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Time elements and oscillatory fluctuations in the Keynesian macroeconomic system

  • Hiroki Murakami EMAIL logo

Abstract

In this paper, we discuss the relationship between three time lags (the consumption lag, the investment decision lag and the gestation lag) and oscillatory economic fluctuations in the Keynesian IS-LM system. We first confirm that in the absence of time lags, the monotone convergence to the unique equilibrium is observed. Next, we demonstrate that, in the Keynesian IS-LM system, in the existence of the investment decision lag and the gestation lag, oscillatory fluctuations are generated around the equilibrium if the gestation lag is relatively long, while in the existence of the consumption lag and the gestation lag, oscillatory behaviors may not occur. We then conclude that the existence of time lags may be one of the major causes of oscillatory fluctuations.

JEL Classification: E12; E21; E22; E31; E32; E41; E44

Corresponding author: Hiroki Murakami, Research Fellow (PD), Japan Society for the Promotion of Science; Graduate School of Economics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Acknowledgments

The author would like to thank Hideyuki Adachi, Kosuke Aoki, Toichiro Asada, Cars Hommes, Akio Matsumoto, Takeshi Nakatani, Yasutaka Niisato, Kenshiro Ninomiya, Takashi Ohno, Hiroshi Osaka, Masahiro Ouchi, Neri Salvadori, Hiroaki Sasaki, Peter Skott, Kazuo Ueda, Hiroyuki Yoshida, Hiroshi Yoshikawa (alphabetical order) and the anonymous referee for their valuable comments on an earlier version of this paper. He is also grateful to Natsuki Kamakura for her help in the presentation of an earlier version of this paper. Needless to say, the author is solely responsible for the possible remaining errors in this paper. This work was financially supported by the Japan Society for the Promotion of Science (Grant in Aid for JSPS Fellows, Grant Number 14J03350).

Appendix

A Stability switching of System (L-I)

In this section, we shall analyze for what values of θd and θg stability switching occurs to (39) of System (L-I).

For this purpose, we shall impose the following assumption.

Assumption 5The following condition is satisfied:

(56)(1CY+IY)Lr+(CrIr)LY<0.

Under Assumption 1, condition (56) is fulfilled either if the marginal effect of the rate of interest on investment is relatively weak or if the marginal effect of aggregate income on liquidity preference (or on demand for money) is weak compared with that of the rate of interest on liquidity preference (at the equilibrium). In particular, in the case of “Keynesian liquidity trap,” in which the magnitude of Lr is large enough, condition (56) is satisfied. In this sense, Assumption 5 does not seem unrealistic.

The characteristic equation (39) can be written as follows:

(57)P0(λ)+P1(λ)eθdλ+P2(λ)eθgλ+P3(λ)e(θd+θg)λ=0,

where

(58)P0(λ)=λ,P1(λ)=αλ,P2(λ)=0,P3(λ)=IK.

Putting λ= in (57), [23] we have the following two expressions, which are mathematically equivalent to each other:

(59)[P0(iω)+P1(iω)eiωθd]+[P2(iω)+P3(iω)eiωθd]eiωθg=0,
(60)[P0(iω)+P2(iω)eiωθg]+[P1(iω)+P3(iω)eiωθg]eiωθd=0.

By definition, stability switching happens to (39) when λ is a root of (39), where i is the imaginary unit and ω is a positive number. By employing the method invented by Lin and Wang (2012), [24] we shall derive the combinations of θd and θg for which stability switching occurs.

Proposition 9.Let Assumptions 1, 2 and 5 hold.

  1. Assume that the following condition is satisfied, which implies that α=0:

    (61)IYLr=IrLY,

    Then, all the roots of Eq. (39) have negative real parts if and only if the following condition is satisfied:

    (62)IK(θd+θg)<π2.
  2. Assume that the following condition is satisfied, which implies that α≠0:

(63)IYLrIrLY.

Then, the stability switching curves of Eq. (39) are the loci of (θd , θg )∈ΘI defined as follows: [25]

(64)ΘI={(φd,m±(ω),φg,n±(ω))R++2:ω[IK1+|α|,IK1|α|],m,nZ},

where

(65)φd,m±(ω)=[±arccos((1+α2)ω2IK22αω2)+2mπ]/ω,
(66)φg,n±(ω)=[±arccos((1α2)ω2IK22αIKω)π2+2nπ]/ω.

