In this article, we consider technology leaders (which are innovators) and technology followers (which are non-innovators) to provide a new theoretical explanation for the well-cited empirical evidence of an inverted-U relationship between competition and aggregate innovation. We consider a two-stage game with a deterministic Research and Development (R&D) process, where the leaders first determine their R&D investments simultaneously and then all leaders and followers determine their outputs simultaneously. We show that the inverted-U relationship between competition and aggregate innovation occurs if competition is affected by the number of technology followers. However, the presence of more technology leaders decreases individual R&D investments while increasing aggregate R&D investments. If the total number of firms remains the same but the composition of technology leaders and followers changes in favor of leaders (followers), individual R&D investments decrease (increase) but aggregate R&D investments increase (decrease). The relationship between competition and R&D investments can be U-shaped if the intensity of competition is measured by product substitutability. Contrary to the standard expectation, the presence of more firms may reduce welfare.