# American Institute of Mathematical Sciences

November  2021, 26(11): 5999-6015. doi: 10.3934/dcdsb.2021143

## A three-country Kaldorian business cycle model with fixed exchange rates: A continuous time analysis

 1 Department of Quantitative Methods and Informatics, Faculty of Economics, Matej Bel University, Tajovského 10,975 50 Banská Bystrica, Slovakia 2 Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina F1, 84 248 Bratislava, Slovakia 3 Faculty of Economics, Chuo University, 742-1 Higashinakano, Tokyo 192-0393, Japan 4 School of Education, Waseda University, 1-6-1 Nishiwaseda, Shinjuku, Tokyo 169-8050, Japan

* Corresponding author: Rudolf Zimka

Received  March 2021 Revised  September 2020 Published  November 2021 Early access  May 2021

This paper analyses a three-country, fixed exchange rates Kaldorian nonlinear macroeconomic model of business cycles. The countries are connected through international trade, and international capital movement with imperfect capital mobility. Our model is a continuous time version of the discrete time three-country Kaldorian model of Inaba and Asada [22]. Their paper provided numerical studies of the dynamics of the three countries under fixed exchange rates. This paper provides analytical examinations of the local stability of the model´s equilibria, and of the existence of business cycles. The results are illustrated by numerical simulations.

Citation: 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
##### References:
 [1] T. Asada, Kaldorian dynamics in an open economy, J. Econ., 62 (1995), 239-269.  doi: 10.1007/BF01238819. [2] T. Asada, A two-regional model of business cycles with fixed exchange rates : A Kaldorian approach, Studies in Regional Sciences, 34 (2004), 19-38.  doi: 10.2457/srs.34.2_19. [3] T. Asada, C. Chiarella, P. Flaschel and R. Franke, Open Economy Macrodynamics: An Integrated Disequilibrium Approach, Springer-Verlag, Berlin, 2003. [4] T. Asada, C. Chiarella, P. Flaschel and R. Franke, Monetary Macrodynamics, Routledge, London, 2010. doi: 10.4324/9780203859964. [5] T. Asada, M. Demetrian and R. Zimka, On dynamics in a Keynesian model of monetary and fiscal policy with debt effect, Commun. Nonlinear Sci. Numer. Simul., 58 (2018), 131-146.  doi: 10.1016/j.cnsns.2017.06.013. [6] T. Asada, M. 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. [7] T. Asada, C. Douskos, V. Kalantonis and P. Markellos, Numerical exploration of Kaldorian interregional macrodynamics: Enhanced stability and predominance of period doubling under flexible exchange rates, Discrete Dyn. Nat. Soc., (2010), Art. ID. 263041, 1–29. doi: 10.1155/2010/263041. [8] T. Asada, C. Douskos and P. Markellos, Numerical exploration of Kaldorian interregional macrodynamics: Stability and the trade threshold for business cycles under fixed exchange rates, Nonlinear Dynamics, Psychology, and Life Sciences, 15 (2011), 105-128. [9] T. Asada, T. Inaba and T. Misawa, A nonlinear macrodynamic model with fixed exchange rates: Its dynamics and noise effects, Discrete Dyn. Nat. Soc., 4 (2000), 319-331.  doi: 10.1155/S1026022600000303. [10] T. Asada, T. Inaba and T. Misawa, An interregional dynamic model: The case of fixed exchange rates, Studies in Regional Science, 31 (2001), 29-41.  doi: 10.2457/srs.31.2_29. [11] T. Asada, V. Kalantonis, M. Markellos and P. Markellos, Analytical expressions of periodic disequilibrium fluctuations generated by Hopf bifurcations in economic dynamics, Appl. Math. Comput., 218 (2012), 7066-7077.  doi: 10.1016/j.amc.2011.12.063. [12] T. Asada, T. Misawa and T. Inaba, Chaotic dynamics in a flexible exchange rate system: A study of noise effects, Discrete Dyn. Nat. Soc., 4 (2000), 309-317. [13] R. Barro and X. Sala-i Martin, Economic Growth, 2$^{nd}$ edition, MIT Press, Cambridge, MA, 2004. [14] J. Benhabib and T. Miyao, Some new results on the dynamics of the generalized Tobin model, International Economic Review, 22 (1981), 589-596.  