Learning to bet (rationally) with logs

In an economy with uncertainty and asymmetric information, suppose that some agents learn the relation between fundamentals and prices by observing past market outcomes. They refine their understanding as they become more experienced, but their past “errors” contaminate the information they receive. Does this process converge to the “perfect” understanding of the market that underlies rational expectation equilibria? We address this question in a simplified setting that allows for explicit computation of the learning process: a two-state economy with logarithmic utilities and no background risk. Our first result is that as long as the wealth of the uninformed agents is less than half the aggregate wealth of the economy, the learning process indeed converges to rational expectations. This convergence, however, is non-monotonic, and the market oscillates between phases of excess price volatility and phases of excess volume of trade. The learning process, in addition, is costly for the uninformed agents. We interpret our results as underscoring the fragility of ree: markets operate orderly only when speculation is less significant than fundamental trade.