Uncomputability and undecidability in economic theory

• Published in: Applied mathematics and computation Vol. 215; no. 4; pp. 1404 - 1416
• Main Author: Vela Velupillai, K.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier Inc 15.10.2009; Elsevier
• Subjects: (Un)computability; (Un)decidability; Calculus of variations and optimal control; Constructivity; Exact sciences and technology; Game theory; General equilibrium theory; Logic and foundations; Mathematical analysis; Mathematical logic, foundations, set theory; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Proof theory and constructive mathematics; Recursion theory; Recursive macroeconomics; Sciences and techniques of general use; General equilibrium theory; (Un)decidability; Recursive macroeconomics; Constructivity; Game theory; (Un)computability; Markov process; Filtering; Decidability; Constructive mathematics; Optimization method; Algorithmics; Computational complexity; Computability; Statistical method; Numerical analysis; Applied mathematics; Algorithm complexity; Dynamic programming; Proof theory; Information theory
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2009.04.051

Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory [Noam Nisan, Tim Roiughgarden, Éva Tardos, Vijay V. Vazirani (Eds.), Algorithmic Game Theory, Cambridge University Press, Cambridge, 2007] and Algorithmic Statistics [Péter Gács, John T. Tromp, Paul M.B. Vitányi, Algorithmic statistics, IEEE Transactions on Information Theory 47 (6) (2001) 2443–2463] 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 (The qualification ‘recursive’ here has nothing to do with ‘recursion theory’. Instead, this is a reference to the mathematical formalizations of the rational economic agent’s intertemporal optimization problems, in terms of Markov Decision Processes, (Kalman) Filtering and Dynamic Programming, where a kind of ‘recursion’ is invoked in the solution methods. The metaphor of the rational economic agent as a ‘signal processor’ underpins the recursive macroeconomic paradigm.), are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.