doi: 10.2307/2526160. [15] Yu. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702. Springer-Verlag, Berlin-New York, 1979. [16] A. Dohtani, T. Misawa, T. Inaba, M. Yokoo and T. Owase, Chaos, complex transients and noise: Illustration with a Kaldor model, Chaos, Solitons and Fractals, 7 (1996), 2157-2174.  doi: 10.1016/S0960-0779(96)00077-X. [17] J. M. Fleming, Domestic financial policies under fixed and floating exchange rates, IMF Stuff Papers, 9 (1962), 369-379. [18] J. A. Frenkel and A. Razin, Fiscal Policies and the World Economy, MIT Press, Cambridge, MA, 1987.  doi: 10.1086/261390. [19] G. Gandolfo, Economic Dynamics, 4$^{th}$ edition, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-03871-6. [20] R. M. Goodwin, Essays in Economic Dynamics, Macmillan, London, 1982. doi: 10.1007/978-1-349-05504-3. [21] K. Hamada, The Political Economy of International Monetary Interdependence, MIT Press, Cambridge, MA, 1985. [22] T. Inaba and T. Asada, On dynamics of a three-country Kaldorian model of business cycles with fixed exchange rates, In Games and Dynamics in Economics: Essays in Honor of Akio Matsumoto, ed. F. Szidarovszky and G. I. Bischi doi: 10.1007/978-981-15-3623-6_6. [23] N. Kaldor, A model of the trade cycle, Econ. J., 50 (1940), 78-92.  doi: 10.2307/2225740. [24] M. Kalecki, Selected Essays on the Dynamics of the Capitalist Economy, Cambridge University Press, Cambridge, UK, 1971. [25] J. M. Keynes, The General Theory of Employment, Interest and Money, Macmillan, London, 1936. [26] P. Krugman, The Self-Organizing Economy, Blackwell, Oxford, UK, 1996. [27] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [28] A. Leijonhufvud, Effective demand failure, Swedish Journal of Economics, 75 (1973), 27-48.  doi: 10.2307/3439273. [29] W. M. Liu, Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256.  doi: 10.1006/jmaa.1994.1079. [30] H.-W. Lorenz, International trade and the possible occurrence of chaos, Economics Letters, 23 (1987), 135-138.  doi: 10.1016/0165-1765(87)90026-7. [31] H.-W. Lorenz, Strange attractors in a multisector business cycle model, Journal of Economic Behavior and Organization, 8 (1987), 397-411.  doi: 10.1016/0167-2681(87)90052-7. [32] P. Maličky and R. Zimka, On the existence of business cycles in Asada's two-regional model, Nonlinear Anal. Real World Appl., 11 (2010), 2787-2795.  doi: 10.1016/j.nonrwa.2009.10.003. [33] P. Maličky and R. Zimka, On the existence of Tori in Asada's two-regional model, Nonlinear Anal. Real World Appl., 13 (2012), 710-724.  doi: 10.1016/j.nonrwa.2011.08.011. [34] P. Medve${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over d} }}$ová, A dynamic model of a small open economy under flexible exchange rates, Acta Polytechnica Hungarica, 8 (2011), 13-26. [35] R. A. Mundell, Capital mobility and stabilization policy under fixed and flexible exchange rates, Canadian Journal of Economics and Political Science, 29 (1963), 475-485.  doi: 10.2307/139336. [36] R. A. Mundell, International Economics, Macmillan, New York, 1968. [37] M. Nakao, Stability of business cycles and economic openness of monetary union: A Kaldorian two-country model, Evolutionary and Institutional Economics Review, 16 (2019), 65-89.  doi: 10.1007/s40844-019-00124-6. [38] S. Newhouse, D. Ruelle and F. Takens, Occurrence of strange axiom A attractors near quasi-periodic flows on $T^m, m\geq 3$, Communications in Mathematical Physics, 64 (1978), 35-40.  