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Chaotic equilibria in models with backward dynamics

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  • Kennedy, Judy A.
  • Stockman, David R.

Abstract

Some economic models like the cash-in-advance model of money or overlapping generations model have the property that the dynamical system characterizing equilibria in the model are multi-valued going forward in time, but single-valued going backward in time, i.e., the model or dynamical system has backward dynamics. In such instances, what does it mean for a dynamical system (set-valued) to be chaotic? Furthermore, under what conditions are such dynamical systems chaotic? In this paper, we provide a definition of chaos that is in the spirit of Devaney for a dynamical system with backward dynamics. We utilize the theory of inverse limits to provide necessary and sufficient conditions for such a dynamical system to be chaotic.

Suggested Citation

  • Kennedy, Judy A. & Stockman, David R., 2008. "Chaotic equilibria in models with backward dynamics," Journal of Economic Dynamics and Control, Elsevier, vol. 32(3), pages 939-955, March.
    • Handle: RePEc:eee:dyncon:v:32:y:2008:i:3:p:939-955
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    References listed on IDEAS

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    1. Michener, Ronald & Ravikumar, B., 1998. "Chaotic dynamics in a cash-in-advance economy," Journal of Economic Dynamics and Control, Elsevier, vol. 22(7), pages 1117-1137, May.
    2. Lucas, Robert E, Jr & Stokey, Nancy L, 1987. "Money and Interest in a Cash-in-Advance Economy," Econometrica, Econometric Society, vol. 55(3), pages 491-513, May.
    3. Christiano, Lawrence J. & G. Harrison, Sharon, 1999. "Chaos, sunspots and automatic stabilizers," Journal of Monetary Economics, Elsevier, vol. 44(1), pages 3-31, August.
    4. David Stockman & Judy Kennedy & James Yorke, 2006. "Inverse Limits and Models with Backward Dynamics," Working Papers 06-12, University of Delaware, Department of Economics.
    5. Gale, David, 1973. "Pure exchange equilibrium of dynamic economic models," Journal of Economic Theory, Elsevier, vol. 6(1), pages 12-36, February.
    6. Grandmont, Jean-Michel, 1985. "On Endogenous Competitive Business Cycles," Econometrica, Econometric Society, vol. 53(5), pages 995-1045, September.
    7. Benhabib, Jess & Day, Richard H., 1982. "A characterization of erratic dynamics in, the overlapping generations model," Journal of Economic Dynamics and Control, Elsevier, vol. 4(1), pages 37-55, November.
    8. Medio, Alfredo & Raines, Brian, 2007. "Backward dynamics in economics. The inverse limit approach," Journal of Economic Dynamics and Control, Elsevier, vol. 31(5), pages 1633-1671, May.
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    Citations

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    Cited by:

    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. Airaudo, Marco, 2016. "Endogenous Stock Price Fluctuations with Dynamic Self-Control Preferences," School of Economics Working Paper Series 2016-2, LeBow College of Business, Drexel University.
    3. Deng, Liuchun & Khan, M. Ali, 2018. "On growing through cycles: Matsuyama’s M-map and Li–Yorke chaos," Journal of Mathematical Economics, Elsevier, vol. 74(C), pages 46-55.
    4. Nika Lazaryan & Thomas A. Lubik, 2019. "Global dynamics in a search and matching model of the labor market," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(2), pages 461-497, September.
    5. David R. Stockman & Judy Kennedy & Brian Raines, 2007. "Expected Utility in Models with Chaos," Working Papers 07-16, University of Delaware, Department of Economics.
    6. Jess Benhabib & Jianjun Miao & Pengfei Wang, 2016. "Chaotic banking crises and regulations," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(2), pages 393-422, February.
    7. David R. Stockman, 2008. "Uniform Measures On Inverse Limit Spaces," Working Papers 08-25, University of Delaware, Department of Economics.
    8. Grandmont, Jean-Michel, 2008. "Nonlinear difference equations, bifurcations and chaos: An introduction," Research in Economics, Elsevier, vol. 62(3), pages 122-177, September.
    9. Alves, P.R.L. & Duarte, L.G.S. & da Mota, L.A.C.P., 2017. "A new characterization of chaos from a time series," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 323-326.
    10. David Stockman & Judy Kennedy & James Yorke, 2006. "Inverse Limits and Models with Backward Dynamics," Working Papers 06-12, University of Delaware, Department of Economics.
    11. Kennedy, Judy & Stockman, David R. & Yorke, James A., 2008. "The inverse limits approach to chaos," Journal of Mathematical Economics, Elsevier, vol. 44(5-6), pages 423-444, April.
    12. Nika Lazaryan & Thomas A. Lubik, 2017. "Global Dynamics in a Search and Matching Model of the Labor Market," Working Paper 17-12, Federal Reserve Bank of Richmond.
    13. Walter Briec & Laurence Lasselle, 2013. "On the Use of the Renormalization Procedure to Estimate the Bifurcation Parameters in Nonlinear Dynamic Models," Computational Economics, Springer;Society for Computational Economics, vol. 41(4), pages 557-574, April.

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