# Dynamic Analysis for a Kaldor–Kalecki Model of Business Cycle with Time Delay and Diffusion Effect

Authors: Wenjie Hu; Hua Zhao; Tao Dong
Journal: Complexity
 (2018)

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## Abstract

The dynamics behaviors of Kaldor–Kalecki business cycle model with diffusion effect and time delay under the Neumann boundary conditions are investigated. First the conditions of time-independent and time-dependent stability are investigated. Then, we find that the time delay can give rise to the Hopf bifurcation when the time delay passes a critical value. Moreover, the normal form of Hopf bifurcations is obtained by using the center manifold theorem and normal form theory of the partial differential equation, which can determine the bifurcation direction and the stability of the periodic solutions. Finally, numerical results not only validate the obtained theorems, but also show that the diffusion coefficients play a key role in the spatial pattern. With the diffusion coefficients increasing, different patterns appear.


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## Body

## 1. Introduction

Recently, business cycle, as one of the important economic phenomena, has received attractive attentions due its widely application in many fields such as economic decisions, macroeconomic regulation, and market regulation [1–8]. In order to understand the mechanisms of business cycle, many models are proposed. One of the most famous business cycle models is the Kaldor–Kalecki business cycle [9, 10], which is described as(1)dYt∂t=αIYt,Kt-γYtdKt∂t=IYt,Kt-qKt,where Y(t) is the gross product, K(t) is the capital stock at time, α is the adjustment coefficient in the goods market, q is the depreciation rate of the capital stock, γ represents the propensity to save, and IYt,Kt is the investment. Under this model, the dynamic behaviors are widely studied such as stability, Hopf bifurcation, codimension-two bifurcation, and chaos [9–15].It is well known that diffusion effects of economic activities are widespread phenomenon that existed all over the world. As a result of the impact of the growth pole, the diffusion effects are the main interactions in economic activities. So, the diffusion effect should be considered in the business cycle model. However, to the best of our knowledge, there are very few works on this field. Inspired by the observation, in this paper, based on the Kaldor–Kalecki model, we propose a novel business cycle with diffusion effect and time delay under the Neumann boundary conditions, which is as follows:(2)∂Yt,x∂t=d1ΔYt,x+αIYt,x-βKt,x-γYt,x∂Kt,x∂t=d2ΔKt,x+IYt-τ,x-q+βKt,x,with its initial and boundary conditions given as follows:(3)∂u1∂x=∂v1∂x=∂u2∂x=∂v2∂x=0,t>0,x∈∂Ωuit,x=ϕit,x,i=1,2,t∈-τ,0,x∈Ω-vit,x=ϕit,x,i=1,2,t∈-τ,0,x∈Ω-,where α>0, β>0, q∈0,1, γ∈0,1, and Ω is the market capacity. There are three contributions of this paper:(1)
Based on the Kaldor–Kalecki model, we propose a novel business cycle under the Neumann boundary conditions. Our model is a spatial-temporal model, which is more general than the existing models.(2)
The time-independent and time-dependent stability are investigated. Moreover, the conditions of the Hopf bifurcation are obtained.(3)
It is found that the diffusion coefficients play a key role in the spatial pattern. With the diffusion coefficients increasing, different patterns appear.The rest paper is organized as follows. In Section2, the time-independent stability, time-dependent stability, and the existence of Hopf bifurcation are obtained. In Section 3, the normal form of Hopf bifurcation is obtained. In Section 4, numerical results are given to validate the obtained theorems.


