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INVARIANT PROBABILITY DISTRIBUTIONS IN ECONOMIC MODELS: A GENERAL RESULT

Published online by Cambridge University Press:  10 June 2004

ALFREDO MEDIO
ALFREDO MEDIO
Affiliation:
University Ca' Foscari of Venice
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Abstract

This paper discusses the asymptotic behavior of distributions of state variables of Markov processes generated by first-order stochastic difference equations. It studies the problem in a context that is general in the sense that (i) the evolution of the system takes place in a general state space (i.e., a space that is not necessarily finite or even countable); and (ii) the orbits of the unperturbed, deterministic component of the system converge to subsets of the state space which can be more complicated than a stationary state or a periodic orbit, that is, they can be aperiodic or chaotic. The main result of the paper consists of the proof that, under certain conditions on the deterministic attractor and the stochastic perturbations, the Markov process describing the dynamics of a perturbed deterministic system possesses a unique, invariant, and stochastically stable probability measure. Some simple economic applications are also discussed.

Keywords

Stochastic StabilityMarkov ProcessesChaotic DynamicsSunspots
Type
Research Article
Information
Macroeconomic Dynamics , Volume 8 , Issue 2 , April 2004 , pp. 162 - 187
Copyright
© 2004 Cambridge University Press

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References

Benhabib J. (ed.) 1992 Cycles and Chaos in Economic Equilibrium. Princeton, NJ: Princeton University Press.
Benhabib J. & R.H. Day 1982 A characterization of erratic dynamics in the overlapping generations model. Journal of Economic Dynamics and Control 4, 37 55.Google Scholar
Blume L.E. 1982 New techniques for the study of stochastic equilibrium processes. Journal of Mathematical Economics 9, 71 82.Google Scholar
Boldrin M. & L. Montrucchio 1986 On the indeterminacy of capital accumulation paths. Journal of Economic Theory 40, 26 39.Google Scholar
Boldrin M. & M. Woodford 1990 Equilibrium models displaying endogenous fluctuations and chaos: A survey. Journal of Monetary Economics 25, 189 223.Google Scholar
Brock W.A. & M. Majumdar 1978 Global asymptotic stability results for multisector models of optimal growth under uncertainty when future utilities are discounted. Journal of Economic Theory 18, 225 243.Google Scholar
Brock W.A. & L.J. Mirman 1972 Optimal economic growth and uncertainty: the discounted case. Journal of Economic Theory 4, 479 513.Google Scholar
Chan K.S. & H. Tong 1994 A note on noisy chaos. Journal of the Royal Statistical Society B 56, 301 311.Google Scholar
Chiappori P.A. & R. Guesnerie 1991 Sunspot equilibria in sequential markets models. In W. Hildenbrand & H. Sonneschein (eds.), Handbook of Mathematical Economics, vol. IV, pp. 1683 1762. New York: Elsevier.
Cugno F. & L. Montrucchio 1998 Scelte Intertemporali: Teoria e Modelli. Rome: Carocci Editore.
Deneckere R. & S. Pelikan 1986 Competitive chaos. Journal of Economic Theory 40, 13 25.Google Scholar
Doob J.L. 1953 Stochastic Processes. New York: John Wiley & Sons.
Duffie D., J. Genakoplos, A. Mas-Colell, & A. MacLennan 1994 Stationary Markov equilibria. Econometrica 62, 745 783.Google Scholar
Farmer R.E.A. & M. Woodford 1997 Self-fulfilling prophecies and the business cycle. Macroeconomic Dynamics 1, 740 770.Google Scholar
Futia C. 1982 Invariant distributions and the limiting behavior of Markov economic models. Econometrica 50, 377 407.Google Scholar
Gale D. 1973 Pure exchange equilibrium of dynamic economic model. Journal of Economic Theory 6, 12 36.Google Scholar
Gordon S.P. 1972 On converses of the stability theorems for difference equations. SIAM Journal of Control 10, 76 81.Google Scholar
Hahn W. 1963 Theory and Applications of Liapunov's Direct Method. Englewood Cliffs, NJ: Prentice-Hall Int.
Hahn W. 1967 Stability of Motion. Berlin: Springer-Verlag.
Joshi S. 1995 Recursive utility and optimal growth under uncertainty. Journal of Mathematical Economics 24, 601 617.Google Scholar
Katok A. & B. Hasselblatt 1995 Introduction to the Modern Theory of Dynamical Systems. Cambridge, UK: Cambridge University Press.
Majumdar M. & I. Zilcha 1987 Optimal growth in a stochastic environment: Some sensitivity and turnpike results. Journal of Economic Theory 43, 116 133.Google Scholar
Massera J.L. 1949 On Liapounoff's condition of stability. Annales of Mathematics (2) 50, 705 721.Google Scholar
May R.M. & G.F. Oster 1976 Bifurcations and dynamic complexity in simple ecological models.American Naturalist 110, 573 594.Google Scholar
Medio A. 1992 Chaotic Dynamics. Theory and Applications to Economics. Cambridge, UK: Cambridge University Press.
Meyn S.P. & P.E. Caines 1991 Asymptotic behavior of stochastic systems processing Markov realizations. SIAM Journal of Control and Optimization 29, 535 561.Google Scholar
Meyn S.P. & R.L. Tweedie 1993 Markov Chains and Stochastic Stability. London: Springer-Verlag.
Montrucchio L. 1986 Optimal decisions over time and strange attractors: An analysis by the Bellman Principle. Mathematical Modelling 7, 341 352.Google Scholar
Radner R. 1973 Optimal stationary consumption with stochastic production and resources. Journal of Economic Theory 6, 68 90.Google Scholar
Reichlin P. 1986 Equilibrium cycles and stabilization policies in an overlapping generations models with production. Journal of Economic Theory 40, 89 102.Google 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, 467 482.Google Scholar
Sharkovsky A.N., S.F. Kolyada, A.G. Sivak, & V.V. Fedorenko 1997 Dynamics of One-Dimensional Maps. Dordrecht, The Netherlands: Kluwer Academic.
Stokey N.L. & R.E. Lucas, Jr. 1989 Recursive Methods in Economic Dynamics. Cambridge, MA: Harvard University Press.
Tong H. 1990 Non-Linear Time Series. A Dynamical System Approach. New York: Oxford University Press.
Tuominen P. & R.L. Tweedie 1979 Markov chains with continuous components. Proceedings of the London Mathematical Society (3) 38, 89 114.Google Scholar
Whitley D. 1983 Discrete dynamical systems in dimensions one and two. Bulletin of London Mathematical Society 15, 177 217.Google Scholar