves are the functions

evaluated on curve 1198712 As for curve 119871

1 stabilization only

occurs on the intermediate region of the segments 1198711(119898 2)

destabilization occurs on the other segments of 1198711 As for

curve 1198712 destabilization occurs on all segments Here we

define a curve formed by connecting curves 119871119895(0 119899) (where

119895 = 1 2 and 119899 = 0 1 2) as 1198980 (Figure 8) We also definea curve formed by connecting the points on curve 1198980 thatminimize the value of 120579

1for a given 120579

2as curve 1198720 Then

we can express the stability condition as follows set (1205791 120579

2)

exists in the left-hand side region of curve1198720On curve 1198720 conditions for Hopf bifurcation are sat-

isfied and therefore a cycle emerges In addition as isobvious from Figure 6 an increase in the fiscal policy lag 120579

1

necessarily destabilizes the steady state However an increasein the monetary policy lag 120579

2has the potential to stabilize

the steady state which depends on the value of 1205791 When

120579

1isin (022 036) we can see that the steady state fluctuates

between stability and instability with an increase in 1205792 Such

a phenomenon cannot be observed in models with only onetime lag Moreover Figure 9 presents a bifurcation diagram

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

L2(0 2)

L1(0 2)

L2(0 1)

L1(0 1)

L2(0 0)

L1(0 0)

L2(1 0)

L1(1 0) L1(2 0)

L2(2 0)

L2(2 1)

L1(2 1)

L2(2 2)

L1(2 2)

L2(1 2)

L1(1 2)

L2(1 1)

L1(1 1)

1205792

1205791

Figure 6 Crossing curves

wherein 1205791= 025 and (28) is used as the investment function

This figure evidently indicates that alternations of stabilityand instability emerge with an increase in 120579

2

In particular when 1205791= 025 stability can be established

by setting 1205792isin (0 106) 120579

2isin (253 433) or 120579

2isin (671 761)

that is for equilibrium stability monetary policy lag shouldbe less than one year between 3 and 45 years or between 68and 75 years when a fiscal policy lag is approximately threemonths

Next we examine how 120573 and 120574 affect the stabilitycondition Both 120573 and 120574 affect 119871

1and 119871

2through 120596 isin Ω It is

difficult to analytically show how 119871

1and 119871

2change Hence

we numerically examine how curve 1198980 (which comprises119871

1(0 119899) and 119871

2(0 119899)) shifts when 120573 and 120574 increase from each

baseline value (120573 = 25 and 120574 = 60)Figures 10 and 11 describe curve 1198980 in cases with 120573 = 8

and 120573 = 15 respectively We can observe that an increasein 120573 shifts curve 1198980 to left Accordingly the stability region119878 reduces Likewise Figures 12 and 13 describe curve 1198980 incaseswith 120574 = 12 and 120574 = 22 respectivelyThe stability regionis also reduced by an increase in 120574 These results suggestthat increases in the activeness levels of fiscal and monetarypolicies increase economic instability as is the case with onlyone policy lag

43 Investment and Stability In general the formulation ofthe investment function has great significance to stabilityin macrodynamic models that is the larger the marginalpropensity to investment is the more likely it becomesunstable In this subsection we examine the effects of changesin the partial derivatives of the investment function 119868

119884and

119868

119870 on stability We compare three cases the case of no policy

lags the case of the existence of onlymonetary policy lag andthe case of the existence of both fiscal and monetary policylags

In the casewith no policy lags 119868119884and 119868119870affect the stability

condition given by Assumption 1 through 1198871 The smaller the

values of 119868119884and 119868119870are (the larger the absolute value of 119868

119870is)

Discrete Dynamics in Nature and Society 9

L1(0 0) and L2(0 0) L1(0 1) and L2(0 1) L1(0 2) and L2(0 2)

L1(1 0) and L2(1 0) L1(1 1) and L2(1 1) L1(1 2) and L2(1 2)

L1(2 0) and L2(2 0) L1(2 1) and L2(2 1) L1(2 2) and L2(2 2)

