Competitive Equilibria: Convergence, Cycles or Chaos
The title of this monograph could have been "What does one do if Anything Goes"; a friend suggested that I should use it as a sub-title instead of the more prosaic one that I have used. There are two basic "Anything Goes" type of results which influence the role of dynamics in economic theory. The first is the Sonnenschein-Debreu-Mantel set of results which indicate that excess demand functions which satisfy only Walras law and the Homogeneity of degree zero postulate do not imply too many restrictions since almost any set of functions could be taken to excess demand functions; the second set of results are due to Boldrin and Montruchhio which show that any dynamical system (continuous or discrete) can describe the time evolution of the optimal paths of an infinite horizon discounted concave maximization problem subject to stationarity constraints. Thus loosely speaking, any dynamical system can be rationalized as occurring in the context of some maximization problem. These two sets of results have had a profound impact on economic theory since they seem to indicate that economic theory is unable to be definitive. A third type of "Anything Goes" result arise from the theory of dynamical systems itself, which economists do not perhaps refer to as much, is due to Smale. To describe the content of this result some notation becomes necessary. Let X be any C1 vector field in the unit simplex of dimension n－1, (delta)n－1; then there exists a C1 vector field F = (Fi) in R of dimension n satisfying Fi = xiMi(x), Mij(x) 2, it would appear that anything goes on account of dynamical systems alone. Thus not only the theory of stability of competitive equilibrium, but the study of growth processes and even the study of dynamical systems and their long term behavior need to be handled and understood carefully. General results would be difficult to obtain; any result will require special and some times, what may appear, to be ad-hoc conditions. As the three sets of results mentioned above seems to indicate, we do not have any choice in the matter and we should thus address ourselves to the nature of conditions which might provide us with results of some interest. This monograph is directed towards this objective. Another aspect that we shall be concerned with is the robustness of results obtained. In economic theory we sometimes know signs of some terms; their magnitude is some thing beyond our grasp. Accordingly we need to worry about our results if they are dependant upon magnitudes of parameters. Or if this is some thing which we cannot overcome, we should look for some ways of obtaining such information. There should be thus better cooperation between researchers in economic theory and applied economics. Finally, it should not be surprising given Smale's results that we have devoted special attention to models where the dynamics involve motion on the plane. It turns out however that even here a rich variety of situations may be exhibited. We hope that this sets out the reason why such a study is being attempted. Currently, the study has been divided up into five chapters. Chapters 1 and 2 contain the basic tools of analysis; the first refers to continuous time processes whereas the second refers to discrete time processes. These chapters contain a summary of definitions and results and some applications of these results. The chapters are by no means a comprehensive account of non-linear dynamic systems; they are there to keep the study self-contained. Chapters 3 and 4 contain an analysis of Stability of Competitive Equilibria; the first refers to Walrasian Tatonnement processes while the second to Non-Walrasian or Non-tatonnement processes. We have tried to make the analysis in these chapters as exhaustive as possible, so that readers may understand and appreciate the different aspects of this problem. In short, we examine the workings of the so-called 'Invisible Han' and obtain conditions when the Invisible Hand is also successful in carrying out the tasks that we usually assume that it is capable of. Chapter 5, currently the last chapter, is devoted to processes of economic growth in one sector models. The aspect studied in some detail is the approach to the question of unemployment cycles.
|Date of creation:||Jul 2003|
|Date of revision:|
|Contact details of provider:|| Postal: 6-1 Mihogaoka, Ibaraki, Osaka 567-0047|
Web page: http://www.iser.osaka-u.ac.jp/index-e.html
More information through EDIRC
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Nishimura Kazuo & Sorger Gerhard, 1996. "Optimal Cycles and Chaos: A Survey," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 1(1), pages 1-20, April.
- Anjan Mukherji, 1973. "On the Sensitivity of Stability Results to the Choice of the Numeraire," Review of Economic Studies, Oxford University Press, vol. 40(3), pages 427-433.
- Mukherji, Anjan, 2002. "An Introduction to General Equilibrium Analysis: Walrasian and Non-Walrasian Equilibria," OUP Catalogue, Oxford University Press, number 9780195659078, May.
- Hildenbrand,Werner & Jerison,Michael, 1988.
"The Demand theory of the Weak axioms of Revealed Preference,"
Discussion Paper Serie A
163, University of Bonn, Germany.
- Hildenbrand, Werner & Jerison, Michael, 1989. "The demand theory of the weak axioms of revealed preference," Economics Letters, Elsevier, vol. 29(3), pages 209-213.
- Kind, Christoph, 1999. "Remarks on the economic interpretation of Hopf bifurcations," Economics Letters, Elsevier, vol. 62(2), pages 147-154, February.
- Lionel McKenzie, 1999. "Equilibrium, Trade, and Capital Accumulation," The Japanese Economic Review, Japanese Economic Association, vol. 50(4), pages 371-397, December.
- Saari, Donald G., 1991. "Erratic behavior in economic models," Journal of Economic Behavior & Organization, Elsevier, vol. 16(1-2), pages 3-35, July.
- Kazuo Nishimura, 2001. "Equilibrium Growth and Nonlinear Dynamics in Continuous-Time Models," The Japanese Economic Review, Japanese Economic Association, vol. 52(1), pages 1-19.
- Amit Bhaduri & Donald J. Harris, 1987. "The Complex Dynamics of the Simple Ricardian System," The Quarterly Journal of Economics, Oxford University Press, vol. 102(4), pages 893-901.
- Smale, Steve, 1976. "A convergent process of price adjustment and global newton methods," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 107-120, July.
