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Bifurcation analysis for the Kaldor–Kalecki model with two delays
 Cao Jianzhi^{1}Email author and
 Sun Hongyan^{1}
https://doi.org/10.1186/s1366201919480
© The Author(s) 2019
 Received: 13 February 2018
 Accepted: 6 January 2019
 Published: 13 March 2019
Abstract
In this paper, a Kaldor–Kalecki model of business cycle with two discrete time delays is considered. Firstly, by analyzing the corresponding characteristic equations, the local stability of the positive equilibrium is discussed. Choosing delay (or the adjustment coefficient in the goods market α) as bifurcation parameter, the existence of Hopf bifurcation is investigated in detail. Secondly, by combining the normal form method with the center manifold theorem, we are able to determine the direction of the bifurcation and the stability of the bifurcated periodic solutions. Finally, some numerical simulations are carried out to illustrate the theoretical results.
Keywords
 Kaldor–Kalecki model
 Hopf bifurcation
 Stability switch
 Periodic solution
 Two delays
1 Introduction
Business cycle (or named economic cycle) is a hot topic in the study of the macroeconomic theory. The definition of the business cycle refers to the overall economic performance in the period of economic expansion appears alternated with economic contraction, a phenomenon of the cycle, expressed as gross domestic product, changes in industrial production, prices, employment and unemployment, and other economic variables. Thus, the study of factors that cause fluctuations in the economic cycle and the duration of the economic cycle have important theoretical and practical significance and will help us to better understand the law of economic operation and to gain a reasonable understanding of the leading role of investment in economic development.
The remaining part is organized as follows: in the next section, employing the characteristic equation, the stability of the positive equilibrium, and the occurrence of local Hopf bifurcation are investigated. In Sect. 3, by using the normal form theory and the center manifold theorem, we derive some formulas that can determine the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions. In Sect. 4, some numerical simulations are carried out to illustrate the main results.
2 Local stability and Hopf bifurcation
In this section, the stability and Hopf bifurcation of the positive equilibrium point will be investigated.
As usual in a Keynesian framework, savings are assumed to be proportional to the current level of income, \(S(Y,K)=\gamma Y\), where the coefficient γ, \(0<\gamma <1\), represents the propensity to save. While in many versions of the Kaldor model the saving function is assumed to be nonlinear, we prefer a linear specification, both for its analytical simplicity and for its sounder microfoundation. Moreover, in our case this assumption does not affect the nonlinearity of the model, which is ensured by the nonlinearity of the investment function.
2.1 Existence and uniqueness of the positive equilibrium
It is easy to verify that System (2.1) has a unique positive equilibrium point \(E(Y^{*},K^{*})\) if the conditions of the following lemma hold.
Lemma 2.1
 (A1)
\(I(0)>0\);
 (A2)
\(I'(Y)<(1+ \frac{\beta }{\delta })\gamma \);
 (A3)
there exists a constant \(L>0\) such that \(\vert I(Y) \vert \leq L\) for all \(Y\in \mathbf{R}\).
Proof
As we all know, \(u(Y)\) is a straight line passing through the origin with the slope of \(\beta \frac{\gamma }{\delta }+\gamma >0\), from (A1) and (A3), \(I(Y)\) is a bounded function on its existence interval, then by intermediate value theorem, the curve \(I(Y)\) and the line \(u(Y)\) must intersect in the first quadrant.
Let \(Y=Y^{*}\) be the unique solution of (2.3), then \(K=K^{*}\) can be given by the formula \(K= \frac{\gamma }{\delta }Y\), one can claim that under hypotheses (A1)–(A3), System (2.1) has a unique equilibrium E. This concludes the proof. □
2.2 Local stability and Hopf bifurcation
 Case I :