## 2. Local Stability and Hopf Bifurcation Analysis

LetY∗,K∗ be an equilibrium of (2) and u1t,x=Yt,x-Y∗, u2t,x=Kt,x-K∗, i(u(t,x))=Iu(t,x)-I∗, and s=S-S∗; (2) can be rewrite as(4)∂u1t,x∂t=d1Δu1t,x+αiu1t,x-βu2t,x-γu1t,x∂u2t,x∂t=d2Δu2t,x+iu1t-τ,x-q+βu2t,x.Taking the Taylor expansion of i at 0 yields(5)iu=c1u+c2u2+c3u3+⋯,where c1=i′0, c2=(1/2)i′′0, and c3=1/3!i′′′0.The linear parts of (4) are as follows:(6)∂u1t,x∂t=d1Δu1t,x+αc1-γu1t,x-αβu2t,x∂u2t,x∂t=d2Δu2t,x+c1u1t-τ,x-q+βu2t,x.Then, the characteristic of (6) is(7)λy-DΔy-Leλy=0,y∈domΔ∖0.Following the method of [16], we define -k2k∈R0=0,1,2,… as the eigenvalue of Δ under the Neumann boundary conditions on the X. Let βk1=γk,0,T, βk2=0,γkT be the corresponding eigenvectors, where γk=cos⁡kx, k=0,1,2,…. We use βk1,βk2k=0∞ to construct a basis of the phase space X. Then, y can be expanded in the following form of Fourier on X:(8)y=∑k=0∞YkTβk1βk2βk3βk4,Yk=y,βk1y,βk2y,βk3y,βk4T.Then, we have(9)LϕTβk1βk2βk3βk4=LϕTβk1βk2βk3βk4,k∈N0.By (7)–(9), we can obtain(10)detλ+d1k200λ+d2k2-αc1-γ-αβc1e-λτ-q+β=0.The characteristic equation of (10) is as follows:(11)λ+d1k2-αc1-γαβ-c1e-λτλ+d2k2+q+β=0.The characteristic equation of (11) is(12)Fλ=λ2+φ1kλ+φ2k+αβc1e-λτ,where (13)φ1k=d1+d2k2+q+β-αc1-γφ2k=d1d2k4+d1q+d1β-d2c1-γαk2-c1-γqα+αβ.Lemma 1.
Ifc1<γ holds, the positive equilibrium of system (2) is locally stable at τ=0.Proof.
Forτ=0, (12) can be rewrite as Fλ=λ2+φ1(k)λ+φ2(k)+αβc1. As c1<γ, α>0, β>0, q∈0,1, and γ∈0,1, one can obtain ∀k∈N, φ1(k)>0 and φ2(k)+αβc1>α(γ-c1)q+γq>0 which means the roots of (12) have negative part. Therefore, the positive equilibrium of system (2) is locally stable at τ=0. The proof is completed.
Substitutingλ=iω into (12) and separating real parts and imaginary parts yield(14)αβc1cos⁡ωτ=ω2-φ2kαβc1sin⁡ωτ=φ1kω.According to sin2ωτ+cos2ωτ=1, from (14), we have(15)Fω=ω4+φ12k-2φ2kω2+φ22k-αβc12=0.By simple calculation, we have(16)φ12k-2φ2k=αγ-c1+d1k22+q+β+d2k22>0φ22k-αβc12=αγ-c1+d1k22q+β+d2k22-αβc12.As di>0,i=1,2, one can obtain (17)φ22k-αβc12=αγ-c1+d1k22q+β+d2k22-αβc12>α2γ-c12q+β2-αβc12.It is easy to see if (γ-c1)(q+β)-(βc1)>0, (15) has no positive roots. Combining with the Lemma 1, we have the following theorem.Theorem 2.
