This study revisits the relationship between competition and innovation by incorporating an endogenous market structure (EMS) in a dynamic general equilibrium model. We consider that both innovative and non-innovative followers engage in Cournot competition with free entry. A competition-enhancing policy, which reduces entry cost, can stimulate the entry of innovative followers when the entry cost is high. However, when the entry cost is sufficiently low, the entry of non-innovative followers crowd-out innovative followers from the market. As a result, there is a non-monotonic relationship (inverted-V shape) between competition and innovation. Furthermore, we show that, while strengthening patent protection positively affects innovation when competition is sufficiently intense, the effect may be negative under milder competition. This suggests that a competition policy could complement a patent policy.