Increasing evidence supports that international trade enhances innovation and productivity growth through an increase in competition. This paper develops a two-country endogenous growth model, with firm specific R&D and a continuum of oligopolistic sectors under Cournot competition to provide a theoretical support to this claim. Since countries are assumed to produce the same set of varieties, trade openness makes markets more competitive, reducing prices and increasing quantities. Since firms undertake cost reducing innovations, the increase in production pushes firms to innovate more. Compared to other oligopolistic competition models, we find a larger pro-competitive effect of trade on innovation under this framework, and this effect is increasing the larger the elasticity of substitution between products.