I examine the question whether a disagreement based speculative trade can persist in an environment in which agents are learning. The speculative trade is understood as an equilibrium outcome, in which agents who buy the asset pay higher price for it, than they would be willing to had they not had the opportunity to re-trade it in the future. The corresponding notion of speculative bubble, which is analyzed in Essay 2, is understood as an excess of the equilibrium price over the current market fundamental.
Harrison and Kreps (QJE 1978) provide an example of a persistent speculative trade. The agents start with different prior beliefs, which are concentrated on disjoint sets. This prevents them from learning. The resulting permanent disagreement leads to a speculative bubble, which is independent on any observed history of data.
I extend this example by adding learning.
In Essay 1, I set up a general equilibrium model of trading with differential beliefs and learning. The dividend process follows a general hidden Markov process.
Using recursive techniques I develop tools to compute and analyze equilibria in this environment. I also provide conditions under which a speculative trade arises.
In Essay 2, I apply those techniques to analyze the dynamics of a speculative bubble in a very special case of a Markov dividend process and the prior beliefs concentrated on two transition matrices.
Those matrices represent two possible theories considered by agents. Agents put positive probabilities on both theories, hence they are learning. The resulting speculative bubble arises whenever the data does not convincingly favor any of the theories used by agents. I give conditions for the data generating process to lead to persistent speculative bubble. I also show that even though the speculative bubble reappears infinitely often it also happens very rarely on a typical sample path.
In fact, the average time in between the periods of high bubble is infinity.