Undecidable Economic Dynamics
In a recent paper (Velupillai, 1999) I discussed the following two propositions (in reverse order):Proposition 1: Assume that the (individual) market excess-demand functions are restricted to be defined on the domain of computable reals. Suppose also that we have an arbitrary exchange economy satisfying (i)~(iii): (i). Market excess-demand functions are homogeneous of degree zero in prices: z(lp) = z(p), l Î Â+ where, z: the vector of market excess-demands p: the vector of prices (ii). Market excess-demand functions satisfy the Walras Law: p.z(p) = 0 (iii). Market excess-demand functions are continuous over their (appropriately dimensioned) domain of definition: and a p*. Then, given any algorithm, initialised at the configuration of the given arbitrary exchange economy, it is undecidable whether it will terminate at p*.Proposition 2: Assume that the market excess-demand functions are computable and, therefore, continuous (hence satisfying (iii), above), on a suitable subset of Sn+ ´ T (T: time axis). Then, there exist exchange economies such that the solution to the associated tŸtonnement process: is uncomputable inside a nontrivial domain of z(p(t),t). These propositions were stated without detailed proofs; only some broad hints were provided. In this paper an attempt is made to provide detailed proofs for the above two propositions. The constructions underlying the detailed proofs makes it possible to seek a closer connection between constructive and computable analysis. In a concluding section I speculate on the possibility that intuitionistic logic may also lie at the foundations of computable analysis.
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