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November  2021, 26(11): 5925-5940. doi: 10.3934/dcdsb.2021145

On dynamics in a medium-term Keynesian model

1. 

Faculty of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan

2. 

Faculty of Economics, Matej Bel University, Tajovského 10,975 90 Banská Bystrica, Slovakia

* Corresponding author: Hiroki Murakami

Received  November 2020 Revised  April 2021 Published  November 2021 Early access  May 2021

This paper rigorously examines the (in)stability of limit cycles generated by Hopf bifurcations in a medium-term Keynesian model. The bifurcation equation of the model is derived and the conditions for stable and unstable limit cycles are presented. Numerical simulations are performed to illustrate the analytical results.

Citation: Hiroki Murakami, Rudolf Zimka. On dynamics in a medium-term Keynesian model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5925-5940. doi: 10.3934/dcdsb.2021145
References:
[1]

T. AsadaM. Demetrian and R. Zimka, On dynamics in a Keynesian model of monetary stabilization policy with debt effect, Communications in Nonlinear Science and Simulation, 58 (2018), 131-146.  doi: 10.1016/j.cnsns.2017.06.013.

[2]

T. AsadaM. Demetrian and R. Zimka, On dynamics in a Keynesian model of monetary and fiscal stabilization policy mix with twin debt accumulation, Metroeconomica, 70 (2019), 365-383. 

[3]

M. Bahnanu-Oskooee and S. Chomsisengphet, Stability of M2 money demand function in industrial countries, Applied Economics, 34 (2010), 2075-2083.  doi: 10.1080/00036840210128744.

[4]

Y. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702. Springer-Verlag, Berlin-New York, 1979.

[5]

A. Dohtani, A growth-cycle model of Solow-Swan type, I, Journal of Economic Behavior and Organization, 76 (2010), 428-444.  doi: 10.1016/j.jebo.2010.07.006.

[6]

P. FlaschelG. Gong and W. Semmer, A Keynesian macroeconometric framework for the analysis of monetary policy rules, Journal of Economic Behavior and Organization, 46 (2001), 101-136.  doi: 10.1016/S0167-2681(01)00189-5.

[7]

P. Flaschel and H. M. Krolzig, Wage-price Phillips curves and macroeconomic stability: Basic structural form, estimation and analysis, Quantitative and Empirical Analysis, 277 (2006), 7-47.  doi: 10.1016/S0573-8555(05)77002-4.

[8]

G. Gandolfo, Economic Dynamics, 4th ed., Springer-Verlag, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-03871-6.

[9]

C. I. Jones, The facts of economic growth, Handbook of Macroeconomics, 2, (2016), 3–69. doi: 10.1016/bs.hesmac.2016.03.002.

[10]

M. Juillard and S. Villemot, Multi-country real business cycle models: Accuracy tests and test bench, Journal of Economic Dynamics and Control, 35 (2010), 178-185.  doi: 10.1016/j.jedc.2010.09.011.

[11]

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

[12]

J. M. Keynes, The General Theory of Employment, Interest and Money, Macmillan, Cham, 1936. doi: 10.1007/978-3-319-70344-2.

[13]

T. Laubach and J. C. Williams, Measuring the natural rate of interest, Review of Economics and Statistics, 85 (2003), 1063-1070.  doi: 10.1162/003465303772815934.

[14] A. Leijonhufvud, On Keynesian Economics and the Economics of Keynes: A Study in Monetary Theory, Oxford University Press, Oxford, 1968. 
[15]

A. Leijonhufvud, Effective demand failures, Scandinavian Journal of Economics, 75 (1973), 27-48.  doi: 10.2307/3439273.

[16]

W. Liu, Criterion of Hopf bifurcations without using eigenvalues, Journal of Mathematical Analysis and Applications, 182 (1994), 250-256.  doi: 10.1006/jmaa.1994.1079.

[17]

H.-W. Lorenz, Nonlinear Dynamical Economics and Chaotic Motion, 2nd ed., Springer, Berlin, 1993. doi: 10.1007/978-3-642-78324-1.

[18]

S. A. Marglin and A. Bhaduri, Profit squeeze and Keynesian theory, The Golden Age of Capitalism: Reinterpreting the Postwar Experience, Clarendon Press, Oxford, (1990), 153–186. doi: 10.1093/acprof:oso/9780198287414.003.0004.

[19]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Springer-Verlag, New York, 1976.

[20]

A. Matsumoto, Note on Goodwin's 1951 nonlinear accelerator model with an investment delay, Journal of Economic Dynamics and Control, 33 (2009), 832-842.  doi: 10.1016/j.jedc.2008.08.013.

[21]

H. Murakami, Keynesian systems with rigidity and flexibility of prices and inflation-deflation expectations, Structural Change and Economic Dynamics, 30 (2014), 68-85.  doi: 10.1016/j.strueco.2014.02.002.