Ifc1<γ and (γ-c1)(q+β)-(βc1)>0, the positive equilibrium of system (2) is locally stable for any τ>0.In the following, one investigates the conditions of Hopf bifurcation of (2). By (14), one can obtain(18)cos⁡ωτ=ω2-φ2kαβc1sin⁡ωτ=φ1kωαβc1.By simple calculation, one has(19)dωdk=-4b1k3+2b2k+b3ω2+8b4k7+6b5k5+4b6k3+2b7k4ω3+2b1k4+b2k2+b3ω<0,where(20)b1=a52-2a1,b2=2a5a6-a2,b3=a62+2a3,b4=a12,b5=2a1a2,b6=a22-2a1a3,b7=-2a2a3,b8=a32-a42,a1=d1d2,a2=d1q+d1β-d2αc1-γ,a3=αq+βc1-γ,a4=αβc1,a5=d1+d2,a6=q+β-αc1-γ.If c1<γ, by simple calculation, one can obtain bi>0,i=1,2,...,7, and then one has dω/dk<0. From (18), one has(21)tan⁡ωτ=φ1kωω2-φ2k.Taking the derivative of k in (21), one can obtain(22)dτdk=-11+e0e1ω+e2k6+e3k4+e4k2+e5ω+τωdωdk+11+e0e6ω2+e7,where(23)e0=a5k2+a6ωω2-a1k4-a2k2+a32>0,e1=a5k2+a6>0,e2=a1a5>0,e3=a2a5+a1a6,e4=a2a6-a3a5,e5=-a3a6,e6=2a5k,e7=2a1a5k5+4a1a6k3+2a2a6+a3a5k.If c1<γ, one can obtain a3<0; by simple calculation, one can get ei>0, i=0,1,…,7. Combining with dω/dk<0, one has(24)dτdk>0,which means τ increase with the increasing of k. So, τ0 must exist in k=0. Now, one considers the case k=0. Let z=ω2; (18) with k=0 is as follows:(25)Fz=z2+α2γ-c12+q+β2z+α2γ-c12q+β2-αβc12.Since α2(γ-c1)2+(q+β)2>0, (25) has only one positive root of (25) if (γ-c1)(q+β)-(βc1)<0. Let z be the positive root of (25); then one can obtain ω=z(26)τj0=1ωarctan⁡φ10ωω2-φ20+1ωjπ,j=1,2,….By (26), one can obtain τ0=τ00.Lemma 3 (see [16, 17]).
Consider the exponential polynomial(27)Pλ,e-λτ,…,e-λτm=λn+P10λn-1+⋯+Pn-10λ+Pn0+P11λn-1+⋯+Pn-11λ+Pn1e-λτ+⋯+P1mλn-1+⋯+Pn-1mλ+Pnme-λτm,where τi≥0i=1,2,…,m and pjij=1,2,…,m are constants. As τ1,τ2,…,τm vary, the sum of the order of the zeroes of Pλ,e-λτ,…,e-λτm in the open right half plane can change only if a zero appears on or crosses the imaginary axis.Theorem 4.
According to Lemmas1 and 3, one has the following.
Ifc1<γ and (γ-c1)(q+β)-(βc1)<0 holds, system (2) is asymptotically stable for τ∈[0,τ0). System (2) undergoes a Hopf bifurcation at the origin when τ=τ0; that is, system (2) has a branch of periodic solutions bifurcating from the trivial solution near τ=τ0.Remark 5.
By incorporate diffusion effect into the Kaldor–Kalecki model of business cycle, a novel Kaldor–Kalecki model of business cycle with diffusion effect and time delay is proposed. Our model is a spatial-temporal model, which is more general than the existing business cycle [8, 9].