doi: 10.1007/BF01940759. [39] J. Niehans, International Monetary Economics, John Hopkins University Press, Baltimore and London, 1984. [40] P. Nijkamp and A. Reggiani, Interaction, Evolution and Chaos in Space, Springer-Verlag, New York, 1992. doi: 10.1007/978-3-642-77509-3. [41] T. Puu, Nonlinear Economic Dynamics, $4^{th}$ edition, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-60775-2. [42] D. Romer, Advanced Macroeconomics, 4$^{th}$ edition, McGraw-Hill, New York, 2012. [43] J. B. Rosser Jr., From Catastrophe to Chaos: A General Theory of Economic Discontinuities, Kluwer Academic Publishers, Boston, 1991. doi: 10.1007/978-1-4613-3796-6. [44] N. Sarantis, Macroeconomic policy and activity in an open economy with oligopoly and collective bargaining, Journal of Economics, 49 (1989), 25-46.  doi: 10.1007/BF01227871. [45] W. Semmler, On nonlinear theories of cycles and the persistence of business cycles, Mathematical Social Sciences, 12 (1986), 47-76.  doi: 10.1016/0165-4896(86)90047-8. [46] R. Sethi, Endogenous growth cycles in an open economy with fixed exchange rates, Journal of Economic Behavior and Organization, 19 (1992), 327-342.  doi: 10.1016/0167-2681(92)90041-9. [47] 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. [48] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, 2. Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

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##### References:
 [1] T. Asada, Kaldorian dynamics in an open economy, J. Econ., 62 (1995), 239-269.  doi: 10.1007/BF01238819. [2] T. Asada, A two-regional model of business cycles with fixed exchange rates : A Kaldorian approach, Studies in Regional Sciences, 34 (2004), 19-38.  doi: 10.2457/srs.34.2_19. [3] T. Asada, C. Chiarella, P. Flaschel and R. Franke, Open Economy Macrodynamics: An Integrated Disequilibrium Approach, Springer-Verlag, Berlin, 2003. [4] T. Asada, C. Chiarella, P. Flaschel and R. Franke, Monetary Macrodynamics, Routledge, London, 2010. doi: 10.4324/9780203859964. [5] T. Asada, M. Demetrian and R. Zimka, On dynamics in a Keynesian model of monetary and fiscal policy with debt effect, Commun. Nonlinear Sci. Numer. Simul., 58 (2018), 131-146.  doi: 10.1016/j.cnsns.2017.06.013. [6] T. Asada, M. 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. [7] T. Asada, C. Douskos, V. Kalantonis and P. Markellos, Numerical exploration of Kaldorian interregional macrodynamics: Enhanced stability and predominance of period doubling under flexible exchange rates, Discrete Dyn. Nat. Soc., (2010), Art. ID. 263041, 1–29. doi: 10.1155/2010/263041. [8] T. Asada, C. Douskos and P. Markellos, Numerical exploration of Kaldorian interregional macrodynamics: Stability and the trade threshold for business cycles under fixed exchange rates, Nonlinear Dynamics, Psychology, and Life Sciences, 15 (2011), 105-128. [9] T. Asada, T. Inaba and T. Misawa, A nonlinear macrodynamic model with fixed exchange rates: Its dynamics and noise effects, Discrete Dyn. Nat. Soc., 4 (2000), 319-331.  doi: 10.1155/S1026022600000303. [10] T. Asada, T. Inaba and T. Misawa, An interregional dynamic model: The case of fixed exchange rates, Studies in Regional Science, 31 (2001), 29-41.  doi: 10.2457/srs.31.2_29. [11] T. Asada, V. Kalantonis, M. Markellos and P. Markellos, Analytical expressions of periodic disequilibrium fluctuations generated by Hopf bifurcations in economic dynamics, Appl. Math. Comput., 218 (2012), 7066-7077.  doi: 10.1016/j.amc.2011.12.063. [12] T. Asada, T. Misawa and T. Inaba, Chaotic dynamics in a flexible exchange rate system: A study of noise effects, Discrete Dyn. Nat. Soc., 4 (2000), 309-317. [13] R. Barro and X. Sala-i Martin, Economic Growth, 2$^{nd}$ edition, MIT Press, Cambridge, MA, 2004. [14] J. Benhabib and T. Miyao, Some new results on the dynamics of the generalized Tobin model, International Economic Review, 22 (1981), 589-596.  