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Research article

Dynamical analysis of Kaldor business cycle model with variable depreciation rate of capital stock

  • Received: 01 December 2019 Accepted: 24 March 2020 Published: 30 March 2020
  • MSC : 34A34

  • In this paper, the Kaldor business cycle model with variable depreciation rate of the capital stock are investigate the existence, uniqueness and stability of the positive equilibrium point, and the existence of the periodic solution and Hopf bifurcation respectively. Finally, we analyze the dynamic behaviors of the specific system and perform numerical simulations.

    Citation: Junjie Mao, Xiaowei Liu. Dynamical analysis of Kaldor business cycle model with variable depreciation rate of capital stock[J]. AIMS Mathematics, 2020, 5(4): 3321-3330. doi: 10.3934/math.2020213

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  • In this paper, the Kaldor business cycle model with variable depreciation rate of the capital stock are investigate the existence, uniqueness and stability of the positive equilibrium point, and the existence of the periodic solution and Hopf bifurcation respectively. Finally, we analyze the dynamic behaviors of the specific system and perform numerical simulations.


    In economics, business cycle model is a very important research direction. The endogenous economic cycle theory believes that the emergence of business cycle may be derived from the nonlinear properties of the economic system and the existence of time delay in the investment process.

    British economist Kaldor proposed a nonlinear business cycle model [1] in 1940, which was the first model to investigate business cycle using nonlinear dynamics. In the literature [1], Kaldor assumed that investment and saving are nonlinear functions of the gross product and the capital stock. After analysis, the economic system can produce long-term cyclical fluctuations without any external shocks. According to Kaldor's model, Chang and Smith [2] proposed the following system in 1971:

    {dYdt=α[I(Y,K)S(Y,K)],dKdt=I(Y,K), (1.1)

    where Y is the gross product, K is the capital stock, α is the adjustment coefficient in the goods market, I(Y,K) is investment function, and S(Y,K) is saving function.

    In 1994, Grasman and Wentzel further improved (1.1) in the literature [3]. They assumed that the saving function is a linear function of the gross product and assumed that the current change in the stock of capital equals the investment minus the depreciation of the capital stock. Then they proposed the following business cycle model:

    {dYdt=α[I(Y,K)γY],dKdt=I(Y,K)δK, (1.2)

    where δ is the depreciation rate of the capital stock and γ is a constant satisfying 0<γ<1. In [4], authors proposed a delayed business cycle model with general investment function.

    In particular, Krawiec and Szydlowski [5] assume that the investment function is I(Y,K)=I(Y)βK, and the system (1.2) becomes:

    {dYdt=α[I(Y)βKγY],dKdt=I(Y)βKδK, (1.3)

    where β>0. In 2011, X. P. Wu [6] investigated Codimension-2 bifurcations of the system (1.3).

    In economics, economists believe that people get experience while they are working, which leads to improved work efficiency. Therefore, in terms of the capital stock, the depreciation rate of the capital stock is not a constant, but will decrease as people's experience increases.

    In this paper we aim to analyze the dynamics of system (1.2) with the variable depreciation rate of the capital stock δ(K) as follows:

    {dYdt=α[I(Y,K)γY],dKdt=I(Y,K)δ(K)K, (1.4)

    where α,γ are constants and 0<α,0<γ<1. Based on reality, we assume I(Y,K)0 and according to the reference [5], the investment function satisfies I(Y,K)=I(Y)βK, where β is a positive constant. In this paper, we assume I(Y,K) is continuously differentiable in R2 with IY>0 and IK<0.

