Homoclinic Bifurcation and Endogenous Cycles in the Dynamic IS-LM Model
DOI:
https://doi.org/10.6000/1929-7092.2015.04.24Keywords:
Multiple steady states, homoclinic bifurcation, oscillating solutionsAbstract
This paper contributes to the new keynesian literature by showing that stable endogenous cycles can emerge as equilibrium solutions of the traditional IS-LM model. The application of the original Bogdanov-Takens theorem allows us to determine the regions of the parametric space where the model exhibits a global indeterminate solution, and a low-growth trapping region, characterized by a continuum of equilibrium trajectories in the proximity of a homoclinic bifurcation.
References
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Bischi, Gian Italo, Roberto Dieci, Giorgio Rodano and Enrico Saltari 2001. “Multiple attractors and global bifurcations in a Kaldor-type business cycle model.” Journal of Evolutionary Economics 11: 527-554.
http://dx.doi.org/10.1007/s191-001-8320-9
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Boldrin, Michele, Kazuo Nishimura, Tadashi Shigoka and Makoto Yano 2001. “Chaotic Equilibrium Dynamics in Endogenous Growth Models.” Journal of Economic Theory 96: 97-132.
http://dx.doi.org/10.1006/jeth.2000.2677
Cai, Jianping 2005. “Hopf bifurcation in the IS-LM business cycle model with time delay.” Electronic Journal of Differential Equations 15: 1-6.
Chamley, Christophe 1993. “Externalities and dynamics in models of learning or doing.” International Economic Review 34: 583-609.
http://dx.doi.org/10.2307/2527183
De Cesare, Luigi and Mario Sportelli 2005. “A dynamic IS-LM model with delayed taxation revenues.” Chaos, Solitons & Fractals 25: 233-44.
http://dx.doi.org/10.1016/j.chaos.2004.11.044
Fanti, Luciano and Piero Manfredi 2007. “Chaotic business cycles and fiscal policy: An IS-LM model with distributed tax collection lags.” Chaos, Solitons and Fractals 32: 736-744.
http://dx.doi.org/10.1016/j.chaos.2005.11.024
Freire, Emilio, Estanislao Gamero, Enrique Ponce, Leopoldo G. Franquelo 1989. “An algorithm for symbolic computation of center manifold.” Pp. 218-230 in Symbolic And Algebraic Computation, edited by P. Gianni. Berlin: Springer-Verlag.
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Gandolfo Giancarlo. 1997. Economic dynamics, Berlin, Springer-Verlag.
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http://dx.doi.org/10.1007/978-1-4612-1140-2
Guirao, Juan L.G., Raquel García-Rubio, Juan A. Vera 2012. “On the dynamics of an inflation IS-LM model.” Economic Modelling 29: 2090-2094.
http://dx.doi.org/10.1016/j.econmod.2012.07.007
Kali?inská, Barbora V. 2012. “Augmented IS-LM model based on particular functions.” Applied Mathematics and Computation 219: 1244-1262.
http://dx.doi.org/10.1016/j.amc.2012.07.033
Lorenz, Hans-Walter. 1993. Nonlinear Dynamical Economics and Chaotic Motion, New York, Springer-Verlag.
http://dx.doi.org/10.1007/978-3-642-78324-1
Makovinyiova, Katarina 2011. “On the existence and stability of business cycles in a dynamic model of a closed economy.” Nonlinear Analysis: Real World Applications 12: 1213-1222.
http://dx.doi.org/10.1016/j.nonrwa.2010.09.016
Makovinyiova, Katarina and Rudolf Zimka 2009. “On stability in generalized Schinasi's macroeconomic model under fixed exchange rates.” Tatra Mt. Math. Publ. 43: 115-122.
http://dx.doi.org/10.2478/v10127-009-0031-8
Mattana, Paolo, Kazuo Nishimura and Tadashi Shigoka 2009. “Homoclinic bifurcation and global indeterminacy of equilibrium in a two-sector endogenous growth model.” International Journal of Economic Theory 5: 1-23.
http://dx.doi.org/10.1111/j.1742-7363.2008.00093.x
Neamtu, Mihaela, Opri? Dimitru and Constantin Chilarescu 2007. “Hopf bifurcation in a dynamic IS-LM model with time delay.” Chaos, Solitons and Fractals 34: 519-530.
http://dx.doi.org/10.1016/j.chaos.2006.03.052
Neri, Umberto and Beatrice Venturi 2007. “Stability and Bifurcations in IS-LM economic models.” International Review of Economics 54: 53-65.
http://dx.doi.org/10.1007/s12232-007-0007-4
Sasakura, Kazuyuki 1994. “On the dynamic behavior of Schinasi's business cycle model.” Journal of Macroeconomics 16: 423-444.