Proof. (i) Since we have α=0 in this case, we can apply the result in Bellman and Cooke (1963, 444, Theorem 13.8) directly to the proof of the assertion.

(ii) To begin, we can confirm from (58) that, for every (θd,θg)R++2:

deg(P0(λ))=1=max{deg(P1(λ)), deg(P2(λ)), deg(P3(λ))}P0(0)+P1(0)+P2(0)+P3(0)=IK>0.limλ(|P1(λ)P0(λ)|,|P2(λ)P0(λ)|,|P3(λ)P0(λ)|)=|α|<1,

and that P0, P1, P2 and P3 have no common zero. Then, we can apply the method by Lin and Wang (2012) to this proof.

Now we shall derive the stability switching curves in (39), following Lin and Wang (2012). Stability switching happens to (39) when there exist (θd , θg ) and ω>0 such that Eqs. (59) and (60) are satisfied. Because of |exp(−iωθd )|=|exp(−iωθg )|=1, for (59) and (60) to hold, we must have:

|P0(iω)+P1(iω)eiωθd|=|P2(iω)+P3(iω)eiωθd|,|P0(iω)+P2(iω)eiωθg|=|P1(iω)+P3(iω)eiωθg|,

which are, under (58), equivalent to:

(67)(1+α2)ω2IK2=2αωcos(ωθd),
(68)(1α2)ω2IK2=2αIKωsin(ωθg)=2αIKωcos(ωθg+π2).

For some positive ω to satisfy (67) and (68), it is necessary and sufficient that the following conditions are fulfilled for some ω>0:

|(1+γ2)ω2IK2|2|α|ω,|(1γ2)ω2IK2|2|α|IKω,

which are equivalent to:

[(1+α)2ω2IK2][(1α)2ω2IK2]0.

or

(69)ω[IK1+|α|,IK1|α|].

Take an arbitrary ω that satisfies (69). Then, one can solve (67) and (68) for θd and θg , respectively, as:

θd=[±arccos((1+α2)ω2IK22αω2)+2mπ]/ω,θg=[±arccos((1α2)ω2IK22αIKω)π2+2nπ]/ω.

Thus, combinations of (θd,θg)R++2 that satisfy (67) and (68) are given by (64). One can confirm that every (θd , θg )∈ΘI solves (59) and (60).

Therefore, we can conclude from Lin and Wang (2012, 526, Theorem 3.1) that the set of all stability switching curves is given by (64).□

Proposition 9 provides the information on the stability switching curves of (36) in System (L-I), i.e. on the combinations of (θd , θg ) for which the characteristic equation (39) has at least one pair of purely imaginary roots. This proposition may suggest the possibility that periodic orbits are generated by Hopf bifurcations along the stability switching curves. [26]

To examine the possibility of periodic orbits generated by Hopf bifurcations in System (L-I), we shall proceed to perform a numerical simulation. For this purpose, we shall specify the functional forms of C, I and L as follows: [27]

(70)L(r,Y)=Y100r+800,
(71)C(r,Y)=0.7Y+120,
(72)I(r,Y,K)=1001+exp(0.01(Y400))5r0.2K+160.

Also, the value of M is fixed as:

(73)M=1000.

Note that, under these specifications, Assumptions 1, 3 and 4 are all fulfilled. [28]

Under the specifications of (70)–(73), the equilibrium point of System (L-I), (r*, Y*, K*), can be calculated as follows: [29]

(74)(r,Y,K)=(2,400,1000).

One can easily confirm that, under (70)–(74), the partial derivatives of C, I and L evaluated at the unique equilibrium coincide with those given in (50)–(52).

By substituting (70)–(72) in (1) and (30) and eliminating r, we obtain:

Y(t)=0.7Y(tθc)+1001+exp(0.01(Y(tθd)400)0.05Y(tθd)0.2K(tθd)+290.

