Consistency and incompleteness in general equilibrium theory

Journal of Evolutionary Economics - Tập 30 - Trang 205-230 - 2018
Simone Landini1, Mauro Gallegati2, J. Barkley Rosser3
1IRES Piemonte, Turin, Italy
2Polytechnic University of Marche, Ancona, Italy
3James Madison University, Harrisonburg, USA

Tóm tắt

We consider the implications for general equilibrium theory of the problems of consistency and completeness as shown in the Gödel-Rosser theorems of the 1930s. That a rigorous consistent formal system is incomplete poses serious problems for dealing with unresolved problems in a fully formal system such as general equilibrium theory. We review the underlying mathematical issues and apply them to this problem for general equilibrium theory. We also consider alternative approaches such as agent-based modeling founded on empirically estimated behavioral assumptions for agents that may allow for a better way to model non-equilibrium evolutionary economic dynamics.

Tài liệu tham khảo

Ackerman F (2002) Still dead after all these years: interpreting the failure of general equilibrium theory. J Econ Methodol 9(2):139–202 Albin PS (1982) The metalogic of economic predictions, calculations and propositions. Math Soc Sci 3(4):329–358 Arrow KJ, Debreu G (1954) Existence of equilibrium for a competitive economy. Econometrica 22:265–290 Arrow KJ, Intrilligator MD (1982) Handbook of mathematical economics. Vol. II. North-Holland, Amsterdam Aumann RJ (1964) Markets with a continuum of traders. Econometrica 32:39–50 Beklemishev LD (2010) Gödel incompleteness theorems and the limits of their applicability. I. Russ Math Surv 65(5):857 Berto F (2008) Tutti Pazzi per Gödel. La guida completa al Teorema del'Incompletezza. Laterza. English translation: There's Something About Gödel: The Complete Guide to the Incompleteness Theorem. Malden, MA., and Oxford: Wiley-Blackwell, 2009 Bewley TF (1972) Existence of equilibrium in economies with infinitely many commodities. J Econ Theory 4:514–545 Blaug M (2003) The formalist revolution of the 1950s. J Hist Econ Thought 25(2):145–156 Blume L, Durlauf S (2001) The interactions-based approach to socioeconomic behavior. In: Durlauf S, Young HP (eds) Social dynamics. MIT Press, Cambridge Boland LA (1979) A critique of Friedman's critics. J Econ Lit 17(2):503–522 Bookstaber R (2017) The end of theory. Financial crises, the failure of economics, and the sweep of human interaction. Princeton University Press, Princeton Bourbaki N (1949) Foundations of mathematics for the working mathematician. J Symb Log 14(1):1–8 Boylan TA, O'Gorman PF (2007) Axiomatization and formalism in economics. J Econ Surv 21(3):426–446 Bryant WDA (2010) General equilibrium: theory and evidence. World Scientific Publishing, Singapore Caldwell BJ (1980) A critique of Friedman's methodological instrumentalism. South Econ J 47(2):366–374 Cartan N (1943) Sur le fondement logique des mathématiques. Revue Scientifique 81:3–11 Debreu G (1959) Theory of value. An axiomatic analysis of economic equilibrium. Wiley, New York Debreu G (1970) Economies with a finite set of equilibria. Econometrica 38:387–392 Debreu G (1974) Excess-demand functions. J Math Econ 1:15–21 Dierker E (1974) Topological Methods in Walrasian Economics, lecture notes in economics and mathematical systems, Vol. 92. Springer, Heidelberg Einstein A (1921) Geometry and Experience, Address to the Prussian Academy of Sciences in Berlin on January 27th 1921, retrieved on September 2017 from http://www.relativitycalculator.com/pdfs/einstein_geometry_and_experience_1921.pdf Fishburn PC (1970) Arrow’s impossibility theorem: concise proof and infinite voters. J Econ Theory 2:103–106 Friedman M (1953) The methodology of positive economics in Essays in Positive Economics. University of Chicago Press, Chicago, pp 3–43 Gintis H (2007) The dynamics of