Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Stochastic equilibria in von Neumann-Gale dynamical systems


Authors: Igor V. Evstigneev and Klaus Reiner Schenk-Hoppé
Journal: Trans. Amer. Math. Soc. 360 (2008), 3345-3364
MSC (2000): Primary 37H99, 37H15; Secondary 91B62, 91B28.
Published electronically: January 11, 2008
MathSciNet review: 2379800
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper examines a class of random dynamical systems related to the classical von Neumann and Gale models of economic dynamics. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. We establish a general existence theorem for equilibrium, which holds under conditions analogous to the standard deterministic ones. Our results answer questions that remained open for more than three decades.


References [Enhancements On Off] (What's this?)

  • 1. V. I. Arkin and I. V. Evstigneev. Stochastic models of control and economic dynamics. Academic Press, 1987.
  • 2. L. Arnold, I. V. Evstigneev, and V. M. Gundlach, Convex-valued random dynamical systems: a variational principle for equilibrium states, Random Oper. Stochastic Equations 7 (1999), no. 1, 23–38. MR 1677758, 10.1515/rose.1999.7.1.23
  • 3. Ludwig Arnold, Volker Matthias Gundlach, and Lloyd Demetrius, Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab. 4 (1994), no. 3, 859–901. MR 1284989
  • 4. M. A. H. Dempster, I. V. Evstigneev and M. I. Taksar. Asset pricing and hedging in financial markets with transaction costs: An approach based on the von Neumann-Gale model. Annals of Finance 2 (2006), 327-355.
  • 5. A. Dvoretzky, A. Wald, and J. Wolfowitz, Elimination of randomization in certain problems of statistics and of the theory of games, Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 256–260. MR 0035979
  • 6. E. B. Dynkin. Some probability models for a developing economy. Soviet Mathematics Doklady 12 (1971), 1422-1425.
  • 7. E. B. Dynkin. Stochastic concave dynamic programming. Math. USSR Sbornik 16 (1972), 501-515.
  • 8. E. B. Dynkin and A. A. Yushkevich, Controlled Markov processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 235, Springer-Verlag, Berlin-New York, 1979. Translated from the Russian original by J. M. Danskin and C. Holland. MR 554083
  • 9. I. V. Evstigneev, Positive matrix-valued cocycles over dynamical systems, Uspehi Mat. Nauk 29 (1974), no. 5(179), 219–220 (Russian). MR 0396906
  • 10. I. V. Evstigneev and S. D. Flåm. Stochastic programming: Non-anticipativity and Lagrange multipliers. In: Encyclopedia of Optimization, Kluwer Academic Publishers, Vol. 4, 2001, pp. 332-338.
  • 11. I. V. Evstigneev and K. R. Schenk-Hoppé. The von Neumann-Gale growth model and its stochastic generalization. In: R. Dana, C. Le Van, T. Mitra, K. Nishimura (eds.), Handbook on Optimal Growth, Vol. I, Chapter 2, Springer, 2006, pp. 337-383.
  • 12. Igor V. Evstigneev and Klaus Reiner Schenk-Hoppé, Pure and randomized equilibria in the stochastic von Neumann-Gale model, J. Math. Econom. 43 (2007), no. 7-8, 871–887. MR 2341683, 10.1016/j.jmateco.2007.04.004
  • 13. Igor V. Evstigneev and Michael I. Taksar, Rapid growth paths in convex-valued random dynamical systems, Stoch. Dyn. 1 (2001), no. 4, 493–509. MR 1875064, 10.1142/S0219493701000242
  • 14. David Gale, The closed linear model of production, Linear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N. J., 1956, pp. 285–303. MR 0085961
  • 15. David Gale, A mathematical theory of optimal economic development, Bull. Amer. Math. Soc. 74 (1968), 207–223. MR 0221835, 10.1090/S0002-9904-1968-11891-9
  • 16. D. Gale. A note on the nonexistence of optimal price vectors in the general balanced-growth model of Gale: Comment. Econometrica 40 (1972), 391-392.
  • 17. J. Hülsmann and V. Steinmetz, A note on the nonexistence of optimal price vectors in the general balanced-growth model of Gale, Econometrica 40 (1972), 387–389. MR 0378726
  • 18. Yuri Kifer, Perron-Frobenius theorem, large deviations, and random perturbations in random environments, Math. Z. 222 (1996), no. 4, 677–698. MR 1406273, 10.1007/PL00004551
  • 19. J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229. MR 0210177
  • 20. Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
  • 21. V. L. Makarov and A. M. Rubinov, Mathematical theory of economic dynamics and equilibria, Springer-Verlag, New York-Heidelberg, 1977. Translated from the Russian by Mohamed El-Hodiri. MR 0439072
  • 22. Hukukane Nikaidô, Convex structures and economic theory, Mathematics in Science and Engineering, Vol. 51, Academic Press, New York-London, 1968. MR 0277233
  • 23. Roy Radner, Balanced stochastic growth at the maximum rate, Contributions to the von Neumann growth model (Proc. Conf., Inst. Advanced Studies, Vienna, 1970) Springer, New York, 1971, pp. 39–52. Zeitschrift für Nationalökonomie, Supplementum 1. MR 0325117
  • 24. Roy Radner, Optimal steady-state behavior of an economy with stochastic production and resources, Mathematical topics in economic theory and computation (Sympos. Math. Econom., SIAM Fall Meeting, Univ. Wisconsin, Madison, Wis., 1971) Soc. Indust. Appl. Math., Philadelphia, Pa., 1972, pp. 99–112. MR 0406415
  • 25. Roy Radner, Optimal stationary consumption with stochastic production and resources, J. Econom. Theory 6 (1973), no. 1, 68–90. MR 0452560
  • 26. R. Radner. Equilibrium under uncertainty. In: K. J. Arrow and M. D. Intrilligator (eds.), Handbook of Mathematical Economics, North-Holland, 1982, pp. 923-1006.
  • 27. R. Tyrrell Rockafellar, Monotone processes of convex and concave type, Memoirs of the American Mathematical Society, No. 77, American Mathematical Society, Providence, R.I., 1967. MR 0225231
  • 28. R. T. Rockafellar and R. J.-B. Wets, Nonanticipativity and \cal𝐿¹-martingales in stochastic optimization problems, Math. Programming Stud. 6 (1976), 170–187. Stochastic systems: modeling, identification and optimization, II (Proc. Sympos., Univ Kentucky, Lexington, Ky., 1975). MR 0462590
  • 29. J. von Neumann. Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. In: Ergebnisse eines Mathematischen Kolloquiums, No. 8, 1935-1936, Franz-Deuticke, 1937, pp. 73-83. [Translated: A model of general economic equilibrium, Review of Economic Studies 13 (1945-1946), 1-9.]
  • 30. Kôsaku Yosida and Edwin Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46–66. MR 0045194, 10.1090/S0002-9947-1952-0045194-X

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Additional Information

Igor V. Evstigneev
Affiliation: Economics Department, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
Email: igor.evstigneev@manchester.ac.uk

Klaus Reiner Schenk-Hoppé
Affiliation: School of Mathematics and Leeds University Business School, Leeds University, Leeds LS2 9JT, United Kingdom
Email: K.R.Schenk-Hoppe@leeds.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04445-0
Keywords: Random dynamical systems, convex multivalued operators, von Neumann--Gale model, rapid paths, convex duality, stochastic equilibrium
Received by editor(s): July 5, 2006
Received by editor(s) in revised form: October 27, 2006
Published electronically: January 11, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.