This section of the article by Jonathan Levin, titled "Bubbles and Crashes," discusses Abreu and Brunnermeier's (2003) paper on the failure of rational arbitrage in asset markets. The "no-trade" theorem states that speculative bubbles cannot exist in a world with only rational traders, even with asymmetric information, as long as they share a common prior. However, the authors argue that even when rational arbitrageurs are aware of mispricing, a lack of common knowledge may prevent them from coordinating their actions, leading to persistent mispricing. The model presented in the paper consists of behavioral traders and rational traders. Behavioral traders cause a bubble in asset prices, which can only be punctured if there is sufficient selling pressure from rational traders. The price process for stocks starts exponentially rising from time \( t = 0 \), coinciding with fundamental values. At a random time \( t_0 \), prices diverge from fundamentals, and the fundamental value is given by \( (1 - \beta(t - t_0)) p_t \), where \( \beta \) is an increasing function. The time \( t_0 \) is exponentially distributed. Rational traders must sell a significant fraction \( \kappa \) of the stock to burst the bubble. If less than \( \kappa \) sell, prices continue to rise. The key feature of the model is that rational traders do not share the same information. Instead, they become aware of the bubble sequentially, and the number of arbitrageurs who become aware before others is unknown. The optimal selling time for each trader is determined by their belief about the start time of the bubble and the random bursting time. The optimal selling time \( T^*(t_0) \) is continuous and strictly increasing, and the function \( T^*(\cdot) \) is its inverse. The expected payoff to a trader who sells at time \( t \) is given by an integral involving the density \( \pi(t|t_i) \). The paper characterizes trading equilibria, showing that there is a unique equilibrium pair \( (\tau, \xi) \). If traders expect the bubble to burst exogenously, it does so at \( t_0 + \bar{\tau} \). If the bubble is expected to burst endogenously, it bursts at \( t_0 + \xi^* \), where \( \xi^* \) depends on the parameters of the model. Even if the bubble bursts endogenously, arbitrage trades are delayed by \( \tau^* \) periods, allowing the bubble to grow significantly above fundamentals.This section of the article by Jonathan Levin, titled "Bubbles and Crashes," discusses Abreu and Brunnermeier's (2003) paper on the failure of rational arbitrage in asset markets. The "no-trade" theorem states that speculative bubbles cannot exist in a world with only rational traders, even with asymmetric information, as long as they share a common prior. However, the authors argue that even when rational arbitrageurs are aware of mispricing, a lack of common knowledge may prevent them from coordinating their actions, leading to persistent mispricing. The model presented in the paper consists of behavioral traders and rational traders. Behavioral traders cause a bubble in asset prices, which can only be punctured if there is sufficient selling pressure from rational traders. The price process for stocks starts exponentially rising from time \( t = 0 \), coinciding with fundamental values. At a random time \( t_0 \), prices diverge from fundamentals, and the fundamental value is given by \( (1 - \beta(t - t_0)) p_t \), where \( \beta \) is an increasing function. The time \( t_0 \) is exponentially distributed. Rational traders must sell a significant fraction \( \kappa \) of the stock to burst the bubble. If less than \( \kappa \) sell, prices continue to rise. The key feature of the model is that rational traders do not share the same information. Instead, they become aware of the bubble sequentially, and the number of arbitrageurs who become aware before others is unknown. The optimal selling time for each trader is determined by their belief about the start time of the bubble and the random bursting time. The optimal selling time \( T^*(t_0) \) is continuous and strictly increasing, and the function \( T^*(\cdot) \) is its inverse. The expected payoff to a trader who sells at time \( t \) is given by an integral involving the density \( \pi(t|t_i) \). The paper characterizes trading equilibria, showing that there is a unique equilibrium pair \( (\tau, \xi) \). If traders expect the bubble to burst exogenously, it does so at \( t_0 + \bar{\tau} \). If the bubble is expected to burst endogenously, it bursts at \( t_0 + \xi^* \), where \( \xi^* \) depends on the parameters of the model. Even if the bubble bursts endogenously, arbitrage trades are delayed by \( \tau^* \) periods, allowing the bubble to grow significantly above fundamentals.