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Chaotic sets and Euler equation branching

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

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.

Suggested Citation

  • Raines, Brian E. & Stockman, David R., 2010. "Chaotic sets and Euler equation branching," Journal of Mathematical Economics, Elsevier, vol. 46(6), pages 1173-1193, November.
    • Handle: RePEc:eee:mateco:v:46:y:2010:i:6:p:1173-1193
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    References listed on IDEAS

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    1. Gardini, Laura & Hommes, Cars & Tramontana, Fabio & de Vilder, Robin, 2009. "Forward and backward dynamics in implicitly defined overlapping generations models," Journal of Economic Behavior & Organization, Elsevier, vol. 71(2), pages 110-129, August.
    2. Benhabib, Jess & Farmer, Roger E. A., 1996. "Indeterminacy and sector-specific externalities," Journal of Monetary Economics, Elsevier, vol. 37(3), pages 421-443, June.
    3. Stockman, David R., 2010. "Balanced-budget rules: Chaos and deterministic sunspots," Journal of Economic Theory, Elsevier, vol. 145(3), pages 1060-1085, May.
    4. Christiano, Lawrence J. & G. Harrison, Sharon, 1999. "Chaos, sunspots and automatic stabilizers," Journal of Monetary Economics, Elsevier, vol. 44(1), pages 3-31, August.
    5. Stockman, David R., 2009. "Chaos and sector-specific externalities," Journal of Economic Dynamics and Control, Elsevier, vol. 33(12), pages 2030-2046, December.
    6. Schmitt-Grohe, Stephanie & Uribe, Martin, 1997. "Balanced-Budget Rules, Distortionary Taxes, and Aggregate Instability," Journal of Political Economy, University of Chicago Press, vol. 105(5), pages 976-1000, October.
    7. W. A. Brock, 1970. "On Existence of Weakly Maximal Programmes in a Multi-Sector Economy," Review of Economic Studies, Oxford University Press, vol. 37(2), pages 275-280.
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    Cited by:

    1. Manjira Datta & Kevin Reffett & Łukasz Woźny, 2018. "Comparing recursive equilibrium in economies with dynamic complementarities and indeterminacy," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(3), pages 593-626, October.
    2. Volná, Barbora, 2015. "Existence of chaos in the plane R2 and its application in macroeconomics," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 237-266.

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