Using a general two-stage framework, this paper gives sufficient conditions for increasing competition to have negative or positive effects on R&D-investment, respectively. Both possibilities arise in plausible situations, even if one uses relatively narrow definitions of increasing competition. The paper also shows that competition is more likely to increase the investments of leaders than those of laggards. When R&D-spillovers are strong, competition is less likely to increase investments. The paper also identifies conditions under which low initial levels of competition make positive effects of competition on investment more likely. Extending the basic framework, the paper shows that separation of ownership and control, endogenous entry and cumulative investments make positive effects of competition on investment more likely. Imperfect upstream competition weakens the effects of competition on investment. Keywords: competition, investment, cost reduction. JEL: L13, L20, L22.