By differentiating both sides with t, we have:

(75)0.3Y˙(t)exp(0.01(Y(tθd)400))[1+exp(0.01(Y(tθd)400))]2Y˙(tθd)+0.05Y˙(tθd)+0.2K˙(tθd)=0.

From (30) and (26), we can find that:

(76)K˙(tθd)=Y(tθdθg)C(r(tθdθg),Y(tθdθg))=0.3Y(tθdθg)120.

Putting (76) in (75) and noting that y=YY*=Y−400, we can derive:

(77)0.3y˙(t)exp(0.01(y(tθd)))[1+exp(0.01y(tθd)))]2y˙(tθd)+0.05y˙(tθd)+0.06y(tθdθg)=0.

We shall perform a numerical simulation on (77).

Under (70)–(74), we have:

α=23.

Then, Proposition 9 implies that, for m=n=0, the following combination of (θd , θg ) and ω gives rise to stability switching in System (L-I):

(78)θd=0.25,θg=2.0445(ω=0.56687).

Finally, we shall set the initial function as follows:

(79)φ(t)=10.

In Figure 3 illustrated is the solution path of (77) with (78) and (79).

Figure 3: Solution path of System (L-I).
Figure 3:

Solution path of System (L-I).

We can find from Figure 3 that a periodic orbit is actually generated by a Hopf bifurcation in System (L-I).

B Stability switching of System (L-C)

In this section, we shall proceed to analyze for what values of θc and θg stability switching occurs to (49) of System (L-C).

We can make use of the same method as utilized in Appendix A by replacing Pi with the following:

(80)P0(λ)=λ,P1(λ)=βλ,P2(λ)=γIK,P3(λ)=βIK.

By putting λ= in (49), we can obtain the following two expressions, which are mathematically equivalent to each other:

(81)[P0(iω)+P1(iω)eiωθc]+[P2(iω)+P3(iω)eiωθc]eiωθg=0,
(82)[P0(iω)+P2(iω)eiωθc]+[P1(iω)+P3(iω)eiωθg]eiωθc=0.

By the same procedure as in the proof of Proposition 9, we can obtain the following result concerning the stability switching of (49) of System (L-C).

Proposition 10.Let Assumptions 1, 2, 4 and 5 hold.

  1. Assume that condition (61) is satisfied, which implies that γ=1. Then, all the roots of Eq. (49) have negative real parts if and only if the following condition is satisfied:

    (83)IKθc<π2.
  2. Assume that condition (63) is satisfied, which implies that γ≠1. Then, the stability switching curves of Eq. (49) are the loci of (θc, θg )∈ΘC defined as follows: [30]

(84)ΘC={(ψc,m±(ω),ψg,n±(ω))R++2:ω[max{(β+γ)IK1+β,(γβ)IK1β},min{(β+γ)IK1+β,(γβ)IK1β}],m,nZ},

where

(85)ψc,m±(ω)=[±arccos((1+β2)ω2(β2+γ2)IK22β(ω2γIK2))+2mπ]/ω,
(86)ψg,n±(ω)=[±arccos((1β2)ω2+(γ2β2)IK22(γβ2)IKω)π2+2nπ]/ω.

Proof. (i) Since we have γ=1 in this case, Eq. (49) can be written as:

(1βeθgλ)(λIKeθcλ)=0,

which is equivalent to [31]

(87)λ=1θglogβ<0,

or

(88)λIKeθcλ=0.

Because of (87), all the roots of (49) have negative real parts if and only if those of (88) do so. According to Bellman and Cooke (1963, 444, Theorem 13.8), it is necessary and sufficient condition for all the roots of (88) to have negative real parts that condition (83) is fulfilled. Hence, the assertion is proved to be true.

(ii) To begin, we can confirm from (80) that, for every (θc,θg)R++2:

deg(P0(λ))=1=max{deg(P1(λ)), deg(P2(λ)), deg(P3(λ))}P0(0)+P1(0)+P2(0)+P3(0)=(γβ)IK>0.limλ(|P1(λ)P0(λ)|,|P2(λ)P0(λ)|,|P3(λ)P0(λ)|)=β<1,

and that P0, P1, P2 and P3 have no common zero. Then, we can apply the method by Lin and Wang (2012) to this proof.