## 3. Direction and Stability of the Hopf Bifurcation

In this section, we give the normal form of Hopf bifurcation of (2) by using the method of [16, 18]. Let ϑ=τ-τ0 and normalize τ by t→t/τ; (2) can be rewritten as(28)R˙t=τ0DUt+Lτ0Ut+FUt,ϑ,where (29)D=diagD1,D2,Lτ0φ=τ0B0φ0+B1φ-τ,B0=αh-αβ0-q+βB1=00c10.Fφ,ϑ=ϑDΔφ0+Lϑφ+f∗φ,ϑf∗φ,ϑ=τ0+ϑαc2φ120+αc3φ130c2φ12-1+c3φ13-1,c1=i′0,c2=12i′′0,c3=13i′′′0.For φ=φ1,φ2T∈ζ, from Theorem 2, we know Λ0-iω0τ0,iω0τ0 are the eigenvalues of the linear part of (28):(30)U˙t=τ0Dut+Lτ0ut,(31)z˙t=Lτ0zt,where Lτ0 is one parameter family of bounded linear operator in C≔C-1,0R2 into R2.Following the method of [16], we define ηθ,τ for θ∈[-1,0], such that(32)L∗τ0φ=∫-10dηθ,τ0φθ,φ∈C,where(33)ηθ,τ0=τ0+ϑB0δθ+B1δθ+1.δθ is Drac-delta function.LetAτ0 be the infinitesimal generator of the semigroup induce by the solution of (28) and A∗ denote the formal adjoint of Aτ0 under the bilinear pairing; we have(34)ψ,φ=ψ0·φ0-∫θ=-10∫ξ=0θψξ-θdηθϕξdξ,for ϕ∈C, ψ∈C∗=C0,1,R2. Then Aτ0 and A∗ are a pair of adjoint operators.It is not hard to see thatiτ0ω0 is an eigenvalue of Aτ0 and -iτ0ω0 is an eigenvalue of A∗. Define q1θ=(1,α1)Teiω0τ0θ and q2θ=q1¯θ; then, one can obtain(35)Aq10=iω0τ0q10.Then, we have(36)α1=αh-iω0τ0αβ=α11+iα12,where(37)α11=αhαβ,α12=-ω0τ0αβ.Defineq1∗θ∗=(1,β1)Te-iω0τ0θ∗ and q2∗θ∗=q-1∗θ∗; then, one can obtain(38)A∗q1∗0=-iω0τ0q1∗0.Then, we have(39)β1=β11+iβ12,where(40)β11=-αhcos⁡ω0τ0-sin⁡ω0τ0ω0τ0c1,β12=-ω0τ0cos⁡ω0τ0+sin⁡ω0τ0αhc1.Letu(θ)=u1(θ),u2(θ) and v(θ)=v1(θ),v2(θ)T with(41)u1θ=q1θ+q2θ2=Reeiω0τ0θ,Reα1eiω0τ0θ=cos⁡ω0τ0θα11cos⁡ω0τ0θ-α12sin⁡ω0τ0θu2θ=q1θ-q2θ2i=lim⁡eiω0τ0θ,lim⁡α1eiω0τ0θ=sin⁡ω0τ0θα11sin⁡ω0τ0θ+α12cos⁡ω0τ0θfor θ∈-1,0 and(42)v1θ∗=q1∗θ∗+q2∗θ∗2=Ree-iω0τ0θ∗,Reβ1e-iω0τ0θ∗=cos⁡ω0τ0θ∗β11cos⁡ω0τ0θ∗+β12sin⁡ω0τ0θ∗v2θ∗=q1∗θ∗-q2∗θ∗2i=lim⁡e-iω0τ0θ∗,lim⁡β1e-iω0τ0θ∗=sin⁡ω0τ0θ∗-β11sin⁡ω0τ0θ∗+β12cos⁡ω0τ0θ∗for θ∗∈-1,0.In the following, we defineu,v∗=uj,v∗kj,k=1,2 and construct a new basis v(43)v=v1,v2=v∗,u-1v∗.Let fk=βk1βk2 and c·f0=c1β01+c2β02 for c=c1,c2T∈ζ; the center space of linear equation (30) is given by(44)PCNζ∗=uv,φ,fk·fkφ∈ζ∗.And ζ∗=PCNζ∗⊕Q, where Q denotes the complement subspace of PCNζ∗ in ζ∗.AsAτ0 is the infinitesimal generator induced by the solution of (32), then (28) can be rewritten as(45)U˙t=Aτ0Ut+X0FUt,ϑ,where(46)X0θ=0,-1≤θ<0Iθ=0.By using the decomposition ζ∗=PCNζ∗⊕Q and (45), (30) can be written as(47)Ut=Ux1tx2t·fk+hx1,x2,ϑ,where x1t,x2tT=v,Ut,fk, hx1,x2,ϑ∈Q, and h0,0,0=Dh0,0,0=0. By using center manifold, we can obtain(48)Ut=ux1tx2t·fk+hx1,x2,0.Let z=x1-ix2 and q1=u1+iu2, from (48), we can