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We study a quality-ladder model of endogenous growth that produces stochastic leadership cycles. Over a cycle, industry leaders can innovate several successive times in the same sector before being replaced by a new entrant. Initially, new leaders do much of the research but they then tend to rest on their laurels and are eventually overtaken. The model generates a skewed firm size distribution and a deviation from Gibrat's law that accord with the empirical evidence. We also find conditions under which policy should favour R&D by incumbents, not outsiders, and show that stronger patent protection may reduce innovation and growth.