    In terms of the depreciation rate of the capital stock, S. Chatterjee [7] assumes that the rate of depreciation of capital is an increasing function of the rate of its utilization. In [8], they assume δ=δ(MtKt1), where Mt is maintenance expenditures and δ(MtKt1)<0,δ(MtKt1)>0,limM0δ(MtKt1)=ˉδ,limMδ(MtKt1)=0. Furthermore, M. Ishaq Nadiri and I. R. Prucha [9] specify a model of factor demand that allows they to estimate the depreciation rates of both physical and R & D capital. Based on their research results, we consider the depreciation rate of the capital stock from another perspective. When the capital stock is small and management ability is weak, the depreciation rate of the capital stock is the largest. However, with the accumulation of the capital stock and experience, the depreciation rate of the capital stock decrease gradually and approach a limit due to limited management ability. Notice that the decay of the depreciation rate also decreases. Thus we suppose that δ(K)<0,δ(K)>0,0<δ(0)=δ0<1,0<limK+δ(K)=δ1<δ0 for some constant K0 in (1.4). For the convenience of analysis, we assume that the depreciation rate of the capital stock satisfies the following relationship:

    δ(K)=δ1+δ0δ11+K.

    Then the system (1.4) can be transformed into the following form:

    {dYdt=α[I(Y)βKγY],dKdt=I(Y)βK(δ1+δ0δ11+K)K, (1.5)

    where 0<α,0<β,0<γ<1,0<δ1<δ0<1. In general, the investment function I(Y) in model (1.5) satisfies the Lipschitz continuity, and thus there exist unique solutions of (1.5).

    The rest of this paper is organized as follows. In section 2, we will present the existence and stability of the unique positive equilibrium point of system (1.5). In addition, the existence of the periodic solution and Hopf bifurcation of system (1.5) are investigated. In Section 3, we will analyze the dynamics of a more specific system (3.2). Further, numerical simulations on system (3.2) are performed in Section 4.

    In this section, we investigated the existence of the unique positive equilibrium point of the system (1.5) and analyzed the stability of the positive equilibrium point. We obtained a sufficient condition that the periodic solution does not exist in the first quadrant. In addition, we analyzed Hopf bifurcation of the system (1.5).

    For an economic system, we would like to know what is the equilibrium state of the system? We can analyze the existence of the unique positive equilibrium point of system (1.5) in the following theorem.

    Theorem 1. System (1.5) has a unique positive equilibrium point E(Y,K), where Y=δ1K2+δ0Kγ(1+K), if the conditions:

    (1) I(0)>0 for all Y0;

    (2) δ1γK2+2γδ1K+δ0γγ2(1+K)2I(Y)δ1γK2+2γδ1K+δ0γγ(1+K)2β<0 for all Y0,K0;

    (3) There exists a constant L>0 such that |I(Y)|L for all Y0;

    hold.

    Proof. Let the right end of the Eq. (1.5) be 0:

    {α[I(Y)βKγY]=0,I(Y)βK(δ1+δ0δ11+K)K=0. (2.1)

    Then we can obtain the relationship between K and Y:

    Y=δ1K2+δ0Kγ(1+K). (2.2)

    Substituting (2.2) into the first equation of (2.1), (2.1) is converted to

    I(δ1K2+δ0Kγ+γK)βKδ1K2+δ0K1+K=0. (2.3)

    Let U(K)=I(δ1K2+δ0Kγ+γK)βKδ1K2+δ0K1+K, then we can get U(K)=δ1γK2+2γδ1K+δ0γγ2(1+K)2I(Y)δ1γK2+2γδ1K+δ0γγ(1+K)2β. According to the conditions (1) and (2), we can know U(0)>0 and U(K)<0 for all K0. Based on the condition (3), we can obtain limK+U(K)=. Then we can know system (1.5) has a unique positive equilibrium point E(Y,K), where Y=δ1K2+δ0Kγ(1+K).

    For business cycle model, the periodic solution represents the economic cycle phenomenon, so the study of the periodic solution is very important. We can analyze existence of the periodic solution of system (1.5) in the following theorem.

    Theorem 2. If α[I(Y)γ]βδ1+(δ1δ0)K(1+K)2<0 or α[I(Y)γ]βδ1+(δ1δ0)K(1+K)2>0 for all Y0,K0, the system (1.5) has no periodic solution in the first quadrant.

