November  2020, 13(11): 3231-3251. doi: 10.3934/dcdss.2020190

How interest rate influences a business cycle model

1. 

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450000, China

2. 

School of Business, Macau University of Science and Technology, Macau 999078, China

3. 

Henan Academy of Big Data, Zhengzhou University, Zhengzhou 450002, China

* Corresponding author: Jingli Ren

Received  November 2018 Revised  May 2019 Published  November 2020 Early access  December 2019

Fund Project: This research is supported by the National Natural Science Foundation of China (11771407), and the Innovative Research Team of Science and Technology in Henan Province (17IRTSTHN007)

We study the effect of interest rate on phenomenon of business cycle in a Kaldor-Kalecki model. From the information of the People's Bank of China and the Federal Reserve System, we know the interest rate is not a constant but with remarkable periodic volatility. Therefore, we consider periodically forced interest rate in the model and study its dynamics. It is found that, both limit cycle through Hopf bifurcation in unforced system and periodic solutions generated by period doubling bifurcation or resonance in periodically forced system, can lead to cyclical economic fluctuations. Our analysis reveals that the cyclical fluctuation of interest rate is one of a key formation mechanism of business cycle, which agrees well with the pure monetary theory on business cycle. Moreover, this fluctuation can cause chaos in a business cycle system.

Citation: Qigang Yuan, Yutong Sun, Jingli Ren. How interest rate influences a business cycle model. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3231-3251. doi: 10.3934/dcdss.2020190
References:
[1]

I. BashkirtsevaL. Ryashko and T. Ryazanova, Stochastic sensitivity analysis of the variability of dynamics and transition to chaos in the business cycles model, Commun. Nonlinear Sci. Numer. Simul., 54 (2018), 174-184.  doi: 10.1016/j.cnsns.2017.05.030.

[2]

D. Besomi, Clément Juglar and the transition from crises theory to business cycle theories, Conference on the Occasion of the Centenary of the Death of Clement Juglar, Paris, 2005.

[3]

F. Cavalli, A. Naimzada and N. Pecora, Real and financial market interactions in a multiplier-accelerator model: Nonlinear dynamics, multistability and stylized facts, Chaos, 27 (2017), 15pp. doi: 10.1063/1.4994617.

[4]

W. W. Chang and D. J. Smyth, The existence and persistence of cycles in a non-linear model: Kaldor's 1940 model re-examined, Rev. Econ. Stud., 38 (1971), 37-44.  doi: 10.2307/2296620.

[5]

J. Greenwood, Z. Hercowitz and G. W. Huffman, Investment, capacity utilization, and the real business cycle, Am. Econ. Rev., (1971), 402-417.

[6]

R. M. Goodwin, The nonlinear accelerator and the persistence of business cycles, Econometrica, 19 (1951), 1-17.  doi: 10.2307/1907905.

[7]

R. G. Hawtrey, The trade cycle, De Economist, 75 (1926), 169-185.  doi: 10.1007/BF02213478.

[8]

R. G. Hawtrey, The monetary theory of the trade cycle and its statistical test, Q. J. ECON., 41 (1927), 471-486.  doi: 10.2307/1883702.

[9]

R. G. Hawtrey, Trade and Credit, Longmans, London, 1928.

[10]

X. He, C. Li and Y. Shu, Bifurcation analysis of a discrete-time Kaldor model of business cycle, Int. J. Bifurcat. Chaos, 22 (2012). doi: 10.1142/S0218127412501866.

[11]

W. S. Jevons, Commercial crises and sun-spots, Nature, 19 (1878), 33-37.  doi: 10.1038/019033d0.

[12]

A. Kaddar and H. T. Alaoui, Hopf bifurcation analysis in a delayed Kaldor-Kalecki model of business cycle, Nonlinear Anal. Model. Control, 13 (2008), 439-449.  doi: 10.15388/NA.2008.13.4.14550.

[13]

N. Kaldor, A model of the trade cycle, Econ. J., 50 (1940), 78-92.  doi: 10.2307/2225740.

[14]

J. M. Keynes, The General Theory of Employment, Interest, and Money, Palgrave Macmillan, Cham, 2018. doi: 10.1007/978-3-319-70344-2.

[15]

N. D. Kondratieff and W. F. Stolper, The long waves in economic life, Review: J. Fernand Braudel Center, 2 (1979), 519-562. 

[16]

A. Krawiec and M. Szydlowski, The Kaldor-Kalecki business cycle model, Ann. Oper. Res., 89 (1999), 89-100.  doi: 10.1023/A:1018948328487.