\(\tau _{1}=\tau _{2}=0\).
 (H)
\(I'(Y^{*})>\gamma \).
From what has been discussed above, taking the adjustment coefficient in the goods market α as the bifurcation parameter, we have the following result.
Theorem 2.1
For System (2.1), \(\tau _{1}=\tau _{2}=0\), if the hypotheses (A1)–(A3) of Lemma 2.1 and (H) are established, then there exists \(\alpha ^{*}\in (0,\infty )\) such that the unique positive equilibrium \(E(Y^{*},K^{*})\) of System (2.1) is locally asymptotically stable when \(0<\alpha <\alpha ^{*}\); \(E(Y^{*},K^{*})\) is unstable when \(\alpha >\alpha ^{*}\); and when \(\alpha =\alpha ^{*}\), the associated characteristic equation has a pair of purely imaginary roots \(\pm i\sqrt{f}\), System (2.1) undergoes a Hopf bifurcation at \(E(Y^{*},K^{*})\).
Remark 2.1
From the economic point of view, when the speed of adjustment of the goods market α is low enough, regardless of any economic system in the initial position, it will eventually converge to a stable equilibrium point; in this equilibrium, the level of output and capital stock is constant. The principle of economics is that when the aggregate demand and aggregate supply gap appears, lower commodity market correction will lead to a more moderate rate of change in output, thus reducing the economic volatility. When the adjustment speed of the commodity market gradually increases and exceeds a certain critical value, the economic system also begins to change from stable to cyclical fluctuations.
Remark 2.2
In [24], taking the savings rate γ as the bifurcation parameter, authors study the stability and Hopf bifurcation of System (1.3) with \(\tau _{1}=\tau _{2}=0\). Comparing with Theorem 2.1, we conclude that the Kaldor–Kalecki model may exhibit various nonlinear dynamic behaviors depending on the choice of parameters.
 (H1)
\(p+s>0\);
 (H2)
\(q+r>0\);
 (H3)
either \(s^{2}p^{2}+2r<0\) and \(r^{2}q^{2}>0\) or \((s^{2}p^{2}+2r)^{2}<4(r^{2}q^{2})\);
 (H4)
either \(r^{2}q^{2}<0\) or \(s^{2}p^{2}+2r>0\) and \((s^{2}p^{2}+2r)^{2}=4(r^{2}q^{2})\);
 (H5)
\(r^{2}q^{2}>0\), \(s^{2}p^{2}+2r>0\) and \((s^{2}p^{2}+2r)^{2}>4(r ^{2}q^{2})\);
Lemma 2.2
([25])
 (i)
if (H1)–(H3) hold, then all roots of Eq. (2.7) have negative real parts for all \(\tau \geq 0\);
 (ii)
if (H1), (H2), and (H4) hold, then when \(\tau \in [0, \tau _{0}^{+})\) all roots of Eq. (2.7) have negative real parts, when \(\tau =\tau _{0}^{+}\), Eq. (2.7) has a pair of purely imaginary roots \(\pm i\omega _{+}\), and when \(\tau >\tau _{0}^{+}\), Eq. (2.7) has at least one root with positive real part;
 (iii)if (H1), (H2), and (H5) hold, then there is a positive integer k such that there are k switches from stability to instability to stability; that is, whenall roots of Eq. (2.7) have negative real parts, when$$\begin{aligned}& \tau \in \bigl[0,\tau _{0}^{+} \bigr], \bigl(\tau _{0}^{},\tau _{1}^{+} \bigr),\ldots, \bigl( \tau _{k1}^{},\tau _{k}^{+} \bigr), \end{aligned}$$Eq. (2.7) has at least one root with positive real part.$$\begin{aligned}& \tau \in [\tau _{0}^{+},\tau _{0}^{}), [\tau _{1}^{+},\tau _{1}^{}),\ldots, [\tau _{k1}^{+},\tau _{k1}^{}) \quad \textit{and} \quad \tau >\tau _{k}^{+}, \end{aligned}$$
 Case II :

