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

Rudolf Zimka; Michal Demetrian; Toichiro Asada; Toshio Inaba

doi:10.3934/dcdsb.2021143

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2021, Volume 26, Issue 11: 5999-6015. Doi: 10.3934/dcdsb.2021143 open access

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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
^* Corresponding author: Rudolf Zimka

Received: February 28, 2021

Revised: August 31, 2020

Early access: May 2021

Published: November 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 37G15; Secondary: 91-10.

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Figure 1. 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^* $

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Figure 2. 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)

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Figure 3. 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)

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Figure 4. 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

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Figure 5. 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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