[22]

H. Murakami, Wage flexibility and economic stability in a non-Walrasian model of economic growth, Structural Change and Economic Dynamics, 32 (2015), 25-41.  doi: 10.1016/j.strueco.2015.01.002.

[23]

H. Murakami, A non-Walrasian microeconomic foundation of the "profit principle" of investment, Essays in Economic Dynamics: Theory, Simulation Analysis, and Methodological Study, Springer, Singapore, (2016), 123–141. doi: 10.1007/978-981-10-1521-2_8.

[24]

H. Murakami, Existence and uniqueness of growth cycles in post Keynesian systems, Economic Modelling, 75 (2018), 293-304.  doi: 10.1016/j.econmod.2018.07.001.

[25]

H. Murakami, A note on the "unique" business cycle in the Keynesian theory, Metroeconomica, 70 (2019), 384-404.  doi: 10.1111/meca.12222.

[26]

H. Murakami, Monetary policy in the unique growth cycle of post Keynesian systems, Structural Change and Economic Dynamics, 52 (2020), 39-49.  doi: 10.1016/j.strueco.2019.10.002.

[27]

H. Murakami, The unique limit cycle in post Keynesian systems, IERCU Discussion Paper, Chuo University, 334 (2020), 1–26.

[28]

H. Murakami and T. Asada, Inflation-deflation expectations and economic stability in a Kaleckian system, Journal of Economic Dynamics and Control, 92 (2018), 183-201.  doi: 10.1016/j.jedc.2017.11.004.

[29]

H. Murakami and H. Sasaki, Economic development with accumulation of public capital: The crucial role of wage flexibility on business cycles, Economic Modelling, 93 (2020), 299-309.  doi: 10.1016/j.econmod.2020.08.005.

[30]

H. Murakami and R. Zimka, On dynamics in a two-sector Keynesian model of business cycles, Chaos, Solitons and Fractals, 130 (2020), 109419, 8 pp. doi: 10.1016/j.chaos.2019.109419.

[31]

E. S. Phelps, Phillips curves, expectations of inflation and optimal unemployment over time, Economica, 34 (1967), 254-281.  doi: 10.2307/2552025.

[32]

J. Tobin, Keynesian models of recession and depression, American Economic Review, 65 (1975), 195-202. 

[33]

J. Tobin, Price flexibility and output stability: An old Keynesian view, Journal of Economic Perspectives, 7 (1993), 45-65.  doi: 10.1257/jep.7.1.45.

[34]

V. Torre, Existence of limit cycles and control in complete Keynesian system by theory of bifurcations, Econometrica, 45 (1977), 1457-1466.  doi: 10.2307/1912311.

[35]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, Vol. 2, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

show all references

References:
[1]

T. AsadaM. Demetrian and R. Zimka, On dynamics in a Keynesian model of monetary stabilization policy with debt effect, Communications in Nonlinear Science and Simulation, 58 (2018), 131-146.  doi: 10.1016/j.cnsns.2017.06.013.

[2]

T. AsadaM. Demetrian and R. Zimka, On dynamics in a Keynesian model of monetary and fiscal stabilization policy mix with twin debt accumulation, Metroeconomica, 70 (2019), 365-383. 

[3]

M. Bahnanu-Oskooee and S. Chomsisengphet, Stability of M2 money demand function in industrial countries, Applied Economics, 34 (2010), 2075-2083.  doi: 10.1080/00036840210128744.

[4]

Y. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702. Springer-Verlag, Berlin-New York, 1979.

[5]

A. Dohtani, A growth-cycle model of Solow-Swan type, I, Journal of Economic Behavior and Organization, 76 (2010), 428-444.  doi: 10.1016/j.jebo.2010.07.006.

[6]

P. FlaschelG. Gong and W. Semmer, A Keynesian macroeconometric framework for the analysis of monetary policy rules, Journal of Economic Behavior and Organization, 46 (2001), 101-136.  doi: 10.1016/S0167-2681(01)00189-5.

[7]

P. Flaschel and H. M. Krolzig, Wage-price Phillips curves and macroeconomic stability: Basic structural form, estimation and analysis, Quantitative and Empirical Analysis, 277 (2006), 7-47.  doi: 10.1016/S0573-8555(05)77002-4.

[8]

G. Gandolfo, Economic Dynamics, 4th ed., Springer-Verlag, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-03871-6.

[9]

C. I. Jones, The facts of economic growth, Handbook of Macroeconomics, 2, (2016), 3–69. doi: 10.1016/bs.hesmac.2016.03.002.

[10]

M. Juillard and S. Villemot, Multi-country real business cycle models: Accuracy tests and test bench, Journal of Economic Dynamics and Control, 35 (2010), 178-185.  doi: 10.1016/j.jedc.2010.09.011.