[17]

W. A. Lewis, Growth and Fluctuations 1870-1913, Routledge, 2009.

[18]

X. P. LiJ. L. RenS. A. CampbellG. S. Wolkowicz and H. P. Zhu, How seasonal forcing influences the complexity of a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 785-807.  doi: 10.3934/dcdsb.2018043.

[19]

T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.  doi: 10.1038/17290.

[20]

A. K. Naimzada and N. Pecora, Dynamics of a multiplier-accelerator model with nonlinear investment function, Nonlinear Dynam., 88 (2017), 1147-1161.  doi: 10.1007/s11071-016-3301-4.

[21]

L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0003-8.

[22]

J. L. Ren and X. P. Li, Bifurcations in a seasonally forced predator-prey model with generalized Holling type â…£ functional response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 19pp. doi: 10.1142/S0218127416502035.

[23]

J. L. Ren and Q. G. Yuan, Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate, Chaos, 27 (2017), 15pp. doi: 10.1063/1.5000152.

[24]

M. T. RosensteinJ. J. Collins and C. J. D. Luca, A practical method for calculating largest Lyapunov exponents from small data sets, Phys. D, 65 (1993), 117-134.  doi: 10.1016/0167-2789(93)90009-P.

[25]

J. A. Schumpeter, Business Cycles, McGraw-Hill, New York, 1939. doi: 10.1522/030021081.

[26]

X. P. Wu and L. Wang, Multi-parameter bifurcations of the Kaldor-Kalecki model of business cycles with delay, Nonlineear Anal. Real World Appl., 11 (2010), 869-887.  doi: 10.1016/j.nonrwa.2009.01.023.

show all references

References:
[1]

I. BashkirtsevaL. Ryashko and T. Ryazanova, Stochastic sensitivity analysis of the variability of dynamics and transition to chaos in the business cycles model, Commun. Nonlinear Sci. Numer. Simul., 54 (2018), 174-184.  doi: 10.1016/j.cnsns.2017.05.030.

[2]

D. Besomi, Clément Juglar and the transition from crises theory to business cycle theories, Conference on the Occasion of the Centenary of the Death of Clement Juglar, Paris, 2005.

[3]

F. Cavalli, A. Naimzada and N. Pecora, Real and financial market interactions in a multiplier-accelerator model: Nonlinear dynamics, multistability and stylized facts, Chaos, 27 (2017), 15pp. doi: 10.1063/1.4994617.

[4]

W. W. Chang and D. J. Smyth, The existence and persistence of cycles in a non-linear model: Kaldor's 1940 model re-examined, Rev. Econ. Stud., 38 (1971), 37-44.  doi: 10.2307/2296620.

[5]

J. Greenwood, Z. Hercowitz and G. W. Huffman, Investment, capacity utilization, and the real business cycle, Am. Econ. Rev., (1971), 402-417.

[6]

R. M. Goodwin, The nonlinear accelerator and the persistence of business cycles, Econometrica, 19 (1951), 1-17.  doi: 10.2307/1907905.

[7]

R. G. Hawtrey, The trade cycle, De Economist, 75 (1926), 169-185.  doi: 10.1007/BF02213478.

[8]

R. G. Hawtrey, The monetary theory of the trade cycle and its statistical test, Q. J. ECON., 41 (1927), 471-486.  doi: 10.2307/1883702.

[9]

R. G. Hawtrey, Trade and Credit, Longmans, London, 1928.

[10]

X. He, C. Li and Y. Shu, Bifurcation analysis of a discrete-time Kaldor model of business cycle, Int. J. Bifurcat. Chaos, 22 (2012). doi: 10.1142/S0218127412501866.

[11]

W. S. Jevons, Commercial crises and sun-spots, Nature, 19 (1878), 33-37.  doi: 10.1038/019033d0.

[12]

A. Kaddar and H. T. Alaoui, Hopf bifurcation analysis in a delayed Kaldor-Kalecki model of business cycle, Nonlinear Anal. Model. Control, 13 (2008), 439-449.  doi: 10.15388/NA.2008.13.4.14550.

[13]

N. Kaldor, A model of the trade cycle, Econ. J., 50 (1940), 78-92.  doi: 10.2307/2225740.

[14]

J. M. Keynes, The General Theory of Employment, Interest, and Money, Palgrave Macmillan, Cham, 2018. doi: 10.1007/978-3-319-70344-2.

