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Chaotic sets and Euler equation branching

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  • Raines, Brian E.
  • Stockman, David R.

Abstract

Abstract Some macroeconomic models may exhibit a type of indeterminacy known as Euler equation branching (e.g., the one-sector growth model with a production externality). The dynamics in such models are governed by a differential inclusion , where H is a set-valued function. In this paper, we introduce the concept of a chaotic set and explore its implications for Devaney chaos, Li-Yorke chaos and distributional chaos (adapted to dynamical systems generated by a differential inclusion). We show that a chaotic set will imply Devaney and Li-Yorke chaos and that a chaotic set with Euler equation branching will imply distributional chaos. We show that the existence of a steady state for a differential inclusion on the plane will generate a chaotic set and hence Devaney and Li-Yorke chaos. As an application, we show how these results can be applied to a one-sector growth model with a production externality - extending the results of Christiano and Harrison (1999). We show that chaotic (Devaney, Li-Yorke and distributional) and cyclic equilibria are possible and that this behavior is not dependent on the steady state being "locally" a saddle, sink or source.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Mathematical Economics.

Volume (Year): 46 (2010)
Issue (Month): 6 (November)
Pages: 1173-1193

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Handle: RePEc:eee:mateco:v:46:y:2010:i:6:p:1173-1193

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Web page: http://www.elsevier.com/locate/jmateco

Related research

Keywords: Indeterminacy Euler equation branching Multiple equilibria Cycles Devaney chaos Li-Yorke chaos Distributional chaos Increasing returns to scale Externality Regime switching;

References

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  1. Schmitt-Grohe, Stephanie & Uribe, Martin, 1997. "Balanced-Budget Rules, Distortionary Taxes, and Aggregate Instability," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 105(5), pages 976-1000, October.
  2. Laura Gardini & Cars Hommes & Fabio Tramontana & Robin de Vilder, 2008. "Forward and Backward Dynamics in Implicitly Defined Overlapping Generations Models," Working Papers, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini 0806, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2008.
  3. Lawrence J. Christiano & Sharon G. Harrison, 1996. "Chaos, sunspots, and automatic stabilizers," Staff Report, Federal Reserve Bank of Minneapolis 214, Federal Reserve Bank of Minneapolis.
  4. Benhabib, Jess & Farmer, Roger E.A., 1995. "Indeterminacy and Sector-Specific Externalities," Working Papers, C.V. Starr Center for Applied Economics, New York University 95-02, C.V. Starr Center for Applied Economics, New York University.
  5. David R. Stockman, 2007. "Chaos and Sector-specific Externalities," Working Papers, University of Delaware, Department of Economics 07-17, University of Delaware, Department of Economics.
  6. Stockman, David R., 2010. "Balanced-budget rules: Chaos and deterministic sunspots," Journal of Economic Theory, Elsevier, Elsevier, vol. 145(3), pages 1060-1085, May.
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