Explorations in Computable Economics

Abstract

Three topics are discussed as pointers towards issues at the frontiers of computable economics: effectivizing the fundamental theorems of welfare economics; computable production theory; the effectivity of tâtonnement. Furthermore, in the appendix, an attempt is made to analyse the notion of finding a formally simplest theory of economic fluctuations. Some discursive notes on multiple equilibria in economic models and their possible generation due to various formal uncomputabilities is also presented.

Axiomatization and Formalism in Economics

“…Poincaré's finitism would, negatively, have excluded recourse to many of the mathematical tools acceptable within Cantor's paradise and, positively, would have exploited the novel mathematical tools of finitist mathematics developed by Church, Turing and others. This difference is succinctly summed up in Velupillai (2000) which in our view is an excellent and sustained effort at economic theorizing utterly faithful to the Poincaré finitist principle.…”

“…The fact that the commodity space has the structure of a real vector space is a basic reason for the success of the mathematization of economic theory. In particular convexity properties of sets in R l , a recurring theme in the theory of general equilibrium, can be fully exploited.’ As we will see in what follows ‘one can also describe the action of an economic agent’ as if ‘it’ were a Turing machine; in this case the commodity space would have an appropriate recursive structure in which computable and combinatorial properties would be exploited (Velupillai, 2000, p. 2, fn 5).…”

“…In our view, Velupillai's Computable Economics is the clearest attempt at the mathematization of economics compatible with the Poincaré finitist principle. Contrary to Debreu, the two programmes are quite divergent: recursion theoretic decision problems replace standard optimization procedures and this leads to undecideable choices, where there ‘is no effective procedure to generate preference orderings’ (Velupillai, 2000, p. 38). ‘Guaranteeing the existence of an optimum is one thing; finding it is another thing; and finding it tractably is yet another thing’ (p. 61, fn 6).…”

VARIATIONS ON THE THEME OFCONNINGaINMATHEMATICAL ECONOMICS

“…Turing machine (or any of its own formal equivalences, by the Church–Turing thesis). Since a complete formalism and the relevant equivalences are described, defined and, where necessary, rigorously proved in Chapter 3 of my Ryde Lectures (Velupillai, 2000), I will simply assume that the interested reader can be trusted to refer to it for any detailed clarification and substantiation.…”

“…The above definitions, descriptions and structures define, therefore, an arithmetical game of length n (cf. Velupillai, 2000, pp. 125–126, for a formal definition).…”