    Proof. Let P(Y,K)=α[I(Y)βKγY],Q(Y,K)=I(Y)βK(δ1+δ0δ11+K)K, we can get PY+QK=α[I(Y)γ]βδ1+δ1δ0(1+K)2. If α[I(Y)γ]βδ1+δ1δ0(1+K)2<0 or α[I(Y)γ]βδ1+δ1δ0(1+K)2>0 always hold for all Y0,K0, according to the Bendixson criterion [10], the system (1.5) has no periodic solution in the first quadrant.

    In this section, we investigated stability of the positive equilibrium point.

    For the convenience of research, we make a transformation: u=YY,v=KK, then the system (1.5) is transformed into the following form:

    {dudt=α[I(u+Y)β(v+K)γ(u+Y)],dvdt=I(u+Y)β(v+K)(δ1+δ0δ11+v+K)(v+K). (2.4)

    The Jacobian matrix of system (2.4) at equilibriun O(0,0) is given by

    J(0,0)=(α[I(Y)γ]αβI(Y)βδ1+δ1δ0(1+K)2). (2.5)

    At the equilibrium point O(0,0), the linearized system of the system (2.4) is

    {dudt=α[I(Y)γ]uαβv,dvdt=I(Y)u+[βδ1+δ1δ0(1+K)2]v. (2.6)

    Then we can know the characteristic equation of J(0,0) is

    λ2+λ[αγαI(Y)+β+δ1+δ0δ1(1+K)2]+αβγ+αγδ1+αγδ0δ1(1+K)2αδ1I(Y)αI(Y)δ0δ1(1+K)2=0.

    Therefore the eigenvalues of J(0,0) are

    λ1=12[αγαI(Y)+β+δ1+δ0δ1(1+K)2]+{[αγαI(Y)+β+δ1+δ0δ1(1+K)2]24α[βγ+γδ1+γδ0δ1(1+K)2δ1I(Y)I(Y)δ0δ1(1+K)2]}12,

    λ2=12[αγαI(Y)+β+δ1+δ0δ1(1+K)2]{[αγαI(Y)+β+δ1+δ0δ1(1+K)2]24α[βγ+γδ1+γδ0δ1(1+K)2δ1I(Y)I(Y)δ0δ1(1+K)2]}12.

    Let ϕ(u,v)=α[I(u+Y)β(v+K)γ(u+Y)],ψ(u,v)=I(u+Y)β(v+K)(δ1+δ0δ11+v+K)(v+K), and based on the above calculation, we can get the local stability of the equilibrium point of the system (1.5) and system (2.4).

    Theorem 3. If 4α[βγ+γδ1+γδ0δ1(1+K)2δ1I(Y)I(Y)δ0δ1(1+K)2]>0, ϕ(u,v), ψ(u,v) are analytic functions, then the following (1) and (2) are hold:

    (1) when αγαI(Y)+β+δ1+δ0δ1(1+K)2<0, the equilibrium point O(0,0) of (2.4) and the equilibrium point E(Y,K) of (1.5) are unstable.

    (2) when αγαI(Y)+β+δ1+δ0δ1(1+K)2>0, the equilibrium point O(0,0) of (2.4) and the equilibrium point E(Y,K) of (1.5) are locally asymptotically stable.

    Proof. If αγαI(Y)+β+δ1+δ0δ1(1+K)2<0, we can know that the real part of the eigenvalues λ1,2 are less than 0. We can get the equilibrium point O(0,0) of (2.4) and the equilibrium point E(Y,K) of (1.5) are unstable. For (2), the proof are similar to (1). Now, the proof of Theorem (2.3) is finished.

    In this section, we use α as a parameter of bifurcation and investigate Hopf bifurcation of the system (2.4).

    For the system (2.4), we can know: ϕ(0,0,α)=0, ψ(0,0,α)=0. At the equilibriun O(0,0), the Jacobian matrix of the system (2.4) is

    J(0,0,α)=(α[I(Y)γ]αβI(Y)βδ1+δ1δ0(1+K)2). (2.7)

    When [αγαI(Y)+β+δ1+δ0δ1(1+K)2]2<4α[βγ+γδ1+γδ0δ1(1+K)2δ1I(Y)I(Y)δ0δ1(1+K)2], the eigenvalues of (2.7) is a pair of conjugate complex numbers:

    λ1,2(α)=m(α)±in(α),

    where:

    m(α)=12[αγαI(Y)+β+δ1+δ0δ1(1+K)2],

    n(α)=12{4α[βγ+γδ1+γδ0δ1(1+K)2δ1I(Y)I(Y)δ0δ1(1+K)2][αγαI(Y)+β+δ1+δ0δ1(1+K)2]2}12.

