Economic equilibrium theory: A computability viewpoint

In the present dissertation, we study economic equilibrium theory from a computability viewpoint. Our formal model is a "computable exchange economy" with bounded rational agents. Computability constraints are imposed on the agents in their perception, communication and decision making processes. First, we require perceptible and communicable objects to be computable by finite algorithms (Turing machines). In particular, we assume the "computable real numbers" constitute the economic quantities in this economy. A real number is computable if its decimal expression can be computed by a finite algorithm. Commodities are divisible and the commodity space is the q-th dimensional space {dollar}IRsbsp{lcub}c{rcub}{lcub}q{rcub}{dollar} of computable real numbers. Second, we require agents' strict preferences on computable commodity bundles to be computable by finite algorithms.; In this setup, we develop a computability economic equilibrium theory. Methods are from recursive analysis--a field in mathematics that synthesizes recursion theory (computation) theory and classical mathematical analysis. The main technical difficulty of this work is that the commodity space {dollar}IRsbsp{lcub}c{rcub}{lcub}q{rcub}{dollar} as a metric space is not complete, but only "recursively complete."; We prove some new results in recursive analysis, including (i) an existence theorem for computable maximizers for computable quasi-concave functions; (ii) a "computable" Minkowski Separation Theorem; (iii) sufficient conditions for preserving the recursive closedness property for subsets of {dollar}IRsp{lcub}q{rcub}{dollar} in operations (e.g. taking convex hull, etc.).; In our computability economic theory, we prove: (i) a computable utility representation theorem; (ii) an existence theorem for computable demand bundles; (iii) a "computable" Second Welfare Theorem; (iv) a computable version of the Debreu-Mas-Colell theorem for Sonnenschein's excess demand question. (v) We provide a constructive proof of the "equivalence" between Brouwer's fixed point theorem and Arrow-Debreu's competitive equilibrium existence theorem. Our theorem applies to computability environments as well as to standard environments (allowing arbitrary real numbers). We give a "computable counterexample" to Arrow-Debreu's competitive equilibrium existence theorem. We also find sufficient conditions for the existence of a competitive equilibrium for computable exchange economies.; Although we concentrate here on exchange economies, a similar approach is also natural for game theory, mechanism design, and other areas of economic theory.