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Chaotic sets and Euler equation branching. (English) Zbl 1232.91467

Summary: 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 \(\dot x \in H(x)\), 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.

MSC:

91B52 Special types of economic equilibria
91B64 Macroeconomic theory (monetary models, models of taxation)
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[1] Benhabib, J.; Farmer, R.E.A., Indeterminacy and sector specific externalities, Journal of monetary economics, 37, 397-419, (1996)
[2] Brock, W.A., On existence of weakly maximal programmes in a multi-sector economy, Review of economic studies, 37, 2, 275-280, (1970) · Zbl 0212.23901
[3] Christiano, L.J.; Harrison, S.G., Chaos, sunspots and automatic stabilizers, Journal of monetary economics, 44, 1, 3-31, (1999)
[4] Devaney, R.L., An introduction to chaotic dynamical systems, (2003), Westview Press Boulder, CO · Zbl 1025.37001
[5] Forti, G.L., Various notions of chaos for discrete dynamical systems. A brief survey, Aequationes mathematicae, 70, 1-2, 1-13, (2005) · Zbl 1080.37010
[6] Gardini, L.; Hommes, C.; Tramontana, F.; de Vilder, R., Forward and backward dynamics in implicitly defined overlapping generations models, Journal of economic behavior & organization, 71, 2, 110-129, (2009)
[7] Kolyada, S.F., 2004. Li-Yorke sensitivity and other concepts of chaos, Natsı¯onal′na Akademı¯ya Nauk Ukraïni. Īnstitut Matematiki. Ukraïns′kiı˘, Matematichniı˘ Zhurnal 56 (8), 1043-1061.
[8] Li, T.-Y.; Yorke, J.A., Period three implies chaos, American mathematical monthly, 82, 985-992, (1975) · Zbl 0351.92021
[9] Oprocha, P., Relations between distributional and Devaney chaos, Chaos, 16, 3, 5, (2006), 033112 · Zbl 1146.37301
[10] Schmitt-Grohé, S.; Uribe, M., Balanced-budget rules, distortionary taxes, and aggregate instability, Journal of political economy, 105, 976-1000, (1997)
[11] Schweizer, B.; Sklar, A.; Smítal, J., Distributional (and other) chaos and its measurement, Real anal. exchange, 26, 2, 495-524, (2000) · Zbl 1012.37022
[12] Schweizer, B.; Smítal, J., Measures of chaos and a spectral decomposition of dynamical systems on the interval, Transactions of the American mathematical society, 344, 2, 737-754, (1994) · Zbl 0812.58062
[13] Smirnov, G.V., 2002. Introduction to the theory of differential inclusions, vol. 41 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI. · Zbl 0992.34001
[14] Stockman, D.R., Chaos and sector-specific externalities, Journal of economic dynamics & control, 33, 12, 2030-2046, (2009) · Zbl 1182.91128
[15] Stockman, D.R., Balanced-budget rules: chaos and deterministic sunspots, Journal of economic theory, 145, 3, 1060-1080, (2010) · Zbl 1245.91065
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