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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

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

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

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

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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