To address the relationship between innovation and competition we jointly estimate the opportunity, production, and impact functions of innovation in a simultaneous system. Based on Swiss micro-data, we apply a 3-SLS system estimation. The findings confirm a robust inverted-U relationship, in which a rise in the number of competitors at low levels of initial competition increases the firm's research effort, but at a diminishing rate, and the research effort ultimately decreases at high levels of competition. When we split the sample by firm types, the inverted-U shape is steeper for creative firms than for adaptive ones. The numerical solution indicates three particular configurations of interest: 1. an uncontested monopoly with low innovation, 2. low competition with high innovation, and 3. a "no innovation trap" at very high levels of competition. The distinction between solution 1. and 2. corresponds to Arrow's positive effect of competition on innovation, whereas the difference between outcomes 2. and 3. captures Schumpeter's positive effect of market power on innovation. Simulating changes of the exogenous variables, technology potential, demand growth, firm size and exports have a positive impact on innovation, while foreign ownership has a negative effect, and higher appropriability has a positive impact on the number of competitors.