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  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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020190
References:
[1]

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

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

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

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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.  Google Scholar

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

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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.  Google Scholar

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

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

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[25]

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

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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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

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
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