general equilibrium. Econ J 117(523):1280–1309 Gintis H, Mandel A (2012) The stability of Walrasian general equilibrium. Documents de travail du Centre d'Économie de la Sorbonne 12065, Université Panthéon-Sorbonne (Paris 1), Centre d'Économie de la Sorbonne Gloria-Palermo S (2010) Introducing formalism in economics: the growth model of John von Neumann. Panoeconomics 2010(2):153–172 Gödel K (1931) “Über formal unentscheidbare Sätze der Principia Mathematica und verwandte Systeme I” with page-facing English translation, in “Kurt Gödel, Collected Works. Vol. I. Publications 1929–1936”, Feferman S et al., (eds), Oxford University Press, New York, 1986. And. “On formally undecidable propositions of Principia Mathematica and related systems”, English translation by Meltzer B and Braithwaite RB, Dover Publications Inc. New York, (1992) Hartley JE (1997) The representative agent in Macroeconomics. Routledge, London, New York Hellman M (2008) Interview, in Hendricks VF, Leitgeb H (eds) Philosophy of Mathematics: 5 Questions. Automatic Press/VIP, Copenhagen Ingrao B, Israel G (1991) The invisible hand, economic equilibrium in the history of science. MIT Press, Cambridge Jerison M (1984) Social welfare and the unrepresentative representative consumer, discussion paper, State University of New York at Albany Jerison M (2006) Nonrepresentative representative consumers. Discussion Papers 06–08, Department of Economics, SUNY-Albany Kaldor N (1972) The irrelevance of equilibrium economics. Econ J 82(328):1237–1255 Katouzian H (1980) Ideology and method in economics. Macmillan, London Kehoe TJ (1985) Multiplicity of equilibria and comparative statics. Q J Econ 100:118–147 Kirman AP (1989) The intrinsic limit of modern economic theory: the emperor has no clothes. Econ J 99(395):126–139 Kirman AP (1992) Whom or what does the representative individual represent? J Econ Perspect 6:117–136 Kirman AP (2004) General equilibrium. In: Bourgine P, Nadal JP (eds) Cognitive economics. An interdisciplinary approach. Springer, Heidelberg Kirman AP, Koch KJ (1986) Market excess demand in exchange economies with identical preferences and collinear endowments. Rev Econ Stud 53(3):457–463 Kirman AP, Sondermann D (1972) Arrow’s theorem, many agents, and invisible dictators. J Econ Theory 5:267–277 Kiyotaki N (2011) A perspective on modern business cycle theory. Econ Q 97(3):195–208 Kleene SC (1967) Mathematical logic. Wiley, New York Kleene SC, Vesley RE (1965) Foundations of intuitionistic mathematics. North- Holland, Amsterdam Koppl R, Rosser JB Jr (2002) All that I have to say has already crossed your mind. Metroeconomica 53:339–360 Krusell P, Smith AA (1998) Income and wealth heterogeneity in the macroeconomy. J Polit Econ 106:867–896 Lehtinen A (2013) Three kinds of 'as-if' claims. J Econ Methodol 20:184–205 Lewis AA (1992) On Turing degrees of Walrasian models and a general impossibility result in the theory of decision making. Math Soc Sci 24:141–171 Lolli G (2002) Filosofia della matematica. L'eredità del Novecento, Il Mulino, Bologna Mandel A, Landini S, Gallegati M, Gintis H (2013) Price dynamics, financial fragility and aggregate volatility. J Econ Dyn Control 51:257–277 Mankiw NG (2006) The macroeconomist and scientist and engineer, NBER Working paper No. w12349; available on SSRN: https://ssrn.com/abstract=913312 Mantel R (1974) On the characterization of aggregate excess-demand. J Econ Theory 7:348–353 Maskin E, Riley J (2003) Uniqueness of equilibrium in sealed high-bid auctions. Games Econ Behav 45:395–409 Mirowski P (2002) Machine dreams: how economics became a cyborg science. Cambridge University Press, New York Móczár J (2017) Arrow-Debreu model versus Kornai-critique. Athens J Bus Econ 3(2):143–170 retrieved on September 2017 from http://zbw.eu/econis-archiv/handle/11159/674 Phuong V, Ma L, Meenagh D, Minford P, Wickens