0 1 2 30

02

04Re

(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

Figure 7 Direction of crossing

0 01 02 03 04 050

1

2

3

4

5

6

7

8

9

S1205792

1205791

Figure 8 Stability region 119878

the more likely the stability condition is satisfied These arefairly typical results

In the case of a monetary policy with a lag 119868119884and 119868119870

affect the value of 120596lowast through 119887

2(119868119870

also directly affects120596

lowast) Accordingly the value of (27) changes In this case itis difficult to analytically show the quality and quantity of

1 2 3 4 5 6 7 8 9350

360

370

380

390

400

410

420

430

440

450

Y

1205792

Figure 9 Bifurcation diagram (1205791= 025)

these effects Hence we resort to a numerical method Basedon the parameter values proposed in Section 32 crossingcurves 120579

20can be described within each range of 119868

119884and

119868

119870that ensure the existence of 120596lowast as shown in Figure 14

The region located below these curves indicates the stabilityregions Similar to the case with no policy lags an increase

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

L1(0 0) and L2(0 0) L1(0 1) and L2(0 1) L1(0 2) and L2(0 2)

L1(1 0) and L2(1 0) L1(1 1) and L2(1 1) L1(1 2) and L2(1 2)

L1(2 0) and L2(2 0) L1(2 1) and L2(2 1) L1(2 2) and L2(2 2)

0 1 2 30

02

04Re

(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

0 1 2 30

02

04

Re(d120582d1205791)minus

1

120596

0 1 2 30

05

1

Re(d120582d1205791)minus

1

120596

0

05

1

Re(d120582d1205791)minus

1

0 1 2 3120596

minus05

Figure 7 Direction of crossing

0 01 02 03 04 050

1

2

3

4

5

6

7

8

9

S1205792

1205791

Figure 8 Stability region 119878

the more likely the stability condition is satisfied These arefairly typical results

In the case of a monetary policy with a lag 119868119884and 119868119870

affect the value of 120596lowast through 119887

2(119868119870

also directly affects120596

lowast) Accordingly the value of (27) changes In this case itis difficult to analytically show the quality and quantity of

1 2 3 4 5 6 7 8 9350

360

370

380

390

400

410

420

430

440

450

Y

1205792

Figure 9 Bifurcation diagram (1205791= 025)

these effects Hence we resort to a numerical method Basedon the parameter values proposed in Section 32 crossingcurves 120579

20can be described within each range of 119868

119884and

119868

119870that ensure the existence of 120596lowast as shown in Figure 14

The region located below these curves indicates the stabilityregions Similar to the case with no policy lags an increase

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

0 005 01 015 02 025 030

05

1

15

2

25

1205792

1205791

S

Figure 10 120573 = 8

0 005 01 015 02 025 030

02

04

06

08

1

12

S

1205792

1205791

Figure 11 120573 = 15

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

9

10

S

1205792

1205791

Figure 12 120574 = 12

0 01 02 03 04 05 060

2

4

6

8

10

12

14

S

1205792

1205791

Figure 13 120574 = 22

in 119868119884causes destabilization However due to the concavity of

the crossing curve there is a possibility that a decrease in 119868119870

causes destabilizationThus the existence of a lag complicatesthe effects of investment on stability

Finally we investigate the case wherein both fiscal andmonetary policy lags exist In this case 119868

119884and 119868119870affect the

stability condition (ie location of curve1198720) through120596 isin Ω119887

1 and 119887

2 According to numerical simulations increases in

the absolute values of 119868119884and 119868119870generally shift the curve1198720

toward the leftTherefore these changes decrease the stabilityregion However if1198720 completely shifts even to the negativeregion of 120579

1 then the curve that specifies the stability region

is no longer 1198720 but is 1198721 For example when 119868

119884= 30

the stability region 119878 is defined as the left-hand side regionof the curve 1198720 in Figure 15 However when 119868