    When α=α=(β+δ1)(1+K)2+δ1δ0[γI(Y)](1+K)2=β(1+K)2+δ1[1(1+K)2]δ0[γI(Y)](1+K)2, we can get:

    m(α)=0,

    n(α)=(β+δ1)(1+K)2+δ1δ0[γI(Y)](1+K)2[βγ+γδ1+γδ0δ1(1+K)2δ1I(Y)I(Y)δ0δ1(1+K)2],

    λ1,2=±in(α).

    Lemma 1 (J. Zhang and B. Feng [11]). Consider the following system:

    {˙x=P(x,y,λ),˙y=Q(x,y,λ), (2.8)

    where P and Q are analytic functions. When parameter λ=0, (0,0) is the center-type stable(unstable) focus. When parameter λ>0, (0,0) is the unstable(stable) focus. The system (2.8) has at least one stable(unstable) limit cycle near point (0,0) for a sufficiently small λ>0.

    Theorem 4. If m(α)>0 for all α>α and uα[I(u+Y)β(v+K)γ(u+Y)]+v[I(u+Y)β(v+K)(δ1+δ0δ11+v+K)(v+K)]<0 for all (u,v)(0,0), then for a sufficiently small α>α, the system (1.5) has at least one stable limit cycle near point E(Y,K).

    Proof. For the system (2.4), we consider the Liapunov function V(u,v)=u2+v2. If 2uα[I(u+Y)β(v+K)γ(u+Y)]+2v[I(u+Y)β(v+K)(δ1+δ0δ11+v+K)(v+K)]<0 for all (u,v)(0,0), we can easily get ˙V(u,v)<0 for all (u,v)(0,0). The equilibriun O(0,0) is a stable focus, when α=α. On the other hand, when α>α, we can get m(α)>0. The equilibriun O(0,0) is an unstable focus. According to the Lemma (1), we can know the system (2.4) has at least one stable limit cycle near point O(0,0) for a sufficiently small α>α. Correspondingly, the system (1.5) has at least one stable limit cycle near point E(Y,K) for a sufficiently small α>α.

    Particularly, we consider the following Kaldor-type investment function:

    I(Y)=eY1+eY. (3.1)

    We introduce (3.1) into the system (1.5), the system (1.5) can be transformed into the following form:

    {dYdt=α[eY1+eYβKγY],dKdt=eY1+eYβK(δ1+δ0δ11+K)K. (3.2)

    For the system (3.2), we can combine Theorem (2.1) to study the existence and uniqueness of the positive equilibrium point of the system (3.2).

    Theorem 5. If eY(1+eY)2β+δ1δ1γ<0 for all Y0, the system (3.2) has a unique positive equilibrium point E(Y,K), where Y=δ1K2+δ0Kγ(1+K).

    Proof. For system (3.2), I(Y)=eY1+eY, we can know I(0)=12>0. Obviously, I(Y)=eY(1+eY)2>0,limY+I(Y)=1, so we can get |I(Y)|<1 for Y0. If (δ1γK2+2γδ1K+δ0γ)eYγ2(1+K)2(1+eY)2δ1γK2+2γδ1K+δ0γγ(1+K)2β<0, for all Y0,K0, then the system (3.2) satisfies the Theorem (2.1). Therefore the system (3.2) has a unique positive equilibrium point E(Y,K), where Y=δ1K2+δ0Kγ(1+K).

    For the system (3.2), we can get the sufficient condition that the system (3.2) has no periodic solution in the first quadrant according to the Theorem (2.2).

