This paper studies the interaction between production subsidies and innovation subsidies. We develop a model which allows us to calculate the socially optimal subsidies (and how they vary with changes in the economic environment), and to understand how firms react to each type of subsidy. In a three-stage game, the government chooses production and innovation subsidies in the first stage to maximize welfare in the presence of a shadow cost of public funds; two firms invest in cost-reducing R&D in the second stage; and the two firms compete in quantities in the last stage. We find that production subsidies crowd out innovation. On the other hand, providing a production subsidy reduces the cost of the innovation subsidy, and vice versa. The optimal production subsidy either increases monotonically with spillovers, or is U-shaped with respect to spillovers, depending on exogenous parameters. The innovation subsidy is increasing in spillovers. The production subsidy is higher for very low spillovers, while the innovation subsidy is higher for moderate/high spillovers. In equilibrium, because of the innovation subsidy, R&D increases with spillovers, and so does welfare. We also consider the case of a financially constrained government, as well as the case of a uniform subsidy to production and innovation costs.