Learning in a Computable Setting: Applications of Gold's Inductive Inference Model

The importance of learning in a dynamical economic system is obvious. How to model the learning process is, however, not so evident. We have chosen to refer to the theory of recursive functions as the source for a rigorous formalization. In line with a famous metaphor like Patinkin's utility computer and Lucas characterization of the economic agent as a collection of procedures, we will concentrate on effective learning procedures. In other words, our stylized learner will be exhaustively described by a Turing Machine (TM). Neuropsychologists and cognitive scientists often adopt this framework in their analysis of first language acquisition by children. Inside economic theory, this choice can be justified from a bounded rationality point of view since the emphasis devoted to algorithms and computable functions is essential to a coherent appraisal of procedural rationality. Spear (1989) asks whether it is possible for a boundedly rational agent to get to know his/her environment so well as to form perfectly rational expectations. The proposed characterisation of bounded rationality is precisely the one suggested by computability theory and the learning model adopted that of inductive inference as given by Gold (1965, 1967). Spear's work has not been followed by other significant efforts in the same direction, even though the theory of recursive functions has proven an important tool of analysis for Microeconomics with Rustem and Velupillai (1990) and Lilly (1993) on the computability of preference orderings. The present note is intended as a reconsideration of Spear's seminal paper with some amendments and generalisations. In particular, concentrating our attention on the two step learning procedure presented in the first part of that article, on the one hand, we show how hysteresis can greatly reduce the positive learning results obtained by Spear; on the other hand, we suggest that institutions may play an important role in making information user friendly hence in facilitating the learning process. Section 2 explicitly presents Gold's model with some more recent results obtained by Osherson, Stob and Weinstein (OSW), Smith (1982), Daley (1983) and others. The purpose of this, perhaps tedious, part is that of describing rather precisely the reference framework so that the conclusions reached in the rest of this study can be put into a correct perspective. Spear's OLG model with rational expectations is given sketchily in section 3. We refer to the original article for further details on the particular structure. In section 4, we employ the inductive inference model in the context of the OLG framework and we point at different economic scenarios that are suitable for its application. Section 5 is devoted to institutions. We present examples and propose some analogies diverting a little from both OLG and Gold's model. We draw our conclusions in the final section.