Innovation, mark-ups and the degree of trade openness vary substantially across sectors. This paper builds a multi-sector endogenous growth model to study the influence of asymmetric trade liberalisation and sectoral differences in the degree of product market competition on the effect that trade has on R&D investments at a firm level. I find that differences in the degree of competition generate large differences in firm innovative responses to trade liberalisation. A movement from autarky to free trade promotes innovation and productivity growth in those sectors which are initially less competitive. However, when the initial tariff level is common across sectors, a homogeneous tariff reduction promotes innovation in those sectors which are initially more competitive. The paper suggests that trade liberalisation could be a source of industry productivity divergence: firms that are located in industries with greater exposure to foreign trade, invest a greater amount in R&D contributing to industry productivity growth. Finally the paper outlines the importance of reallocation effects within industry and across industries that are the result of these asymmetries. An asymmetric trade liberalisation has a small but negative impact on aggregate productivity growth.