标题： 关于股息支付资产上的理性泡沫：对蒂罗尔（1985）的评论 显示英文标题

标题： Rational Bubbles on Dividend-Paying Assets: A Comment on Tirole (1985)

摘要： 蒂罗尔（1985）研究了一个具有资本积累的重叠世代模型，并表明资产泡沫的出现解决了资本过度积累问题。 他的命题1(c)声称，如果股息增长率高于无泡沫利率（即没有泡沫的迪亚蒙德经济中的稳态利率），但低于人口增长率，则泡沫是必要的，因为在这种情况下不存在无泡沫均衡，但存在唯一的有泡沫均衡。 我们提供了一个反例，基于一个分析例子，但在额外假设下证明了该命题，这些假设包括初始资本足够大且股息足够小。 显示英文摘要

摘要： Tirole (1985) studied an overlapping generations model with capital accumulation and showed that the emergence of asset bubbles solves the capital over-accumulation problem. His Proposition 1(c) claims that if the dividend growth rate is above the bubbleless interest rate (the steady-state interest rate in the Diamond economy without bubbles) but below the population growth rate, then bubbles are necessary in the sense that there exists no bubbleless equilibrium but there exists a unique bubbly equilibrium. We provide a counterexample to this claim based on an analytical example but prove the claim under the additional assumptions that initial capital is sufficiently large and dividends are sufficiently small.