Notes on Learning Rational Expectations under Computability Constraints

Much of the learning literature in orthodox frameworks, and with standard purposes, has been expertly summarized by Sargent in his recent Ryde Lectures (Sargent, 1993). Sargent's work, in turn, has been brilliantly reviewed by DeVany (1994). In recent growth models, traditional OLG models, in the experimental economics literature, in game-theoretic settings and in many other areas at the frontiers of economic theory there is a learning component in crucial senses. But there is a more fundamental approach to learning, than those employed in the above areas, and it has only recently been applied in an economic setting. This approach has been used by Spear in his pioneering piece (Spear, 1989). The whole learning problem is imaginatively recast in a recursion theoretic setting and computability and decidability results are invoked to discuss the feasibility of learning rational expectations equilibria (REE). There are, however, some minor infelicities in Spear's discussion of the recursion theoretic basis and some of the remarks and suggestions on the analytics of his model. The main purpose of this note will be to clarify some of these infelicities without, however, questioning the basic message in Spear's important paper. Indeed the hope is that these clarifying notes will strengthen the message in Spear's fertile suggestions for modelling learning. These notes are structured as follows. In 2 we attempt to clarify and streamline the nature of the computability constrants that have to be placed on the economic underpinnings of Spear's model(s). In 3 a similar attempt is made for the pure recursion theoretic assumptions and results. The concluding sections pulls the threads together and suggests directions in which Spear's work can be extended.