obtain(49)Rt=12q1z+q-1z-·fk+Wz,z-,where(50)Wz,z-=hz+z-2,iz-z-2,0≜W20z22+W11zz-+W02z-22+⋯.Following [18], we have(51)z˙=iω0τ0z+gz,z-,where(52)gz,z-=v1-iv2fUt,0,fk≜g20z22+g11zz-+g20z-22+⋯for v0=v10,v20T.By calculation, from (52), we have(53)fRt,0,fk=τ041π∫0πcos3kxdxd11d21z22+τ041π∫0πcos3kxdxd12d22zz-+τ041π∫0πcos3kxdxd13d23z-22+τ02d14d24z2z-2,where(54)d11=ac2,d12=c2e-iω0τ0,d21=ac2,d22=c2e-iω0τ0d31=ac2,d32=c2e-iω0τ0d41=ac22W1110+W2010cos⁡kx,cos⁡kx+14ac31π∫0πcos4kxdxd42=c22W1110e-iω0τ0+W2010eiω0τ0cos⁡kx,cos⁡kx+14c31π∫0πcos4kxdxWijnθ,cos⁡kx=1π∫0πWijnθxdx,i+j=2,n=1,2,3,4.It is easy to see1/π∫0πcos3kxdx=0 when k≠0. Letρ1,ρ2=v10-iv20; compare the coefficients with (52); we can obtain(55)g20=0k=Nτ4d11ρ1+d21ρ2k=0g11=0k=Nτ4d12ρ1+d22ρ2k=0g02=0k=Nτ4d13ρ1+d23ρ2k=0.g21=τ2d14ρ1+d24ρ2Because ofW20θ and W11θ in the g21, we need to determine them. It follows from (52) that(56)W˙z,z-=W20zz˙+W11z˙z-+W11zz-˙+W02z-z-˙+⋯,(57)Aτ0W˙z,z-=Aτ0W20z22+Aτ0W11zz-+Aτ0W02z-22+⋯.According to [18], we have(58)W˙=Aτ0W+Hz,z-,where(59)Hz,z-=H20z22+H11zz-+H02z-22+⋯=X0fUt,0-uv,X0fUt,0,fk·fk.By comparing (56), (58), and (59), we have(60)Aτ0-2iω0τ0W20θ=-H20θ,Aτ0W11θ=-H11θ.Expanding (59), we can obtain(61)Hz,z-=-12q1θg20+q2θg-02·fkz22-12q1θg11+q2θg-11·fkzz-+⋯.According to (60), the following can be obtained:(62)H20θ=0k∈N-12q1θg20+q2g11θ·f0k=0H11θ=0k∈N-12q1θg11+q2g-11θ·f0k=0.According to (60), (62), and q1θ=q10eiω0τ0, we can obtain(63)W20θ=12ig20ω0τ0q1θ+ig-203ω0τ0q2θ·fk+E1e2iω0τ0θW11θ=12-ig11ω0τ0q1θ+ig-11ω0τ0q2θ·fk+E2,where E1=E11,E12∈R2, E2=E21,E22∈R2, and E1,E2 are constant vector.In order to seek appropriateE1,E2, we can obtain the following by the definition of Aτ0 and (60)(64)H200=2iω0τ0W200-τ0DΔW200-Lτ0W20θH110=-τ0DΔW110-Lτ0W11θ,where(65)H200=τ04d11,d21Tcos2kxk∈Nτ04d11,d21T-12q10g20+q20g-02·fkk=0H110=τ04d12,d22Tcos2kxk∈Nτ04d12,d22T-12q10g11+q20g-11·fkk=0.Then, we can obtain(66)E1=142iω0+d1k2-αc1-γαβ-c1e-2iω0τ2iω0+d2k2+q+β-1d11d21cos2kxE2=14d1k2-αc1-γαβ-c1d2k2+q+β-1d11d21cos2kx.Now, we can calculate W20θ and W11θ; g21 is also expressed, and then the following important parameter can be obtained(67)C10=g212,μ2=-ReC10Reλ′τ0,β2=2ReC10,T2=-ImC10+μ2Imλ′τ0ω0τ0.Theorem 6.
By Theorem4, one has the following results:(i)
The sign ofμ2 can determine the direction of Hopf bifurcation: if μ2>0μ2<0, the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for τ>τ0τ<τ0.(ii)
The sign ofβ2 determines the stability of the bifurcating periodic solutions: if β2<0, β2>0 the bifurcation periodic solutions are stable (unstable).(iii)
The sign ofT2 determines the period of the bifurcating periodic solutions: if T2>0T2<0 the period increases (decreases).