M, Xu Y (2016) Testing macro-models by indirect inference: a survey for users. Open Econ Rev 27:1–38 Piñero GE (2014) Gödel. I Teoremi dell'Incompletezza, RBA Italia S.r.l. Poinsot L (1842) Éléments de statique, 8th edition, Bachelier, Paris Rabin MO (1957) Effective computability of winning strategies. In: Dresher M, Tucker AW, Wolfe P (eds) Annals of mathematics studies, no. 39,Contributions to the theory of games, vol III. Princeton University Press, Princeton, pp 147–157 Richter MKA, Wong KC (1999) Non-computability of competitive equilibrium. Economic Theory 14:1–27 Rizvi SAT (2006) The Sonneschein-Mantel-Debreu results after thirty years. Hist Pol Econ 38(1):228–245 Robinson A (1966) Non-standard analysis. Princeton Landmarks in Mathematics, Princeton University Press, Princeton Rosser JB (1936) Extensions of some theorems of Gödel and Church. J Symb Log 1:87–91 Rosser JB Jr (2012) On the foundations of mathematical economics. New Math Nat Comput 8:53–72 Rosser JB Jr, Rosser MV (2015) Complexity and behavioral economics. Nonlinear Dynamics Psychol Life Sci 19:201–226 Rosser JB, Turquette AR (1952) Many valued logics. North-Holland, Amsterdam Samuelson PA (1947) Foundations of economic analysis. Harvard University Press, Cambridge Scarf HE (1960) Some examples of global instability of the competitive equilibrium. Int Econ Rev 1:157–172 Shafer W, Sonnenschein H (1989) Market demand and excess demand functions. In: Arrow KJ, Intriligaror MD (eds) Handbook of mathematical economics, vol II. North-Holland, Amsterdam Simon HA (1957) Models of man. Wiley, New York Smith JW, Lyons G, Sauer-Thompson F (1999) The bankruptcy of economics. Ecology, economics and the sustainability of the Earth. MacMillan Press, London Smolin L (2006) The trouble with physics: the rise of string theory, the fall of a science, and what comes next. Houghton Mifflin Harcourt, Boston Smorynski C (1977) The incompleteness theorems. In: Barwise J (ed) Handbook of mathematical logic. North-Holland, Amsterdam Solow RM (1985) Economic history of economics. Am Econ Rev 75:328–331 Sonnenschein H (1972) Market excess-demand functions. Econometrica 40(3):549–563 Sonnenschein H (1973) Do Walras' identity and continuity characterize the class of community excess-demand functions? J Econ Theory 6:345–354 Sornette D, Davis AB, Ide K, Vixie KR, Pisarenko V, Kamm JR (2017) Algorithm for model validation: theory and applications. PNAS-Proc Nat Acad Sci USA 104(16):6562–6567 Retrieved on September 2017 from http://www.pnas.org/content/104/16/6562.short Tarski A (1956) The concept of truth in formalized languages’, logic, semantics and metamathematics. Oxford University Press (1935), Oxford JH Woodger (trans.); First published as ‘Der Wahrheitsbegriff in Den Formaliserten Sprachen’, Studia Philosophica I (1935) Tsuji M, da Costa NCA, Doria FA (1992) The incompleteness theories of games. J Philos Log 27:553–568 Turing AM (1936) On computable numbers with an application to the Entscheidungsproblem. Proc Lond Math Soc 2:230–265 Veblen TB (1898) Why is economics not an evolutionary science? Q J Econ 12:373–397 Velupillai KV (2000) Computable Economics. Oxford University Press, Oxford Velupillai KV (2009) Uncomputability and undecidability in economic theory. Appl Math Comput 215:1404–1416 von Neumann J (1937) English version (1945-46) a model of general economic equilibrium. Rev Econ Stud 14:1–9 Walras L (1874) Éléments d'économie politique pure, ou théorie de la richesse sociale, Imprimiere L. Corbaz & Cie editeurs, Lausanne; translated by W Jaffé, 1954, Elements of pure economics, or the theory of social wealth, Allen & Unwin, London Weintraub ER (2002) How economics became a mathematical science. Duke University Press, Durham Winrich JE (1984) Self-reference and the incomplete structure of neoclassical economics. J Econ Issues (4):987–1005