    Theorem 6. If α[eY(1+eY)2γ]βδ1+(δ1δ0)K(1+K)2<0 or α[eY(1+eY)2γ]βδ1+(δ1δ0)K(1+K)2>0 hold for all Y0,K0, then the system (3.2) has no periodic solution in the first quadrant.

    We can make a transformation: y=YY, k=KK, then the system (3.2) is transformed into the following form:

    {dydt=α[ey+Y1+ey+Yβ(k+K)γ(y+Y)],dkdt=ey+Y1+ey+Yβ(k+K)(δ1+δ0δ11+k+K)(k+K). (3.3)

    The Jacobian matrix of the system (3.3) at equilibriun O(0,0) is given by

    J(0,0)=(α[eY(1+eY)2γ]αβeY(1+eY)2βδ1+δ1δ0(1+K)2). (3.4)

    At the equilibrium point O(0,0), the linearized system of the system (3.3) is

    {dydt=α[eY(1+eY)2γ]yαβk,dkdt=eY(1+eY)2y+[βδ1+δ1δ0(1+K)2]k. (3.5)

    The characteristic equation of J(0,0) is

    λ2+λ[αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2]+αβγ+αγδ1+αγδ0δ1(1+K)2αδ1eY(1+eY)2α(δ0δ1)eY(1+eY)2(1+K)2=0.

    Therefore the eigenvalues of J(0,0) are

    λ1=12[αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2]+{[αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2]24α[βγ+γδ1+γδ0δ1(1+K)2δ1eY(1+eY)2(δ0δ1)eY(1+eY)2(1+K)2]}12,

    λ2=12[αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2]{[αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2]24α[βγ+γδ1+γδ0δ1(1+K)2δ1eY(1+eY)2(δ0δ1)eY(1+eY)2(1+K)2]}12.

    Let φ(y,k)=α[ey+Y1+ey+Yβ(k+K)γ(y+Y)],Ψ(y,k)=ey+Y1+ey+Yβ(k+K)(δ1+δ0δ11+k+K)(k+K). Since I(Y)=eY(1+eY)2, we can substitute I(Y) into the theorem (2.3) to get the local stability of the equilibrium point of the system (3.2) and system (3.3) in the following theorem.

    Theorem 7. If 4α[βγ+γδ1+γδ0δ1(1+K)2δ1eY(1+eY)2(δ0δ1)eY(1+eY)2(1+K)2]>0, φ(y,k),Ψ(y,k) are analytic functions, then the following (1) and (2) are hold:

    (1) when αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2<0, the equilibrium point O(0,0) of the system (3.3) and the equilibrium point E(Y,K) of the system (3.2) are unstable.

    (2) when αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2>0, the equilibrium point O(0,0) of the system (3.3) and the equilibrium point E(Y,K) of the system (3.2) are locally asymptotically stable.

    For the system (3.3), we can get: φ(0,0,α)=0,Ψ(0,0,α)=0.

    At the equilibrium point O(0,0), we can get the Jacobian matrix of system (3.3):

    J(0,0,α)=(α[eY(1+eY)2γ]αβeY(1+eY)2βδ1+δ1δ0(1+K)2). (3.6)

    When [αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2]2<4α[βγ+γδ1+γδ0δ1(1+K)2δ1eY(1+eY)2(δ0δ1)eY(1+eY)2(1+K)2], the eigenvalues of (3.6) is a pair of conjugate complex numbers:

    λ1,2(α)=w(α)±id(α),

    where:

    w(α)=12[αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2],

    d(α)=12{4α[βγ+γδ1+γδ0δ1(1+K)2δ1eY(1+eY)2(δ0δ1)eY(1+eY)2(1+K)2][αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2]2}12.

    When α=α=(βδ1)(1+K)2(1+eY)2+(δ1δ0)(1+eY)2γ(1+K)2(1+eY)2eY(1+K)2, we can know:

    w(α)=0,

    d(α)={(βδ1)(1+K)2(1+eY)2+(δ1δ0)(1+eY)2γ(1+K)2(1+eY)2eY(1+K)2[βγ+γδ1+γδ0δ1(1+K)2δ1eY(1+eY)2(δ0δ1)eY(1+eY)2(1+K)2]}12,

    λ1,2=±id(α).

