We build a model of firm-level innovation, productivity growth and reallocation featuring endogenous entry and exit. A key feature is the selection between high- and low-type firms, which differ in terms of their innovative capacity. We estimate the parameters of the model using detailed US Census micro data on firm-level output, R&D and patenting. The model provides a good fit to the dynamics of firm entry and exit, output and R&D, and its implied elasticities are in the ballpark of a range of micro estimates. We find industrial policy subsidizing either the R&D or the continued operation of incumbents reduces growth and welfare. For example, a subsidy to incumbent R&D equivalent to 53 of GDP reduces welfare by about 1.53 because it deters entry of new high-type firms. On the contrary, substantial improvements (of the order of 53 improvement in welfare) are possible if the continued operation of incumbents is taxed while at the same time R&D by incumbents and new entrants is subsidized. This is because of a strong selection effect: R&D resources (skilled labor) are inefficiently used by low-type incumbent firms. Subsidies to incumbents encourage the survival and expansion of these firms at the expense of potential high-type entrants. We show that optimal policy encourages the exit of low-type firms and supports R&D by high-type incumbents and entry.