[11]

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

[12]

J. M. Keynes, The General Theory of Employment, Interest and Money, Macmillan, Cham, 1936. doi: 10.1007/978-3-319-70344-2.

[13]

T. Laubach and J. C. Williams, Measuring the natural rate of interest, Review of Economics and Statistics, 85 (2003), 1063-1070.  doi: 10.1162/003465303772815934.

[14] A. Leijonhufvud, On Keynesian Economics and the Economics of Keynes: A Study in Monetary Theory, Oxford University Press, Oxford, 1968. 
[15]

A. Leijonhufvud, Effective demand failures, Scandinavian Journal of Economics, 75 (1973), 27-48.  doi: 10.2307/3439273.

[16]

W. Liu, Criterion of Hopf bifurcations without using eigenvalues, Journal of Mathematical Analysis and Applications, 182 (1994), 250-256.  doi: 10.1006/jmaa.1994.1079.

[17]

H.-W. Lorenz, Nonlinear Dynamical Economics and Chaotic Motion, 2nd ed., Springer, Berlin, 1993. doi: 10.1007/978-3-642-78324-1.

[18]

S. A. Marglin and A. Bhaduri, Profit squeeze and Keynesian theory, The Golden Age of Capitalism: Reinterpreting the Postwar Experience, Clarendon Press, Oxford, (1990), 153–186. doi: 10.1093/acprof:oso/9780198287414.003.0004.

[19]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Springer-Verlag, New York, 1976.

[20]

A. Matsumoto, Note on Goodwin's 1951 nonlinear accelerator model with an investment delay, Journal of Economic Dynamics and Control, 33 (2009), 832-842.  doi: 10.1016/j.jedc.2008.08.013.

[21]

H. Murakami, Keynesian systems with rigidity and flexibility of prices and inflation-deflation expectations, Structural Change and Economic Dynamics, 30 (2014), 68-85.  doi: 10.1016/j.strueco.2014.02.002.

[22]

H. Murakami, Wage flexibility and economic stability in a non-Walrasian model of economic growth, Structural Change and Economic Dynamics, 32 (2015), 25-41.  doi: 10.1016/j.strueco.2015.01.002.

[23]

H. Murakami, A non-Walrasian microeconomic foundation of the "profit principle" of investment, Essays in Economic Dynamics: Theory, Simulation Analysis, and Methodological Study, Springer, Singapore, (2016), 123–141. doi: 10.1007/978-981-10-1521-2_8.

[24]

H. Murakami, Existence and uniqueness of growth cycles in post Keynesian systems, Economic Modelling, 75 (2018), 293-304.  doi: 10.1016/j.econmod.2018.07.001.

[25]

H. Murakami, A note on the "unique" business cycle in the Keynesian theory, Metroeconomica, 70 (2019), 384-404.  doi: 10.1111/meca.12222.

[26]

H. Murakami, Monetary policy in the unique growth cycle of post Keynesian systems, Structural Change and Economic Dynamics, 52 (2020), 39-49.  doi: 10.1016/j.strueco.2019.10.002.

[27]

H. Murakami, The unique limit cycle in post Keynesian systems, IERCU Discussion Paper, Chuo University, 334 (2020), 1–26.

[28]

H. Murakami and T. Asada, Inflation-deflation expectations and economic stability in a Kaleckian system, Journal of Economic Dynamics and Control, 92 (2018), 183-201.  doi: 10.1016/j.jedc.2017.11.004.

[29]

H. Murakami and H. Sasaki, Economic development with accumulation of public capital: The crucial role of wage flexibility on business cycles, Economic Modelling, 93 (2020), 299-309.  doi: 10.1016/j.econmod.2020.08.005.

[30]

H. Murakami and R. Zimka, On dynamics in a two-sector Keynesian model of business cycles, Chaos, Solitons and Fractals, 130 (2020), 109419, 8 pp. doi: 10.1016/j.chaos.2019.109419.

[31]

E. S. Phelps, Phillips curves, expectations of inflation and optimal unemployment over time, Economica, 34 (1967), 254-281.  doi: 10.2307/2552025.

[32]

J. Tobin, Keynesian models of recession and depression, American Economic Review, 65 (1975), 195-202. 

[33]

J. Tobin, Price flexibility and output stability: An old Keynesian view, Journal of Economic Perspectives, 7 (1993), 45-65.  doi: 10.1257/jep.7.1.45.

[34]

V. Torre, Existence of limit cycles and control in complete Keynesian system by theory of bifurcations, Econometrica, 45 (1977), 1457-1466.  doi: 10.2307/1912311.

[35]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, Vol. 2, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

Figure 1.  Solution path with (72)

Figure 2.  Solution path with (45)

Figure 3.  Solution paths projected on $ K $-$ P $ plane
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