By the same method used in the proof of Proposition 9, one can easily find that, when stability switching happens to (49), the following conditions hold:

(89)(1+β2)ω2(β2+γ2)IK2=2β(ω2γIK2)cos(ωθc),
(90)(1β2)ω2+(β2γ2)IK2=2IK(γβ2)ωsin(ωθg)=2IK(γβ2)ωcos(ωθg+π2).

For some positive ω to fulfill (89) and (90), the following condition is satisfied:

|(1+β2)ω2(β2+γ2)IK2|=2β|ω2γIK2||(1β2)ω2+(β2γ2)IK2|=2(γβ2)IKω,

which are equivalent to:

[(1+β)2ω2(β+γ)2IK2][(1β)2ω2(γβ)2IK2]0,

or

(91)ω[max{(β+γ)IK1+β,(γβ)IK1β},min{(β+γ)IK1+β,(γβ)IK1β}].

Take an arbitrary ω that satisfies (91). Then, one can solve (89) and (90) for θc and θg , respectively, as: [32]

θc=[±arccos((1+β2)ω2(β2+γ2)IK22β(ω2γIK2))+2mπ]/ω,θg=[±arccos((1β2)ω2+(γ2β2)IK22(γβ2)IKω)π2+2nπ]/ω

Thus, combinations of (θc,θg)R++2 that satisfy (89) and (90) are given by (84). One can confirm that every (θc , θg )∈ΘI solves (81) and (82).

Therefore, we can conclude from Lin and Wang (2012, 526, Theorem 3.1) that the set of all stability switching curves is given by (84).□

Proposition 10 provides the information on the stability switching curves of (36) in System (L-C), i.e. on the combinations of (θd , θg ) for which the characteristic equation (39) has at least one pair of purely imaginary roots. This proposition may suggest the possibility that periodic orbits are generated by Hopf bifurcations along the stability switching curves.

To examine the possibility of periodic orbits generated by Hopf bifurcations in System (L-C), we shall also perform a numerical simulation. For this purpose, we shall specify the functional forms of C, I and L and the value of M as given in (70)–(73). Note that, under these specifications, Assumptions 1, 3 and 4 are all fulfilled.

Under the specifications of (70)–(73), the equilibrium point of System (L-C), (r*, Y*, K*), is also given by (74). Of course, under (70)–(74), the partial derivatives of C, I and L evaluated at the unique equilibrium coincide with those given in (50)–(52).

By substituting (70)–(72) in (1) and (40), eliminating r and differentiating both sides, we obtain:

(92)1.05Y˙(t)exp(0.01(Y(t)400))[1+exp(0.01(Y(t)400))]2Y˙(t)Y˙(tθc)+0.2K˙(t)=0.

From (30) and (26), we can find that:

(93)K˙(t)=Y(tθg)C(r(tθcθg),Y(tθcθg))=Y(tθg)0.7Y(tθcθg)120.

Putting (93) in (92) and noting that y=YY*=Y−400, we can derive:

(94)1.05y˙(t)exp(0.01(y(t)))[1+exp(0.01(y(t)))]2y˙(t)0.7y˙(tθc)+0.2y(tθg)0.14y(tθcθg)=0.

We shall perform a numerical simulation on (94).

Under (70)–(74), we have:

β=0.875,γ=1.25.

Then, Proposition 10 implies that, for m=n=0, the following combination of (θc , θg ) and ω gives rise to stability switching in System (L-I):

(95)θc=0.25,θg=2.4690(ω=0.473172).

Finally, we shall set the initial function as given in (79).

In Figure 4 illustrated is the solution path of (94) with (95) and (79).

Figure 4: Solution path of System (L-C).
Figure 4:

Solution path of System (L-C).

We can find from Figure 4 that a periodic orbit is actually generated by a Hopf bifurcation in System (L-C).

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Supplemental Material:

The online version of this article (DOI: 10.1515/snde-2015-0052) offers supplementary material, available to authorized users.


Published Online: 2017-2-15
Published in Print: 2017-4-1

©2017 Walter de Gruyter GmbH, Berlin/Boston

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