- Paul A. Samuelson, 1958. "An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money," Journal of Political Economy, University of Chicago Press, vol. 66, pages 467.
- Lionel W. McKenzie, 2005. "Classical General Equilibrium Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262633302, June.
- Hirota, Masayoshi, 1985. "Global stability in a class of markets with three commodities and three consumers," Journal of Economic Theory, Elsevier, vol. 36(1), pages 186-192, June.
- Majumdar, Mukul, 1994. "Chaotic Dynamical Systems: An Introduction," Economic Theory, Springer, vol. 4(5), pages 641-48, August.
- Day, Richard H. & Pianigiani, Giulio, 1991. "Statistical dynamics and economics," Journal of Economic Behavior & Organization, Elsevier, vol. 16(1-2), pages 37-83, July.
- H. Rose, 1967. "On the Non-Linear Theory of the Employment Cycle," Review of Economic Studies, Oxford University Press, vol. 34(2), pages 153-173.
- T. W. Swan, 1956. "ECONOMIC GROWTH and CAPITAL ACCUMULATION," The Economic Record, The Economic Society of Australia, vol. 32(2), pages 334-361, November.
- Herbert E. Scarf, 1959. "Some Examples of Global Instability of the Competitive Equilibrium," Cowles Foundation Discussion Papers 79, Cowles Foundation for Research in Economics, Yale University.
- Montrucchio, L., 1988. "Dynamical Systems That Solve Continuous-Time Concave Optimization Problems Anything Goes," UFAE and IAE Working Papers 109-89, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
- Mitra, Tapan, 2001. "A Sufficient Condition for Topological Chaos with an Application to a Model of Endogenous Growth," Journal of Economic Theory, Elsevier, vol. 96(1-2), pages 133-152, January.
- Saari, Donald G & Simon, Carl P, 1978. "Effective Price Mechanisms," Econometrica, Econometric Society, vol. 46(5), pages 1097-1125, September.
- Venkatesh Bala, 1997. "A pitchfork bifurcation in the tatonnement process," Economic Theory, Springer, vol. 10(3), pages 521-530.
- Boldrin, Michele & Montrucchio, Luigi, 1986. "On the indeterminacy of capital accumulation paths," Journal of Economic Theory, Elsevier, vol. 40(1), pages 26-39, October.
- Mas-Colell,Andreu, 1985.
"The Theory of General Economic Equilibrium,"
Cambridge University Press, number 9780521265140.
- Debreu, Gerard, 1974. "Excess demand functions," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 15-21, March.
- Anderson, Christopher M. & Plott, Charles R. & Shimomura, K.-I.Ken-Ichi & Granat, Sander, 2004.
"Global instability in experimental general equilibrium: the Scarf example,"
Journal of Economic Theory,
Elsevier, vol. 115(2), pages 209-249, April.
- Anderson, Christopher M. & Granat, Sander & Plott, Charles R. & Shimomura, Ken-Ichi, 2000. "Global Instability in Experimental General Equilibrium: The Scarf Example," Working Papers 1086, California Institute of Technology, Division of the Humanities and Social Sciences.
- Day, Richard H. & Pianigiani, Giulio, 1991. "Statistical Dynamics and Economics," Working Paper Series 293, Research Institute of Industrial Economics.
- Mukherji, Anjan, 1974. "The Edgeworth-Uzawa Barter Stabilizes Prices," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 15(1), pages 236-41, February.
- Bala, Venkatesh & Majumdar, Mukul & Mitra, Tapan, 1998. "A note on controlling a chaotic tatonnement," Journal of Economic Behavior & Organization, Elsevier, vol. 33(3-4), pages 411-420, January.
- Hirota, Masayoshi, 1981. "On the Stability of Competitive Equilibrium and the Patterns of Initial Holdings: An Example," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 22(2), pages 461-67, June.
- Nishimura, Kazuo & Yano, Makoto, 1994. "Optimal Chaos, Nonlinearity and Feasibility Conditions," Economic Theory, Springer, vol. 4(5), pages 689-704, August.
- Kihlstrom, Richard E & Mas-Colell, Andreu & Sonnenschein, Hugo, 1976. "The Demand Theory of the Weak Axiom of Revealed Preference," Econometrica, Econometric Society, vol. 44(5), pages 971-78, September.
- Jordan, J. S., 1983. "Locally stable price mechanisms," Journal of Mathematical Economics, Elsevier, vol. 11(3), pages 235-259, July.
- Swan, Trevor W, 2002. "Economic Growth," The Economic Record, The Economic Society of Australia, vol. 78(243), pages 375-80, December.
- Anjan Mukherji, 1999. "A simple example of complex dynamics," Economic Theory, Springer, vol. 14(3), pages 741-749.
- Tjalling C. Koopmans, 1963. "On the Concept of Optimal Economic Growth," Cowles Foundation Discussion Papers 163, Cowles Foundation for Research in Economics, Yale University.
- Silvestre, Joaquim, 1982. "Fixprice analysis in exchange economies," Journal of Economic Theory, Elsevier, vol. 26(1), pages 28-58, February.
- Majumdar, Mukul & Mitra, Tapan, 1994. "Periodic and Chaotic Programs of Optimal Intertemporal Allocation in an Aggregative Model with Wealth Effects," Economic Theory, Springer, vol. 4(5), pages 649-76, August.
- Kamiya, Kazuya, 1990. "A Globally Stable Price Adjustment Process," Econometrica, Econometric Society, vol. 58(6), pages 1481-85, November.
When requesting a correction, please mention this item's handle: RePEc:dpr:wpaper:0591. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Fumiko Matsumoto)
If references are entirely missing, you can add them using this form.