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Uncomputability and Undecidability in Economic Theory

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  • K. Vela Velupillai

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    Abstract

    Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory ([28]) and Algorithmic Statistics ([13]) are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory - and its applied branches, say computational complexity theory or algorithmic information theory - and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory, are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.

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    Bibliographic Info

    Paper provided by Department of Economics, University of Trento, Italia in its series Department of Economics Working Papers with number 0806.

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    Date of creation: 2008
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    Handle: RePEc:trn:utwpde:0806

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    Keywords: General Equilibrium Theory; Game Theory; Recursive Macro-economics; (Un)computability; (Un)decidability; Constructivity;

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    References

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    1. K. Vela Velupillai, 2004. "The Unreasonable Ineffectiveness of Mathematics in Economics," Working Papers, National University of Ireland Galway, Department of Economics 0080, National University of Ireland Galway, Department of Economics, revised 2004.
    2. K. Vela Velupillai, 2007. "A Computable Economist’s Perspective on Computational Complexity," Department of Economics Working Papers, Department of Economics, University of Trento, Italia 0723, Department of Economics, University of Trento, Italia.
    3. K. Vela Velupillai, 2007. "Taming the Incomputable, Reconstructing the Nonconstructive and Deciding the Undecidable in Mathematical Economics," Department of Economics Working Papers, Department of Economics, University of Trento, Italia 0722, Department of Economics, University of Trento, Italia.
    4. Arrow, Kenneth J, 1986. "Rationality of Self and Others in an Economic System," The Journal of Business, University of Chicago Press, University of Chicago Press, vol. 59(4), pages S385-99, October.
    5. (Vela) Velupillai, K., 1997. "Expository notes on computability and complexity in (arithmetical) games," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 21(6), pages 955-979, June.
    6. K. Vela Velupillai, 2005. "The foundations of computable general equilibrium theory," Department of Economics Working Papers, Department of Economics, University of Trento, Italia 0513, Department of Economics, University of Trento, Italia.
    7. Marimon, Ramon & Scott, Andrew (ed.), 1999. "Computational Methods for the Study of Dynamic Economies," OUP Catalogue, Oxford University Press, Oxford University Press, number 9780198294979, October.
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    Cited by:
    1. K.Vela Velupillai, 2011. "Negishi's Theorem and Method," ASSRU Discussion Papers, ASSRU - Algorithmic Social Science Research Unit 1129, ASSRU - Algorithmic Social Science Research Unit.
    2. K. Vela Velupillai, 2012. ""Dear Professor Mankiw..."- Reflections on Four Decades of Learning and Teaching Macroeconomics," ASSRU Discussion Papers, ASSRU - Algorithmic Social Science Research Unit 1206, ASSRU - Algorithmic Social Science Research Unit.
    3. K. Vela Velupillai, 2011. "DSGE And Beyond – Computable And Constructive Challenges," ASSRU Discussion Papers, ASSRU - Algorithmic Social Science Research Unit 1122, ASSRU - Algorithmic Social Science Research Unit.
    4. K. Vela Velupillai, 2011. "Foley's Thesis, Negishi's Method, Existence Proofs and Computation," ASSRU Discussion Papers, ASSRU - Algorithmic Social Science Research Unit 1124, ASSRU - Algorithmic Social Science Research Unit.
    5. K. Vela Velupillai, 2012. "The Chimera of a Complete Analysis of Economic Dynamics," ASSRU Discussion Papers, ASSRU - Algorithmic Social Science Research Unit 1207, ASSRU - Algorithmic Social Science Research Unit.
    6. K. Vela Velupillai, 2011. "The Fundamental Theorems of Welfare Economics, DSGE and the Theory of Policy - Computable & Constructive Foundations," ASSRU Discussion Papers, ASSRU - Algorithmic Social Science Research Unit 1125, ASSRU - Algorithmic Social Science Research Unit.

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