\(\tau _{1}=0\), \(\tau _{2}>0\).
We summarize the above analysis in the following theorem for model (2.1).
Theorem 2.2
 (i)
if \(\delta >\max \{\delta ^{*},\delta ^{**}\}\), then conditions (H1)–(H3) in Lemma 2.2 are satisfied. Furthermore, the unique positive equilibrium \(E(Y^{*},K^{*})\) of System (2.1) is locally asymptotically stable for all \(\tau _{2}>0\);
 (ii)
if \(\delta ^{*}<\delta _{1}^{*}\), \(\delta \in (\delta _{1} ^{*},\delta _{1}^{**})\) or \(\delta _{1}^{*}<\delta ^{*}<\delta _{1}^{**}\), \(\delta \in (\delta ^{*},\delta _{1}^{**})\) and \((s^{2}p^{2}+2r)^{2}=4(r ^{2}q^{2})\) hold, then (H1), (H2), and (H4) in Lemma 2.2 are satisfied. \(E(Y^{*},K^{*})\) is locally asymptotically stable when \(\tau _{2} \in [0,\tau _{20}^{+})\); \(E(Y^{*},K^{*})\) is unstable, when \(\tau > \tau _{20}^{+}\); Eq. (2.8) has a pair of purely imaginary roots \(\pm i\omega _{+}\), when \(\tau =\tau _{20}^{+}\). Furthermore, if \(b+2a(b\delta )<0\), then the transversality condition is established, System (2.1) undergoes a Hopf bifurcation at \(E(Y^{*},K^{*})\), where \(\omega _{+}\) and \(\tau _{20}^{+}\) can be calculated by the formula in Lemma 2.2.
Remark 2.3
From Theorem 2.2, we find that conditions for generating oscillation are far more stringent than stable. From the viewpoint of economics, if the time delay for investment \(\tau _{1}\) is ignored and the depreciation rate of capital stock δ is high enough, no matter what the value of \(\tau _{2}>0\), the gross product and the capital stock will eventually converge to a stable equilibrium point. On the other hand, if δ is small enough, with finite time \(\tau _{20}^{+}\), when the time delay for capital stock in the past \(\tau _{2}<\tau _{20}^{+}\), the economic system is stable, when \(\tau _{2}>\tau _{20}^{+}\), the economic operation can appear unstable fluctuation.
 Case III :

\(\tau _{1}>0\), \(\tau _{2}=0\).
Theorem 2.3
 (i)
when \(0<\delta <\delta _{2}^{*}\), the unique positive equilibrium \(E(Y^{*},K^{*})\) of System (2.1) is unstable for all \(\tau _{1}>0\);
 (ii)
when \(\delta >\delta _{2}^{*}\), the unique positive equilibrium \(E(Y^{*},K^{*})\) of System (2.1) is locally asymptotically stable when \(\tau _{1}\in [0,\tau _{1}^{(0)})\); when \(\tau _{1}>\tau _{1}^{(0)}\), \(E(Y^{*},K^{*})\) is unstable; when \(\tau =\tau _{1}^{(k)}\), \(k=0,1,2,\ldots \) , the characteristic equation (2.11) has a pair of purely imaginary roots \(\pm i\omega _{0}\), System (2.1) undergoes a Hopf bifurcation at \(E(Y^{*},K^{*})\).
Remark 2.4
In [5], different from the method in this paper, model (2.1) for \(\tau _{1}>0\), \(\tau _{2}=0\) is formulated in terms of a secondorder nonlinear delay differential equation, the Hopf bifurcation theorem is obtained by computing the normal form on the center manifold, which requires tedious calculation.
Remark 2.5
The theorem of stability and Hopf bifurcation of System (2.1) for \(\tau _{1}>0\), \({\tau _{2}=0}\) in the present paper is obtained by the existence and uniqueness of the equilibrium point. However, in Ref. [6] the existence of equilibrium point is only a hypothesis, thus the conclusion we obtained is more clear.
 Case IV :

