Equivalence results when firms compete in prices/quantities and innovation effort

I consider the classical oligopoly game in which firms choose prices or quantities at the same time each chooses an innovation effort that directly establishes its (constant) marginal cost of production. I show how this game can be viewed as completely equivalent to a standard oligopoly game in which firms simply choose prices or quantities given a strictly concave cost of producing output. Hence, the effect of changing parameters unrelated to technology (e.g., the number of competing firms, the size of the market, or the number of merging firms) can be studied with no loss in a setting in which firms cannot innovate and produce using an increasing marginal returns technology. Exactly the same equivalence persists when performing welfare analysis, so the comparison between welfare benchmarks and equilibrium outcomes is the same as when firms choose prices or quantities given a strictly concave cost of producing output.