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We present a model in which an asset bubble can persist despite the presence of rational arbitrageurs. The resilience of the bubble stems from the inability of arbitrageurs to temporarily coordinate their selling strategies. This synchronization problem together with the individual incentive to time the market results in the persistence of bubbles over a substantial period. Since the derived trading equilibrium is unique, our model rationalizes the existence of bubbles in a strong sense. The model also provides a natural setting in which news events, by enabling synchronization, can have a disproportionate impact relative to their intrinsic informational content.

This paper presents a systematic framework for studying infinitely repeated games with discounting, focussing on pure strategy (subgame) perfect equilibria. It introduces a number of concepts which organize the theory in a natural way. These include the idea of an optimal penal code, and the related notions of simple penal codes and simple stratea profiles. I view a strategy profile as a rule specifying an initial path (i.e., an infinite stream of one-period action profiles), and punishments (also paths, and hence infinite streams) for any deviations from the initial path, or from a previously prescribed punishment. An arbitrary strategy profile may involve an infinity of punishments and complex historydependent prescriptions. The main result of this paper is that much of this potential strategic complexity is redundant: every perfect equilibrium path is the outcome of some perfect simple smtegy profile. A simple strategy profile is independent of history in the following strong sense: it specifies the same player specific punishment after any deviation by a particular player. Thus simple strategy profiles have a parsimonious description in terms of (n + 1) paths where n is the number of players. Unlike the undiscounted case there is no need to "make the punishment fit the crime." In particular, a player who has a "myopic" incentive to deviate from his own punishment may be deterred from doing so simply by restarting the punishment already in effect. The key to the above result is that, with discounting, worst perfect equilibria exist for each player. These define an optimalpenal code. The notion of a simplepenal code yields an elementary proof of the existence of an optimal penal code and leads directly to the theorem on the "sufficiency" of simple strategy profiles.

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There exist optimal symmetric equilibria in the Green-Porter model [4, S] having an elementary intertemporal structure. Such an equilibrium is described entirely by two subsets of price space and two quantities, the only production levels used by firms in any contingency. The central technique employed in the analysis is the reduction of the repeated game to a family of static games. Journal of Economic Literature Classification Numbers: 026, 611. Q 1986 Academic press. IW.