\(\tau _{1}=\tau _{2}>0\).
By similar discussion to [26, 27], we arrive at the following theorem.
Theorem 2.4

assume \(\delta ^{*}<\delta <\beta \), then we have
 (i)
if \(I'(Y^{*})<\min \{\gamma (1 \frac{\beta }{\delta }),\gamma \frac{\sqrt{\beta ^{2}\delta ^{2}}}{ \alpha } \}\), then conditions (H1)–(H3) hold, the unique positive equilibrium \(E(Y^{*},K^{*})\) of System (2.1) is locally asymptotically stable for all \(\tau _{1}\geq 0\);
 (ii)
if \(\vert I'(Y^{*})\gamma \vert < \frac{\beta }{\delta }\gamma \), then conditions (H1), (H2), and (H4) hold, \(E(Y^{*},K^{*})\) is locally asymptotically stable for \(\tau _{1}\in [0,\tau _{10}^{+})\). System (2.1) undergoes a Hopf bifurcation at \(E(Y^{*},K^{*})\) when \(\tau _{1}=\tau _{1j}^{+}\), \(j=0,1,2,\ldots \) .
 (iii)if \((s^{2}p^{2}+2r)^{2}>4(r^{2}q^{2})\) and \(\frac{\beta \gamma }{\delta }< \vert I'(Y^{*})\gamma \vert <\frac{\sqrt{\beta ^{2}\delta ^{2}}}{\alpha }\), then conditions (H1), (H2), and (H5) hold. System (2.1) undergoes k (a finite number) switches from stability to instability to stability when the parameters are such thatand eventually it becomes unstable.$$\begin{aligned}& \tau _{10}^{}< \tau _{10}^{+}< \tau _{11}^{}< \cdots < \tau _{1,k1}^{}< \tau _{1,k1}^{+} < \tau _{1k}^{}< \tau _{1,k+1}^{}< \tau _{1k}^{+}\cdots , \end{aligned}$$
 (i)

assume \(\delta >\max \{\beta ,\delta ^{*}\}\), then we have
 (i)
if \(I'(Y^{*})<\gamma (1 \frac{\beta }{\delta })\), then conditions (H1)–(H3) hold, the unique positive equilibrium \(E(Y^{*},K^{*})\) of System (2.1) is locally asymptotically stable for all \(\tau _{1}\geq 0\);
 (ii)
if \(\vert I'(Y^{*})\gamma \vert < \frac{\beta }{\delta }\gamma \), then conditions (H1), (H2), and (H4) hold, \(E(Y^{*},K^{*})\) is locally asymptotically stable for \(\tau _{1}\in [0,\tau _{10}^{+})\). System (2.1) undergoes a Hopf bifurcation at \(E(Y^{*},K^{*})\) when \(\tau _{1}=\tau _{1j}^{+}\), \(j=0,1,2,\ldots \) .
 (i)
 Case V :

\(\tau _{1}=2\tau _{2}>0\).

if \(\beta \delta >0\), then \(ba\delta +a\beta <0\) when \(I'(Y^{*})< \gamma (1 \frac{\beta }{2\beta \delta })\);

if \(\beta \delta <0\) and \(2\beta \delta >0\), then \(ba\delta +a \beta >0\) all the time;

if \(2\beta \delta <0\), then \(ba\delta +a\beta <0\) when \(I'(Y^{*})> \gamma (1+ \frac{\beta }{\delta 2\beta })\).
Lemma 2.3
Remark 2.6
The conditions in Lemma 2.3 guarantee that \(ba\delta +a\beta <0\). However, \(\frac{\omega \tau _{2}}{2}=\frac{\pi }{2}+j\pi \), \(j\in \mathbf{Z}\) is a root of (2.18) if and only if \(\omega ^{2}=ba\delta +a\beta >0\), this is a contradiction, in other words, \(\omega \tau _{2}=\pi +2j \pi \), \(j\in \mathbf{Z}\) is not a root of (2.18).
One has the following theorem by the Hopf bifurcation.
Theorem 2.5
For System (2.1), \(\tau _{1}=2\tau _{2}>0\), if \(\delta ^{*}<\delta <\beta \), \(I'(Y^{*})<(1 \frac{\beta }{2\beta \delta })\gamma \), (A1)–(A3) of Lemma 2.1 and the transversality condition (2.24) are satisfied, then the unique positive equilibrium \(E(Y^{*},K^{*})\) of System (2.1) is locally asymptotically stable when \(\tau _{2}\in [0,\tau _{20})\); when \(\tau _{2}>\tau _{20}\), \(E(Y^{*},K^{*})\) is unstable; when \(\tau =\tau _{2j}\), \(j\in \mathbf{Z}\), the characteristic equation (2.17) has a pair of purely imaginary roots \(\pm i\omega _{*}\), System (2.1) undergoes a Hopf bifurcation at \(E(Y^{*},K^{*})\).
 Case VI :

