c ○ World Scientific Publishing Company CRASHES AS CRITICAL POINTS

We study a rational expectation model of bubbles and crashes. The model has two components: (1) our key assumption is that a crash may be caused by local self-reinforcing imitation between noise traders. If the tendency for noise traders to imitate their nearest neighbors increases up to a certain point called the “critical ” point, all noise traders may place the same order (sell) at the same time, thus causing a crash. The interplay between the progressive strengthening of imitation and the ubiquity of noise is characterized by the hazard rate, i.e. the probability per unit time that the crash will happen in the next instant if it has not happened yet. (2) Since the crash is not a certain deterministic outcome of the bubble, it remains rational for traders to remain invested provided they are compensated by a higher rate of growth of the bubble for taking the risk of a crash. Our model distinguishes between the end of the bubble and the time of the crash: the rational expectation constraint has the specific implication that the date of the crash must be random. The theoretical death of the bubble is not the time of the crash because the crash could happen at any time before, even though this is not very likely. The death of the bubble is the most probable time for the crash. There also exists a finite probabilit