[15]

N. D. Kondratieff and W. F. Stolper, The long waves in economic life, Review: J. Fernand Braudel Center, 2 (1979), 519-562. 

[16]

A. Krawiec and M. Szydlowski, The Kaldor-Kalecki business cycle model, Ann. Oper. Res., 89 (1999), 89-100.  doi: 10.1023/A:1018948328487.

[17]

W. A. Lewis, Growth and Fluctuations 1870-1913, Routledge, 2009.

[18]

X. P. LiJ. L. RenS. A. CampbellG. S. Wolkowicz and H. P. Zhu, How seasonal forcing influences the complexity of a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 785-807.  doi: 10.3934/dcdsb.2018043.

[19]

T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.  doi: 10.1038/17290.

[20]

A. K. Naimzada and N. Pecora, Dynamics of a multiplier-accelerator model with nonlinear investment function, Nonlinear Dynam., 88 (2017), 1147-1161.  doi: 10.1007/s11071-016-3301-4.

[21]

L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0003-8.

[22]

J. L. Ren and X. P. Li, Bifurcations in a seasonally forced predator-prey model with generalized Holling type â…£ functional response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 19pp. doi: 10.1142/S0218127416502035.

[23]

J. L. Ren and Q. G. Yuan, Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate, Chaos, 27 (2017), 15pp. doi: 10.1063/1.5000152.

[24]

M. T. RosensteinJ. J. Collins and C. J. D. Luca, A practical method for calculating largest Lyapunov exponents from small data sets, Phys. D, 65 (1993), 117-134.  doi: 10.1016/0167-2789(93)90009-P.

[25]

J. A. Schumpeter, Business Cycles, McGraw-Hill, New York, 1939. doi: 10.1522/030021081.

[26]

X. P. Wu and L. Wang, Multi-parameter bifurcations of the Kaldor-Kalecki model of business cycles with delay, Nonlineear Anal. Real World Appl., 11 (2010), 869-887.  doi: 10.1016/j.nonrwa.2009.01.023.

Figure 1.  S-shape of $ I(Y) $
Figure 2.  Fluctuations of interest rate. (a) Interest rate of FED from September 1986 to February 2007; (b) Interest rate of PBC from July 1998 to October 2015
Figure 3.  Phase portraits of a fold bifurcation of system (5) for $ \alpha = 2 $, $ \beta = 0.6 $, $ \gamma = 0.283368 $, $ \delta = 0.3 $, $ m = 2.294 $, $ n = 4.612 $, $ C_{0} = 0.2 $. (a) $ d = 0.6 $. (b) $ d = 0.750017 $. (c) $ d = 0.8 $
Figure 4.  Bifurcation diagram of system (5) in $ d-\gamma $ plane for $ \alpha = 3 $, $ \beta = 0.6 $, $ \delta = 0.3 $, $ m = 3 $, $ n = 5.6 $, $ C_{0} = 0.6 $
Figure 5.  Phase portrait of Hopf bifurcation of system (5) for $ \alpha = 3 $, $ \beta = 0.6 $, $ d = 0.195 $, $ \delta = 0.3 $, $ m = 3 $, $ n = 5.6 $, $ C_{0} = 0.6 $. (a) Supercritical Hopf bifurcation for $ \gamma = 1.175 $. (b) Subcritical Hopf bifurcation for $ \gamma = 0.66 $
Figure 6.  Time series of limit cycles corresponding to the cases in Fig. 5. (a) Stable limit cycle generated by supercritical Hopf bifurcation for $ \gamma = 1.175 $. (b) Stable limit cycle in system (5) for case $ \gamma = 0.66 $. (c) Unstable limit cycle generated by subcritical Hopf bifurcation for $ \gamma = 0.66 $
Figure 7.  (b) Bifurcation diagrams of the forced system in with $ \alpha = 3, \beta = 0.6, \gamma = 0.4, \delta = 0.3, m = 3, n = 5.6, C_{0} = 0.2 $. (c) partial enlargements of (b)
Figure 8.  (a) Bifurcation diagrams of the forced system in with $ \alpha = 3, \beta = 0.6, \gamma = 0.7, \delta = 0.3, m = 3.6, n = 10.58, C_{0} = 0.2 $. (b) partial enlargements of (a)
Figure 9.  (a) Bifurcation diagrams of the forced system in $ (\epsilon - d) $ plane with $ \alpha = 5, \beta = 0.6, \gamma = 0.85, \delta = 0.3, m = 3.211, n = 3.368, C_{0} = 1 $
Figure 10.  (a) Bifurcation diagrams of the forced system in $ (\epsilon - d) $ plane with $ \alpha = 5, \beta = 0.6, \gamma = 0.8, \delta = 0.3, m = 3.211, n = 3.368, C_{0} = 1 $
Figure 11.  Phase portrait of different solutions. (a) A stable period-two orbit for $ d = 0.3, \epsilon = 0.8 $. (b), (c) Poincaré section and Time series of the stable period-two orbit. (d) A stable period-four orbit for $ \ d = 0.18, \epsilon = 0.8 $. (e), (f) Poincaré section and Time series of the stable period-four orbit. (g) Phase portrait of torus for $ d = 0.2, \epsilon = 0.1 $. (h), (i) Poincaré section and Time series of the torus
Figure 12.  Phase portrait of chaotic attractor. (a) Chaotic attractor through cascade of period doublings for $ d = 0.25, \epsilon = 0.5 $. (b), (c) Corresponding Poincaré section and Time series of (a). (d) Chaotic attractor through torus destruction for $ d = 0.2, \epsilon = 0.2 $. (e), (f) Corresponding Poincaré section and Time series of (d)
Figure 13.  (a) Spectrum of largest Lyapunov exponents for $ \epsilon = 0.5 $. (b) Spectrum of largest Lyapunov exponents for $ \epsilon = 0.2 $
Figure 14.  (a) Bifurcation diagram in $ (Y-d) $ plane for $ \epsilon = 0.5 $. (b) Bifurcation diagram in $ (Y-d) $ plane for $ \epsilon = 0.2 $
Figure 15.  (a) A stable period-three orbit for $ \gamma = 0.85,d = 0.45, \epsilon = 0.2 $. (b), (c) Poincaré section and Time series of the stable period-three orbit. (d) A stable period-five orbit for $ \gamma = 0.8,d = 0.235, \epsilon = 0.7 $. (e), (f) Poincaré section and Time series of the stable period-five orbit
[1]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208