\(\tau _{1}\neq \tau _{2}\).
We conclude the discussions above as follows.
Theorem 2.6
 (i)
if \(h(\omega )=0\) exhibits no positive root, the unique positive equilibrium \(E(Y^{*},K^{*})\) of System (2.1) is locally asymptotically stable for all \(\tau _{2}>0\);
 (ii)
if \(h(\omega )=0\) has \(N_{1}\) positive roots, then there exists a positive number \(\tau _{20}^{*}=\min \{\tau _{2i}^{(0)},i=1,2,\ldots,N_{1} \}\) such that \(E(Y^{*},K^{*})\) is locally asymptotically stable for \(\tau _{2}\in [0,\tau _{20}^{*})\) and unstable for \(\tau _{2}>\tau _{20}^{*}\). Furthermore, System (2.1) undergoes a Hopf bifurcation at \(E(Y^{*},K^{*})\) when \(\tau _{2}=\tau _{20}^{*}\).
Remark 2.7
Since the set \(\{(\tau _{1}, \tau _{2}) \tau _{1}=2\tau _{2}\}\) belongs to \(\{(\tau _{1}, \tau _{2}) \tau _{1}\neq \tau _{2}, \tau _{1}>0, \tau _{2}>0 \}\), it can be seen from the above discussion that Case V just is a special case of Case VI.
Now we can state the following result.
Theorem 2.7
 (i)
the unique positive equilibrium \(E(Y^{*},K^{*})\) of System (2.1) is locally asymptotically stable if and only if \(b\in (b_{0}^{},b_{0}^{+})\). If \(b\in (\infty ,b_{0}^{})\) or \(b\in (b_{0}^{+},+\infty )\), \(E(Y^{*},K^{*})\) is unstable;
 (ii)
Eq. (2.1) undergoes Hopf bifurcations at the equilibrium \(E(Y^{*},K^{*})\) when \(b=b_{j}\), \(j=1,2,\ldots \) ,
3 Direction and stability of the Hopf bifurcation
In this section, by using the algorithm developed in Hassard et al. [30], we will study the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions when \(\tau _{1}\neq \tau _{2}\) (Case VI). For the other five cases, most of the derivations are nearly the same steps, hence we omit them.
Theorem 3.1
 (i)
\(\mu _{2}\) determines the direction of the Hopf bifurcation: if \(\mu _{2}>0\) (<0), then the Hopf bifurcation is supercritical(subcritical) and the bifurcating periodic solutions exist for \(\tau _{2}>\tau _{20}\) (\(<\tau _{20}\));
 (ii)
\(\beta _{2}\) determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are orbitally stable (unstable) if \(\beta _{2}<0\) (>0);
 (iii)
\(T_{2}\) determines the period of the bifurcating periodic solutions: the period increases (decreases) if \(T_{2}>0\) (<0).
4 Numerical examples
4.1 \(\tau _{1}=\tau _{2}=0\)
Since \(I'(Y^{*})= \frac{e^{0.659}}{(1+e^{0.659})^{2}}=0.2247>0.2=\gamma \), then condition (H) is satisfied. Furthermore, we can obtain \(\alpha ^{*}=10.1215\) and \(\sqrt{f}=0.6267\).
4.2 \(\tau _{1}=0\), \(\tau _{2}>0\)
4.3 \(\tau _{1}=2\tau _{2}>0\)
4.4 \(\tau _{1}\neq \tau _{2}>0\)
Declarations
Acknowledgements
The authors are grateful to both reviewers for their helpful suggestions and comments.
Funding
This work was supported by the National Natural Science Foundation of China (No: 11771115), Natural Science Foundation of Hebei Province of China (No: A2016201206), and Hebei Education Department (No.: QN2016030, 2017018).
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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