    Theorem 8. If w(α)>0 for all α>α and yα[ey+Y1+ey+Yβ(k+K)γ(y+Y)]+k[ey+Y1+ey+Yβ(k+K)(δ1+δ0δ11+k+K)(k+K)]<0 for all (y,k)(0,0), then for a sufficiently small α>α, the system (3.2) has at least one stable limit cycle near point E(Y,K).

    Proof. When α=α, we consider the Liapunov function V(y,k)=y2+k2. If yα[ey+Y1+ey+Yβ(k+K)γ(y+Y)]+k[ey+Y1+ey+Yβ(k+K)(δ1+δ0δ11+k+K)(k+K)]<0 for all (y,k)(0,0), we can easily get ˙V(y,k)<0 for all (y,k)(0,0). The equilibrium O(0,0) is a stable focus. When α>α, we can get w(α)>0. The equilibrium O(0,0) is an unstable focus. According to the Lemma (1), we can know the system (3.3) has at least one stable limit cycle near point O(0,0) for a sufficiently small α>α. Correspondingly, the system (3.2) has at least one stable limit cycle near point E(Y,K) for a sufficiently small α>α.

    Let γ=0.3,β=0.5,δ1=0.2,δ0=0.8, then the system (3.2) can be converted to:

    {dYdt=α[eY1+eY0.5K0.3Y],dKdt=eY1+eY0.7K3K5+5K. (3.7)

    Let f(Y)=eY(1+eY)22120. When Y0, we can know f(Y)<0. Based on Theorem (2.4), the system (3.7) has a unique positive equilibrium point E(1.382467594,0.7692934721). When γ=0.3,β=0.5,δ1=0.2,δ0=0.8, we can get α[eY(1+eY)2γ]βδ1+(δ1δ0)K(1+K)2<0 for all α>0,Y0,K0. According to the Theorem (3.1) to get the system (3.7) has no periodic solution in the first quadrant. We take α=5 and numerically simulate the system (3.7). When α=5, we can get 4α[βγ+γδ1+γδ0δ1(1+K)2δ1eY(1+eY)2(δ0δ1)eY(1+eY)2(1+K)2]=4.093794628>0 and αγαeY(1+eY)2+β+δ1+δ0δ1(1+K)2=1.589831756>0. According to the Theorem (3.2), we know E(1.382467594,0.7692934721) is locally asymptotically stable. The above dynamics analysis can be verified in Figure 1, Figure 2 and Figure 3. In this special case, we can know that Kaldor business cycle model with variable depreciation rate of capital stock does not have a limit cycle and the economic system is stable in the long run.

    Figure 1.  Phase diagram.
    Figure 2.  Time series diagram of Y(t).
    Figure 3.  Time series diagram of K(t).

    Due to the complexity of economic phenomena, the depreciation rate of the capital stock is variable. This paper investigated the dynamic behaviors of Kaldor business cycle model with variable depreciation rate of capital stock. First, we investigated the general system's existence of the unique positive equilibrium point and the periodic solution. Second, we analyzed the stability of the equilibrium point and investigated Hopf bifurcation. In addition, we introduced the Kaldor-type investment function to the general system. We analyzed the dynamic behaviors of the specific system and performed numerical simulations. From the results of the study, we can see that the economic system is complex and sensitive. Under different actual conditions, economic systems can evolve into different states. Sometimes the economic system will tend to a stable state, sometimes it will be unstable, and sometimes it will even be in a cyclical state.

    We thank the unknown referees for their helpful comments that led us to improve this paper. This research was partially supported by NSF of China (11601251, 11771257).

    The authors declare no conflict of interest.



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  • This article has been cited by:

    1. S. Lasfar, K. Hattaf, N. Yousfi, Dynamics of a diffusive business cycle model with two delays and variable depreciation rate, 2024, 11, 23129794, 617, 10.23939/mmc2024.03.617
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