http://dx.doi.org/10.1016/0164-0704(94)90015-9
Schinasi, Garry J. 1981. “A nonlinear dynamic model of short run fluctuations.” Review of Economic Studies 48: 649-653.
http://dx.doi.org/10.2307/2297204
Schinasi, Garry J. 1982. Fluctuations in a dynamic, intermediate-run IS-LM model: applications of the Poincaré-Bendixon theorem.” Journal of Economic Theory 28: 369-375.
http://dx.doi.org/10.1016/0022-0531(82)90068-0
Slobodyan, Sergey 2007. “Indeterminacy and stability in a modified Romer model.” Journal of Macroeconomics 29: 169-177.
http://dx.doi.org/10.1016/j.jmacro.2005.08.001
Tu, Hongliang, Xiaoli He and Junhai Ma 2013. “Stability and Hopf Bifurcation of a 3-dimensional Dynamic IS-LM Model with Delays.” Journal of Information & Computational Science 10: 2581-2590.
http://dx.doi.org/10.12733/jics20101773
Wiggins, Stephen. 1991. Introduction to applied nonlinear dynamical systems and chaos, New York, Springer-Verlag.
http://dx.doi.org/10.2307/1059575
Baffigi, Alberto 2011. “Italian National Accounts.” Economic History Working Papers, Banca d'Italia No. 18.
Bella, Giovanni and Paolo Mattana 2014. “Global indeterminacy of the equilibrium in the Chamley model of endogenous growth in the vicinity of a Bogdanov-Takens bifurcation.” Mathematical Social Sciences 71: 69-79.
http://dx.doi.org/10.1016/j.mathsocsci.2014.05.001
Benhabib, Jess and Roger E.A. Farmer 1994. “Indeterminacy and increasing returns.” Journal of Economic Theory 63: 19-46.
http://dx.doi.org/10.1006/jeth.1994.1031
Benhabib, Jess and Roger E.A. Farmer 1996. “Indeterminacy and sector-specific externalities.” Journal of Monetary Economics 17: 421-43.
http://dx.doi.org/10.1016/0304-3932(96)01257-3
Benhabib, Jess, Quinglai Meng and Kazuo Nishimura 2000. “Indeterminacy under constant returns to scale in multisector economies.” Econometrica 68: 1541-1548.
http://dx.doi.org/10.1111/1468-0262.00173
Benhabib, Jess, Kazuo Nishimura and Tadashi Shigoka 2008. “Bifurcation and sunspots in the continuous time equilibrium model with capacity utilization.” International Journal of Economic Theory 4: 337-355.
http://dx.doi.org/10.1111/j.1742-7363.2008.00083.x
Benhabib, Jess and Roberto Perli 1994. “Uniqueness and indeterminacy: On the dynamics of endogenous growth.” Journal of Economic Theory 63: 113-142.
http://dx.doi.org/10.1006/jeth.1994.1035
Benhabib, Jess, Roberto Perli and Danyang Xie 1994. “Monopolistic Competition, Indeterminacy and Growth.” Ricerche Economiche 48: 279-298.
http://dx.doi.org/10.1016/0035-5054(94)90009-4
Benhabib, Jess, Stephanie Schmitt-Grohé and Martin Uribe 2001. “The perils of Taylor rules.” Journal of Economic Theory 96: 40-69.
http://dx.doi.org/10.1006/jeth.1999.2585
Bischi, Gian Italo, Roberto Dieci, Giorgio Rodano and Enrico Saltari 2001. “Multiple attractors and global bifurcations in a Kaldor-type business cycle model.” Journal of Evolutionary Economics 11: 527-554.
http://dx.doi.org/10.1007/s191-001-8320-9
Blanchard, Olivier J. and Stanley Fischer. 1989. Lectures on Macroeconomics, Cambridge, MIT Press.
Boldrin, Michele, Kazuo Nishimura, Tadashi Shigoka and Makoto Yano 2001. “Chaotic Equilibrium Dynamics in Endogenous Growth Models.” Journal of Economic Theory 96: 97-132.
http://dx.doi.org/10.1006/jeth.2000.2677
Cai, Jianping 2005. “Hopf bifurcation in the IS-LM business cycle model with time delay.” Electronic Journal of Differential Equations 15: 1-6.
