Remarks on criticality and crisis in pure exchange economies
DOI:
https://doi.org/10.12775/TMNA.2021.065Keywords
Walras equilibria, intrinsic derivative, branching, crisisAbstract
In the framework of the theory of Equilibrium Manifold of a Pure Exchange Economy, introduced by Balasko, we give a rigorous definition of a crisis transition and establish some infinitesimal criteria which distinguish unavoidable crises from other critical equilibria. Our approach builds on the relationship between the mathematical notions of branching, envelopes, and the intrinsic derivative.References
E. Accinelli, The equilibrium set of the infinite dimensional Walrasian economies and the natural projection, J. Math. Econom. 49 (2013), 435–440.
E. Accinelli and M. Puchet, A characterization of the singular economies of the infinite dimensional models in general equilibrium theory, Documento No: 07/1998, Documentos de trabajo, Departamento de Economia, Facultad de Ciencias Universidad de La Republica Oriental del Uruguay.
E. Accinelli, A. Piria and M. Puchet, Taste and singular economies, Rev. Mex. de Economia y Finanzas 2 (2003), 35–48.
A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press, 1993.
V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of Differentiable Maps, vol. 1, Monographs in Math. 82, 1985.
Y. Balasko, The graph of the Walras correspondence, Econometrica 43 (1975), 907–912.
Y. Balasko, Equilibrium analysis and envelope theory, J. Math. Econom. 5 (1978), 153–172.
Y. Balasko, Fondements de la Théorie de l’Équilibre Général, Économica, Paris, 1988.
Y. Balasko, The natural projection approach to the infinite horizont models, J. Math. Econom. 27 (1997), 251–265.
Y. Balasko, The Equilibrium Manifold: Postmodern Developments in the Theory of General Economic Equilibrium, The MIT Press Cambridge, 2009.
Y. Balasko, Foundations of the Theory of General Equilibrium, second edition, World Scientific Pub, 2016.
O. Bolza, Lectures on the Calculus of Variations, third edition, Chelsea Pub. Co., New York, 1973.
J.W. Bruce and P.J. Giblin, Cuves and Singularities, Cambridge University Press, 1984.
R. Caccioppoli, Sulle corrispondenze funzionali inverse diramate: teoria generale e applicazioni ad alcune equazioni funzionali nonlineari e al problema di Plateau, I, II, Rend. Accad. Naz. Lincei 24 (1936), 258–263, 416–421; Opere Scelte, vol. 2, Edizioni Cremonese, Roma.
P.A. Chiappori and I. Ekeland, Aggregation and market demand: an exterior differential calculus viewpoint, Econometrica 67 (1999), 1435–1457.
E. Covarrubias, Regular Infinite Economies, The B.E. Journal of Theoretical Economics 10 (2010), 251–265.
G. Debreu, Economies with a finite set of equilibria, Econometrica 38 (1970), 387–392.
P.M. Fitzpatrick, Homotopy, linearization, and bifurcation, Nonlinear Anal. 12 (1988), 171–184.
P.M. Fitzpatrick and J.Pejsachowicz, Branching and bifurcation, Discrete Contin. Dyn. Syst. Ser. S 12 (2019), 1955–1975.
P.M. Fitzpatrick, J. Pejsachowicz and P. Rabier, Degree theory for proper C 2 Fredholm maps, J. Reine Angew. Math. 427 (1992), 1–33.
M. Hirsch, Differential Topology, Graduate Text in Mathematics, Springer–Verlag, Berlin, 1976.
C.-W. Ho, A note on proper maps, Proc. Amer. Math. Soc. 51 (1975), 237–241.
J. Ize, Topological bifurcation, Topological Nonlinear Analysis, Birkhäuser, Progress in Nonlinear Differential Equations 15 (1995), 341– 463.
C.G.J. Jacobi, Vorlesungen über Dynamik, 1884, elibrary.matf.bg.ac.rs.
F. Kubler, I. Ekeland, P.A. Chiappori and H. Polemarchakis, The identification of preferences from equilibrium prices (1999), https://basepub.dauphine.fr/handle/123456789/6360.
A. Loi and S. Matta, A note on the structural stability of the equilibrium manifold, J. Math.l Econom. 46 (2010), 591–594.
G. Marchesi, A. Portaluri and N. Waterstraat, Not every conjugate point of a semiRiemannian geodesic is a bifurcation point, Differential Integral Equations 31 (2018), 871–880.
A. Mas-Colell, J.R. Green and M.D. Whinston, Microeconomic Theory, Oxford University Press, New York, 1995.
W.S. Massey, Algebraic Topology, an Introduction, Springer, New York, 1967.
J. Milnor, Topology from Differentiable View-Point, Princeton University Press, 1968.
M. Musso, J. Pejsachowicz and A. Portaluri, A Morse Index Theorem and bifurcation for perturbed geodesics on semi-Riemannian manifolds, Topol. Methods Nonlinear Anal. 25 (2005), 69–99.
G. Peano, Lezioni di Analisi Infinitesimale, vol. 2, (Candeletti, ed.), Turin, 1893.
J. Pejsachowicz and P.J. Rabier, Degree theory for C 1 -Fredholm mappings of index 0, J. Anal. Math. 76 (1998), 289–319.
I.R. Porteous, Simple singularities of maps, Lecture Notes in Mathematics, vol. 192, Springer–Verlag, pp. 286–311.
R.A.Plastock, Nonlinear Fredholm maps of index zero, Proc. Amer. Math. Soc. 68 (1978), 317–322.
S. Smale, Global Analysis and Economics I: Pareto Optimum and a Generalization of Morse Theory, Dynamical Systems Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, 1971.
S. Smale, Global analysis and economics IIA, J. Math. Economics 1 (1974), 1–14.
R. Thom, Sur la theorie des enveloppes, J. Math. Pures Appl. 41 (1962), 177–192.
T. Uchiyama, A geometric approach to the transfer problem for a finite number of traders (2017), 1701.04491, arXiv.org.
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Jacobo Pejsachowicz
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
