Ex ante versus interim rationality and the existence of bubbles

Tirole (1982) is commonly interpreted as proving that bubbles are impossible with finitely many rational traders with common priors. We study a simple variation of his model in which bubbles can occur, even though traders have common priors and common knowledge that the asset has no fundamental value. In equilibrium, agents purchase the asset at successively higher prices until the bubble "bursts" and no subsequent trade occurs. Each trader's initial wealth determines the last date at which he could possibly trade. The date at which the bubble bursts is a function of these finite "truncation dates" for the individual traders. Since initial wealth is private information, no trader knows when the bubble will burst. There are two key differences between our model and Tirole's which enable us to construct equilibrium bubbles this way. First, Tirole requires ex ante optimality, while we only require every trader's strategy to be optimal conditional on his information-i.e., interim optimal. As we argue in the text, this would seem to be the relevant definition of optimality. Second, Tirole considers competitive equilibria, while we analyze a simple bargaining game.