Chamley, Christophe 1993. “Externalities and dynamics in models of learning or doing.” International Economic Review 34: 583-609.
http://dx.doi.org/10.2307/2527183
De Cesare, Luigi and Mario Sportelli 2005. “A dynamic IS-LM model with delayed taxation revenues.” Chaos, Solitons & Fractals 25: 233-44.
http://dx.doi.org/10.1016/j.chaos.2004.11.044
Fanti, Luciano and Piero Manfredi 2007. “Chaotic business cycles and fiscal policy: An IS-LM model with distributed tax collection lags.” Chaos, Solitons and Fractals 32: 736-744.
http://dx.doi.org/10.1016/j.chaos.2005.11.024
Freire, Emilio, Estanislao Gamero, Enrique Ponce, Leopoldo G. Franquelo 1989. “An algorithm for symbolic computation of center manifold.” Pp. 218-230 in Symbolic And Algebraic Computation, edited by P. Gianni. Berlin: Springer-Verlag.
http://dx.doi.org/10.1007/3-540-51084-2_20
Gamero, Estanislao, Emilio Freire, Enrique Ponce 1991. “Normal forms for planar systems with nilpotent linear part.” Pp. 123-127 in Bifurcation and Chaos: Analysis, Algorithms, Applications, edited by R. Seydel, F.W. Schneider, T. Küpper and H. Troger. Basel: Birkhäuser.
http://dx.doi.org/10.1007/978-3-0348-7004-7_14
Gandolfo Giancarlo. 1997. Economic dynamics, Berlin, Springer-Verlag.
Guckenheimer, John and Philip Holmes. 1983. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, New York, Springer-Verlag.
http://dx.doi.org/10.1007/978-1-4612-1140-2
Guirao, Juan L.G., Raquel García-Rubio, Juan A. Vera 2012. “On the dynamics of an inflation IS-LM model.” Economic Modelling 29: 2090-2094.
http://dx.doi.org/10.1016/j.econmod.2012.07.007
Kali?inská, Barbora V. 2012. “Augmented IS-LM model based on particular functions.” Applied Mathematics and Computation 219: 1244-1262.
http://dx.doi.org/10.1016/j.amc.2012.07.033
Lorenz, Hans-Walter. 1993. Nonlinear Dynamical Economics and Chaotic Motion, New York, Springer-Verlag.
http://dx.doi.org/10.1007/978-3-642-78324-1
Makovinyiova, Katarina 2011. “On the existence and stability of business cycles in a dynamic model of a closed economy.” Nonlinear Analysis: Real World Applications 12: 1213-1222.
http://dx.doi.org/10.1016/j.nonrwa.2010.09.016
Makovinyiova, Katarina and Rudolf Zimka 2009. “On stability in generalized Schinasi's macroeconomic model under fixed exchange rates.” Tatra Mt. Math. Publ. 43: 115-122.
http://dx.doi.org/10.2478/v10127-009-0031-8
Mattana, Paolo, Kazuo Nishimura and Tadashi Shigoka 2009. “Homoclinic bifurcation and global indeterminacy of equilibrium in a two-sector endogenous growth model.” International Journal of Economic Theory 5: 1-23.
http://dx.doi.org/10.1111/j.1742-7363.2008.00093.x
Neamtu, Mihaela, Opri? Dimitru and Constantin Chilarescu 2007. “Hopf bifurcation in a dynamic IS-LM model with time delay.” Chaos, Solitons and Fractals 34: 519-530.
http://dx.doi.org/10.1016/j.chaos.2006.03.052
Neri, Umberto and Beatrice Venturi 2007. “Stability and Bifurcations in IS-LM economic models.” International Review of Economics 54: 53-65.
http://dx.doi.org/10.1007/s12232-007-0007-4
Sasakura, Kazuyuki 1994. “On the dynamic behavior of Schinasi's business cycle model.” Journal of Macroeconomics 16: 423-444.
http://dx.doi.org/10.1016/0164-0704(94)90015-9
Schinasi, Garry J. 1981. “A nonlinear dynamic model of short run fluctuations.” Review of Economic Studies 48: 649-653.
http://dx.doi.org/10.2307/2297204
Schinasi, Garry J. 1982. Fluctuations in a dynamic, intermediate-run IS-LM model: applications of the Poincaré-Bendixon theorem.” Journal of Economic Theory 28: 369-375.
http://dx.doi.org/10.1016/0022-0531(82)90068-0
Slobodyan, Sergey 2007. “Indeterminacy and stability in a modified Romer model.” Journal of Macroeconomics 29: 169-177.
http://dx.doi.org/10.1016/j.jmacro.2005.08.001
Tu, Hongliang, Xiaoli He and Junhai Ma 2013. “Stability and Hopf Bifurcation of a 3-dimensional Dynamic IS-LM Model with Delays.” Journal of Information & Computational Science 10: 2581-2590.
http://dx.doi.org/10.12733/jics20101773
Wiggins, Stephen. 1991. Introduction to applied nonlinear dynamical systems and chaos, New York, Springer-Verlag.
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Published
2015-12-14
How to Cite
Bella, G. (2015). Homoclinic Bifurcation and Endogenous Cycles in the Dynamic IS-LM Model. Journal of Reviews on Global Economics, 4, 242–250. https://doi.org/10.6000/1929-7092.2015.04.24
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Special Issue - Hayek, Keynes and the Crisis: Analyses and Remedies. An Introduction
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