We study the optimal design of R&D policies and corporate taxation when the outputs of innovation are not appropriable in the absence of intellectual property rights policies and there are non-internalized technology spillovers across firms. Firms are heterogeneous in their research productivity, i.e., in the efficiency with which they convert a given set of R&D inputs into successful innovations. There is asymmetric information about firm productivity and about its stochastic evolution over time that prevents the first best solution to the technology spillover. The problem is thus posed as one of dynamic mechanism design with externalities. We characterize the optimal constrained efficient allocations over firms' life cycles and for firms of different productivities. We show that the constrained efficient allocations can be implemented either by a patent system plus a price subsidy for the monopolists' products, together with a parsimonious R&D subsidy function or, equivalently, by a prize mechanism. We estimate our model using firm-level data matched to patent data and quantify the optimal policies. Simpler innovation policies, such as linear R&D subsidies and linear profit taxes, lead to large revenue losses relative to the optimal mechanism.