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IMPLICIT EQUILIBRIUM DYNAMICS

Published online by Cambridge University Press:  03 November 2011

Alfredo Medio  and
Brian Raines
Alfredo Medio*
Affiliation:
University of Udine
Brian Raines
Affiliation:
Baylor University
*
Address correspondence to: Alfredo Medio, 54 Route de la Pauvetta, 06140 Tourrettes Sur Loup, France; e-mail: alfredomedio@gmail.com
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Abstract

We discuss the problem known in economics as backward dynamics occurring in models of perfect foresight, intertemporal equilibrium described mathematically by implicit difference equations. In a previously published paper [Journal of Economic Dynamics and Control 31 (2007), 1633–1671], we showed that by means of certain mathematical methods and results known as inverse limits theory it is possible to establish a correspondence between the backward dynamics of a noninvertible map and the forward dynamics of a related invertible map acting on an appropriately defined space of sequences, each of whose elements corresponds to an intertemporal equilibrium. We also proved the existence of different types of topological attractors for one-dimensional models of overlapping generations. In this paper, we provide an extension of those results, constructing a Lebesgue-like probability measure on spaces of infinite sequences that allows us to distinguish typical from exceptional dynamical behaviors in a measure–theoretical sense, thus proving that all the topological attractors considered in MR07 are also metric attractors. We incidentally also prove that the existence of chaos (in the Devaney–Touhey sense) backward in time implies (and is implied by) chaos forward in time.

Keywords

Backward DynamicsOverlapping GenerationsInverse LimitsAttractorsLebesgue-Like Measure
Type
Articles
Information
Macroeconomic Dynamics , Volume 16 , Issue 4 , September 2012 , pp. 518 - 555
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Avila, A., Lyubich, M., and de Melo, W. (2004) Regular or stochastic dynamics in real analytic families of unimodal maps. Annals of Mathematics 156, 1–78.Google Scholar
Avila, A. and Moreira, C. G. (2005a) Phase–parameter relation and sharp statistical properties for general families of unimodal map. Contemporary Mathematics 389, 1–42.CrossRefGoogle Scholar
Avila, A. and Moreira, C. G. (2005b) Statistical properties of unimodal maps: Physical measures, periodic points and pathological laminations. Publications Mathématiques de l'IHÉS 101, 1–67.CrossRefGoogle Scholar
Banks, J., Brooks, J., Cairns, G., Davis, G., and Stacey, P. (1992) On Devaney's definition of chaos. American Mathematical Monthly 99 (4), 332–334.CrossRefGoogle Scholar
Barge, M. and Diamond, B. (1994) The dynamics of continuous maps of finite graphs through inverse limits. Transactions of the American Mathematical Society 344 (2), 773–790.CrossRefGoogle Scholar
Benhabib, J. and Day, R. (1982) A characterization of erratic dynamics in the overlapping generations model. Journal of Economic Dynamics and Control 4, 37–55.CrossRefGoogle Scholar
Boldrin, M. and Woodford, M. (1990) Equilibrium models displaying endogenous fluctuations and chaos: A survey. Journal of Monetary Economics 25 (2), 189–222.CrossRefGoogle Scholar
Devaney, R. L. (2003) An Introduction to Chaotic Dynamical Systems, 2nd ed. Boulder, CO: Westview Press.Google Scholar
Gale, D. (1973) Pure exchange equilibrium of dynamic economic models. Journal of Economic Theory 6, 12–36.CrossRefGoogle Scholar
Gardini, L., Hommes, C., Tramontana, F., and de Vilder, R. (2009) Forward and backward dynamics in implicitly defined overlapping generations models. Journal of Economic Behavior and Organization 71 (2), 110–129.CrossRefGoogle Scholar
Grandmont, J-M. (1983) On endogenous competitive business cycles. In Sonnenschein, H. F. (ed.), Models of Economic Dynamics, Lecture Notes in Economics and Mathematical Systems, Vol. 264. New York: Springer–Verlag.Google Scholar
Grandmont, J-M. (1985) On endogenous competitive business cycles. Econometrica 53, 995–1045.CrossRefGoogle Scholar
Grandmont, J-M. (1989) Local bifurcations and stationary sunspots. In Barnett, W. A., Geweke, J., and Shell, K. (eds.), Economic Complexity: Chaos, Sunspots, Bubbles and Nonlinearity. Cambridge, UK: Cambridge University Press.Google Scholar
Ingram, W. T. (1995) Periodicity and indecomposability. Proceedings of the American Mathematical Society 123, 1907–1916.CrossRefGoogle Scholar
Ingram, W. T. (2000a) Inverse limits on [0,1] using piecewise linear unimodal bonding maps. Proceedings of the American Mathematical Society 128 (1), 279–286.CrossRefGoogle Scholar
Ingram, W. T. (2000b) Inverse limits. Aportaciones Mathemática 15, 1–80.Google Scholar
Katok, A. and Hasselblatt, B. (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Kennedy, J. A. and Stockman, D. R. (2008) Chaotic equilibria in models with backward dynamics. Journal of Economic Dynamics and Control 32, 939–955.CrossRefGoogle Scholar
Kennedy, J., Stockman, D., and Yorke, J. (2007) Inverse limits and an implicitly defined difference equation from economics. Topology and Its Applications 154, 2533–2552.CrossRefGoogle Scholar
Kennedy, J., Stockman, D. R., and Yorke, J. A. (2008) The inverse limits approach to chaos. Journal of Mathematical Economics 44, 423–444.CrossRefGoogle Scholar
Keynes, J. M. (1921) A Treatise on Probability. London: MacMillan.Google Scholar
Kraft, R. L. (1999) Chaos, cantor sets and hyperbolicity for logistic maps. American Mathematical Monthly 106 (5), 400–408.CrossRefGoogle Scholar
Li, S. H. (1992) Dynamical properties of the shift maps on the inverse limit spaces. Ergodic Theory and Dynamical System 12, 95–108.CrossRefGoogle Scholar
Medio, A. (1998) The Problem of Backward Dynamics in Economics. Working Paper 98.05, University Ca' Foscari, Venice.Google Scholar
Medio, A. and Raines, B. (2006) Inverse limit spaces arising from problems in economics. Topology and Its Applications 153 (18), 3437–3449.CrossRefGoogle Scholar
Medio, A. and Raines, B. (2007) [MR07] Backward dynamics in economics: The inverse limit approach. Journal of Economic Dynamics and Control 31, 1633–1671.CrossRefGoogle Scholar
Ott, E. (2006) Basin of attraction. Scholarpedia 1 (8), 1701.CrossRefGoogle Scholar
Oxtoby, J. C. (1971) Measure and Category. New York: Springer-Verlag.CrossRefGoogle Scholar
Samuelson, P. A. (1958) An exact consumption–loan model of interest with or without the social contrivance of money. Journal of Political Economy 66 (6), 467–482.CrossRefGoogle Scholar
Shultz, F. (2007) Chaotic unimodal and bimodal maps, preprint. http://palmer.wellesley.edu/~fshultz/ChaoticMapsShultz.pdf.Google Scholar
Touhey, P. (1997) Yet another definition of chaos. American Mathematical Monthly 104 (5), 411–414.CrossRefGoogle Scholar
Vellekoop, M. and Berglund, R. (1994) On intervals: Transitivity is chaos. American Mathematical Monthly 101 (4), 353–355.Google Scholar
Walters, P. (1982) An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79. New York/Berlin: Springer–Verlag.CrossRefGoogle Scholar