I study an oligopolistic growth model in which firms can innovate by creatively destroying their competitors’ goods, improving their own goods, and developing new ones. A large firm is equivalent to a mass of small firms that can coordinate their activities to maximize joint profits. Larger firms adapt their innovation mix to avoid cannibalization, and as a result they impose a high rate of creative destruction risk on their competitors without generating much growth. A tax on large firm acquisitions of their smaller competitors’ goods may backfire by encouraging large firms to creatively destroy those goods instead. In a special case of the model with only creative destruction, I prove conditions so that a rise in large firm profitability leads to a fall in growth, and so that a fall in taxes on acquisitions leads to an increase in growth. In the full quantitative model, a fall in large firm fixed costs, calibrated to match the recent rise in concentration in the US, explains 41% of the fall in growth from the 1990s to the 2010s, as well as the burst in growth during the late 1990s, the positive across industry correlation between changes in concentration and growth, and the fall in growth relative to R&D expenditures. Dispersion in large firm innovation costs or profitability across industries yields a novel theory of the inverted-U relationship between concentration and growth.