doi: 10.2307/2526160. [15] Yu. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702. Springer-Verlag, Berlin-New York, 1979. [16] A. Dohtani, T. Misawa, T. Inaba, M. Yokoo and T. Owase, Chaos, complex transients and noise: Illustration with a Kaldor model, Chaos, Solitons and Fractals, 7 (1996), 2157-2174.  doi: 10.1016/S0960-0779(96)00077-X. [17] J. M. Fleming, Domestic financial policies under fixed and floating exchange rates, IMF Stuff Papers, 9 (1962), 369-379. [18] J. A. Frenkel and A. Razin, Fiscal Policies and the World Economy, MIT Press, Cambridge, MA, 1987.  doi: 10.1086/261390. [19] G. Gandolfo, Economic Dynamics, 4$^{th}$ edition, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-03871-6. [20] R. M. Goodwin, Essays in Economic Dynamics, Macmillan, London, 1982. doi: 10.1007/978-1-349-05504-3. [21] K. Hamada, The Political Economy of International Monetary Interdependence, MIT Press, Cambridge, MA, 1985. [22] T. Inaba and T. Asada, On dynamics of a three-country Kaldorian model of business cycles with fixed exchange rates, In Games and Dynamics in Economics: Essays in Honor of Akio Matsumoto, ed. F. Szidarovszky and G. I. Bischi doi: 10.1007/978-981-15-3623-6_6. [23] N. Kaldor, A model of the trade cycle, Econ. J., 50 (1940), 78-92.  doi: 10.2307/2225740. [24] M. Kalecki, Selected Essays on the Dynamics of the Capitalist Economy, Cambridge University Press, Cambridge, UK, 1971. [25] J. M. Keynes, The General Theory of Employment, Interest and Money, Macmillan, London, 1936. [26] P. Krugman, The Self-Organizing Economy, Blackwell, Oxford, UK, 1996. [27] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [28] A. Leijonhufvud, Effective demand failure, Swedish Journal of Economics, 75 (1973), 27-48.  doi: 10.2307/3439273. [29] W. M. Liu, Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256.  doi: 10.1006/jmaa.1994.1079. [30] H.-W. Lorenz, International trade and the possible occurrence of chaos, Economics Letters, 23 (1987), 135-138.  doi: 10.1016/0165-1765(87)90026-7. [31] H.-W. Lorenz, Strange attractors in a multisector business cycle model, Journal of Economic Behavior and Organization, 8 (1987), 397-411.  doi: 10.1016/0167-2681(87)90052-7. [32] P. Maličky and R. Zimka, On the existence of business cycles in Asada's two-regional model, Nonlinear Anal. Real World Appl., 11 (2010), 2787-2795.  doi: 10.1016/j.nonrwa.2009.10.003. [33] P. Maličky and R. Zimka, On the existence of Tori in Asada's two-regional model, Nonlinear Anal. Real World Appl., 13 (2012), 710-724.  doi: 10.1016/j.nonrwa.2011.08.011. [34] P. Medve${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over d} }}$ová, A dynamic model of a small open economy under flexible exchange rates, Acta Polytechnica Hungarica, 8 (2011), 13-26. [35] R. A. Mundell, Capital mobility and stabilization policy under fixed and flexible exchange rates, Canadian Journal of Economics and Political Science, 29 (1963), 475-485.  doi: 10.2307/139336. [36] R. A. Mundell, International Economics, Macmillan, New York, 1968. [37] M. Nakao, Stability of business cycles and economic openness of monetary union: A Kaldorian two-country model, Evolutionary and Institutional Economics Review, 16 (2019), 65-89.  doi: 10.1007/s40844-019-00124-6. [38] S. Newhouse, D. Ruelle and F. Takens, Occurrence of strange axiom A attractors near quasi-periodic flows on $T^m, m\geq 3$, Communications in Mathematical Physics, 64 (1978), 35-40.  