## 4. Numerical Simulations

In this section, two simulations are given to validate the obtained theorems. Letd1=d2=0.1, α=3, β=0.2, γ=0.25, q=0.05, iYt,x=eYt,x/(1+eYt,x), and system (2) can be rewrite as follows:(68)∂Yt,x∂t=0.1∗ΔYt,x+3eYt,x1+eYt,x-0.2Yt,x-0.25Kt,x∂Kt,x∂t=0.1∗ΔKt,x+eYt,x1+eYt,x-0.25Kt,x,the Neumann boundary conditions of system (2) is as follows:(69)∂u1∂x=∂v1∂x=∂u2∂x=∂v2∂x=0,t>0,x∈∂Ω,and the initial conditions is as follows:(70)u1t,x=v1t,x=u2t,x=v2t,x=0.1.By simple calculations, we can obtain the equilibrium is(0.498,2.489),  c1<γ(γ-c1)(q+β)-(βc1)=-0.0198<0,  ω0=0.366, and τ0≈2.3805. Consider τ=2.2<τ0, by Theorem 4, system (2) is asymptotically stable, which is verified in Figures 1 and 2.Figure 1
The temporal solution of system (2) with τ=2.2.
(a)
(b)Figure 2
The temporal solution of system (2) with τ=2.2, x=5.
(a)
(b)
(c)Considerτ=2.6>τ0; by Theorem 4, system (2) is unstable and a Hopf bifurcation occurs. By simple calculation, we can obtain the parameter C10=7.0694-0.892020i,  μ2=7.3686>0, and β2=14.1388>0 which means Hopf bifurcation is supercritical since μ2>0. The simulations are shown in Figures 3 and 4, which indicate that there is a stable limit cycle.Figure 3
The periodic solutions diagram of system (2) with τ=2.6.
(a)
(b)Figure 4
The periodic solutions diagram of system (2) with τ=2.6, x=5.
(a)
(b)
(c)In the following, we investigate the effect of diffusion on the dynamics of system (2). Let di=0.1,0.5 and τ=2.8>τ0; the diagrams are shown in Figures 4–7. From Figure 4, we can see that system (2) have a periodic solution when di=0.1. With D increasing to 0.5, the periodic solution disappear, system (2) becomes divergent, which is shown in Figure 6. Moreover, from Figures 5–7, we can see that the spatial pattern with di=0.1 is different from the spatial pattern with di=0.5. To summarize, it can be seen the diffusion coefficient affects the pattern formation of system (2).Figure 5
The spatial pattern of system (2) with τ=2.6, di=0.1.Figure 6
The temporal solution of system (2) with τ=2.6, di=0.5.
(a)
(b)Figure 7
The spatial pattern of system (2) with τ=2.6, di=0.5.


## 5. Conclusions

It is well known that diffusion effects of economic activities are widespread phenomenon that existed all over the world. As a result of the impact of the growth pole, the diffusion effects are the main interactions in economic activities. So, the diffusion effect should be considered in the business cycle model. In this paper, we consider a Kaldor–Kalecki business cycle model with diffusion effect and time delay under the Neumann boundary conditions. First the time-independent and time-dependent stability are investigated. Then, we find that the time delay can give rise to the Hopf bifurcation when the time delay passes a critical value. Moreover, the normal form of Hopf bifurcation is obtained. Finally, numerical results not only validate the obtained theorems, but also show that the diffusion coefficients play a key role in the spatial pattern. With the diffusion coefficients increasing, different patterns appear.



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