119884= 40 the

stability region 119878 is defined as the left-hand side region of thecurve1198721 in Figure 16 Thus an increase in 119868

119884may increase

the stability region

5 Conclusion

In this study we analyzed the effects of fiscal and monetarypolicy lags on economic stability In Section 3 we examinedthe case of a monetary policy with a lag We confirmed thatthe results of Asada and Yoshida [12] that examined thecase of a fiscal policy with a lag also basically hold in ourmodel that is policy lags may cause economic instability andsuppress the effect of expeditious policies aimed at economicstability (a model with a fiscal policy that does not consider apolicy lag is proposed by Asada [21])

Moreover in Section 4 we examined the case whereinboth fiscal and monetary policy lags exist and demonstratedthat in the neighborhood of a stability crossing curve amonetary policy lag can have a stabilizing effect

According to Friedman [22] policy lags can generally beclassified into three types recognition implementation anddiffusion lags Unlike recognition and diffusion lags imple-mentation lag can be considered as purposefully adjustable to

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

34 36 380

05

1

15

2

25

3

045

5

55

6

65

7

75

8

85

9

95

12057920

12057920

minus01 minus005

IY IK

Figure 14 Crossing curve 12057920

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

M0 M1

S

1205792

1205791

Figure 15 119868119884= 30

some extentTherefore this study suggests that an adjustmentof the timing of policy implementation can be a means toachieve stabilization

Appendix

Direction of Crossing

Differentiating (19) with respect to 1205792 we obtain

2120582 + (119887

1+ 120572120573) minus 120572119860

2119890

minus1205792120582+ 120572119860

2120582120579

2119890

minus1205792120582

119889120582

119889120579

2

= minus120572119860

2120582

2119890

minus1205792120582

(A1)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

M1M0

S

minus2

1205792

1205791

Figure 16 119868119884= 40

or equivalently

(

119889120582

119889120579

2

)

minus1

=

minus (2120582 + 119887

1+ 120572120573) 119890

1205792120582

120572119860

2120582

2+

1

120582

2minus

120579

2

120582

(A2)

From (19) we can express 1198901205792120582 as follows

119890

1205792120582=

120572119860

2120582

120582

2+ (119887

1+ 120572120573) 120582 + (119887

2minus 120572120573119868

119870)

(A3)

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Discrete Dynamics in Nature and Society

Substituting (A3) into (A2) and by evaluation at 120582 = 119894120596lowastwe obtain

Re( 119889120582119889120579

2

)

minus110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816120582=119894120596lowast

= Re[minus (2119894120596

lowast+ 119887

1+ 120572120573)

119894120596

lowastminus120596

lowast2+ (119887

1+ 120572120573) 119894120596

lowast+ (119887

2minus 120572120573119868

119870)

minus

1

120596

lowast2

minus

120579

2

119894120596

lowast] =

(119887

1+ 120572120573)

2120596

lowast2+ 2120596 120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

minus

1

120596

lowast2

=

2120596

lowast120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast minus 120596

lowast2minus (119887

2minus 120572120573119868

119870)

2

120596

lowast3minus (119887

2minus 120572120573119868

119870) 120596

lowast

2

+ (119887

1+ 120572120573)120596

lowast2

2

(A4)

The denominator of the right-hand side of this expressionis positive Thus we only have to observe the sign of thenumerator The numerator is expanded as 120596lowast4 minus (119887

2minus 120572120573119868

119870)

Furthermore using (26) it is rewritten as follows

120596

lowast4minus (119887

2minus 120572120573119868

119870) = 120596

lowast2radic

119873(A5)

As119873 gt 0 we obtain Re(1198891205821198891205792)

minus1|

120582=119894120596lowast gt 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthor appreciates the helpful comments and suggestionsreceived from an anonymous referee and Professors AkioMatsumoto and Toichiro Asada (Chuo University)

References

[1] M Kalecki ldquoAmacrodynamic theory of business cyclesrdquo Econo-metrica vol 3 no 3 pp 327ndash344 1935