[2]

John Leventides, Iraklis Kollias. Optimal control indicators for the assessment of the influence of government policy to business cycle shocks. Journal of Dynamics and Games, 2014, 1 (1) : 79-104. doi: 10.3934/jdg.2014.1.79

[3]

Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437

[4]

Rudolf Zimka, Michal Demetrian, Toichiro Asada, Toshio Inaba. A three-country Kaldorian business cycle model with fixed exchange rates: A continuous time analysis. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5999-6015. doi: 10.3934/dcdsb.2021143

[5]

Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz, Huaiping Zhu. How seasonal forcing influences the complexity of a predator-prey system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 785-807. doi: 10.3934/dcdsb.2018043

[6]

Paolo Ubezio. Unraveling the complexity of cell cycle effects of anticancer drugs in cell populations. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 323-335. doi: 10.3934/dcdsb.2004.4.323

[7]

Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009

[8]

Bin Li, Hai Huyen Dam, Antonio Cantoni. A low-complexity zero-forcing Beamformer design for multiuser MIMO systems via a dual gradient method. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 297-304. doi: 10.3934/naco.2016012

[9]

Fang Wu, Lihong Huang, Jiafu Wang. Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021264

[10]

Julián López-Góme, Andrea Tellini, F. Zanolin. High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Communications on Pure and Applied Analysis, 2014, 13 (1) : 1-73. doi: 10.3934/cpaa.2014.13.1

[11]

Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104

[12]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 589-597. doi: 10.3934/dcds.2014.34.589

[13]

Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846

[14]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129

[15]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, 2021, 29 (5) : 3069-3079. doi: 10.3934/era.2021026

[16]

Karsten Matthies. Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 585-602. doi: 10.3934/dcds.2003.9.585

[17]

Jianqin Zhou, Wanquan Liu, Xifeng Wang. Structure analysis on the k-error linear complexity for 2n-periodic binary sequences. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1743-1757. doi: 10.3934/jimo.2017016

[18]

Wenjun Zhang, Bernd Krauskopf, Vivien Kirk. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2825-2851. doi: 10.3934/dcds.2012.32.2825

[19]

Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985

[20]

Jianqin Zhou, Wanquan Liu, Xifeng Wang. Complete characterization of the first descent point distribution for the k-error linear complexity of 2n-periodic binary sequences. Advances in Mathematics of Communications, 2017, 11 (3) : 429-444. doi: 10.3934/amc.2017036

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (387)
  • HTML views (348)
  • Cited by (0)

Other articles
by authors

[Back to Top]