doi: 10.1007/BF01940759. [39] J. Niehans, International Monetary Economics, John Hopkins University Press, Baltimore and London, 1984. [40] P. Nijkamp and A. Reggiani, Interaction, Evolution and Chaos in Space, Springer-Verlag, New York, 1992. doi: 10.1007/978-3-642-77509-3. [41] T. Puu, Nonlinear Economic Dynamics, $4^{th}$ edition, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-60775-2. [42] D. Romer, Advanced Macroeconomics, 4$^{th}$ edition, McGraw-Hill, New York, 2012. [43] J. B. Rosser Jr., From Catastrophe to Chaos: A General Theory of Economic Discontinuities, Kluwer Academic Publishers, Boston, 1991. doi: 10.1007/978-1-4613-3796-6. [44] N. Sarantis, Macroeconomic policy and activity in an open economy with oligopoly and collective bargaining, Journal of Economics, 49 (1989), 25-46.  doi: 10.1007/BF01227871. [45] W. Semmler, On nonlinear theories of cycles and the persistence of business cycles, Mathematical Social Sciences, 12 (1986), 47-76.  doi: 10.1016/0165-4896(86)90047-8. [46] R. Sethi, Endogenous growth cycles in an open economy with fixed exchange rates, Journal of Economic Behavior and Organization, 19 (1992), 327-342.  doi: 10.1016/0167-2681(92)90041-9. [47] 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. [48] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, 2. Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7.
Solution starts 'inside' the unstable cycle and goes to the equilibrium. $\alpha_1 = 0.98\times\alpha_1^*$. Initial values: $Y_{10} = 1.3\times Y_1^*,Y_{20} = 1.3\times Y_2^*,Y_{30} = Y_3^*,K_{i0} = K_i^*,\ i = 1,2,3,\ M_{10} = M_1^*, M_{20} = M_2^*$. We see that all variables of the solution go to the equilibrium. This is in compliance with the result from the bifurcation equation (29) - that the equilibrium is locally stable on the left side of the bifurcation value $\alpha_1^*$
Solution starts 'outside' the unstable cycle and goes out of the cycle. $\alpha_1 = 0.98\times\alpha_1^*$. Initial values: $Y_{10} = 1.395\times Y_1^*,Y_{20} = 1.395\times Y_2^*,Y_{30} = Y_3^*,K_{i0} = K_i^*,\ i = 1,2,3,\ M_{10} = M_1^*, M_{20} = M_2^*$. We see that all variables of the solution which starts a little further from the equilibrium as the solution in Figure 1, go out of the equilibrium. Comparing the initial values of both solutions and their subsequent paths, we observe that these two solutions are 'separated' by a cycle that is unstable. This is in compliance with the result obtained from bifurcation equation (29), that the cycle is unstable (subcritical)
Solution goes out of the equilibrium. $\alpha_1 = 1.1\times\alpha_1^*$. Initial values: $Y_{10} = 1.1\times Y_1^*,Y_{20} = 1.1\times Y_2^*,Y_{30} = Y_3^*,K_{i0} = K_i^*,\ i = 1,2,3,\ M_{10} = M_1^*, M_{20} = M_2^*$. It is clear that if $\alpha_1>\alpha_1^*$ the equilibrium is unstable. This is consistent with our theoretical results. A similar result holds for the value $\alpha_1 = \alpha_1^*$. Though in this case the solution goes out of equilibrium more slowly than when $\alpha_1>\alpha_1^*$. This follows immediately from the first differential equation in model (28)
The dependence of equilibrium values on parameter $\delta$ at $\beta = \frac{5000}{3}$. These graphs show how equilibrium varies with parameter $\delta$, and how parameter $\delta$ influences real national incomes, real physical capital stocks and nominal money stocks in the selected economies
The dependence of $K_1^*,K_2^*,K_3^*$ on parameter $\beta$ at $\delta = 1$. The graphs show that the dependence of the equilibrium values $K_1^*,K_2^*,K_3^*$ on $\beta$ is almost negligible. The same holds for the equilibrium values of $M_1^*,M_2^*$. As the dependence of equilibrium values $K_1^*,K_2^*,K_3^*,M_1^*,M_2^*$ on the changes of parameter $\beta$ is negligible, we do not present considerations on the impact of simultaneous changes in both parameters $\delta$ and $\beta$ in this paper
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