[2] M Szydłowski ldquoTime-to-build in dynamics of economic mod-els I Kaleckirsquos modelrdquo Chaos Solitons amp Fractals vol 14 no 5pp 697ndash703 2002

[3] M Szydłowski ldquoTime to build in dynamics of economicmodels II Models of economic growthrdquo Chaos Solitons ampFractals vol 18 no 2 pp 355ndash364 2003

[4] L Fanti and P Manfredi ldquoChaotic business cycles and fiscalpolicy an IS-LM model with distributed tax collection lagsrdquoChaos Solitons amp Fractals vol 32 no 2 pp 736ndash744 2007

[5] H Yoshida and T Asada ldquoDynamic analysis of policy lag in aKeynes-Goodwin model stability instability cycles and chaosrdquoJournal of Economic Behavior amp Organization vol 62 no 3 pp441ndash469 2007

[6] G J Schinasi ldquoA nonlinear dynamic model of short runfluctuationsrdquoReview of Economic Studies vol 48 no 4 pp 649ndash656 1981

[7] K Sasakura ldquoOn the dynamic behavior of Schinasirsquos businesscycle modelrdquo Journal of Macroeconomics vol 16 no 3 pp 423ndash444 1994

[8] L De Cesare and M Sportelli ldquoA dynamic IS-LM model withdelayed taxation revenuesrdquo Chaos Solitons amp Fractals vol 25no 1 pp 233ndash244 2005

[9] A Matsumoto and F Szidarovszky ldquoDynamics in delay IS-LM model with tax collectionsrdquo IERCU Discussion Paper 208Chuo University Hachioji Japan 2013

[10] N Kaldor ldquoA model of the trade cyclerdquo The Economic Journalvol 50 no 197 pp 78ndash82 1940

[11] WW Chang and D J Smyth ldquoThe existence and persistence ofcycles in a non-linearmodel Kaldorrsquos 1940model re-examinedrdquoThe Review of Economic Studies vol 38 no 1 pp 37ndash44 1971

[12] T Asada and H Yoshida ldquoStability instability and complexbehavior in macrodynamic models with policy lagrdquo DiscreteDynamics in Nature and Society vol 5 no 4 pp 281ndash295 2001

[13] L Zhou and Y Li ldquoA dynamic IS-LM business cycle model withtwo time delays in capital accumulation equationrdquo Journal ofComputational andAppliedMathematics vol 228 no 1 pp 182ndash187 2009

[14] J Cai ldquoHopf bifurcation in the IS-LMbusiness cyclemodel withtime delayrdquo Electronic Journal of Differential Equations vol 15pp 1ndash6 2005

[15] M Sportelli L De Cesare and M T Binetti ldquoA dynamic IS-LM model with two time delays in the public sectorrdquo AppliedMathematics and Computation vol 243 pp 728ndash739 2014

[16] A Matsumoto and F Szidarovszky ldquoNonlinear multiplier-accelerator model with investment and consumption delaysrdquoStructural Change andEconomicDynamics vol 33 pp 1ndash9 2015

[17] K Gu S-I Niculescu and J Chen ldquoOn stability crossing curvesfor general systems with two delaysrdquo Journal of MathematicalAnalysis and Applications vol 311 no 1 pp 231ndash253 2005

[18] G Gandolfo Economic Dynamics Springer Berlin Germany4th edition 2010

[19] A Matsumoto and F Szidarovszky ldquoAn elementary study of aclass of dynamic systems with two time delaysrdquo Cubo vol 14no 3 pp 103ndash113 2012

[20] R E Bellman and K L Cooke Differential-Difference Equa-tions Academic Press New York NY USA 1963

[21] T Asada ldquoGovernment finance and wealth effect in a Kaldoriancycle modelrdquo Journal of Economics vol 47 no 2 pp 143ndash1661987

[22] M Friedman ldquoA monetary and fiscal framework for economicstabilityrdquoTheAmerican Economic Review vol 38 no 3 pp 245ndash264 1948

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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