We develop a general equilibrium model of vertical innovation in which multiple firms compete monopolistically in the quality space. The model features many firms, each of which holds the monopoly to produce a unique quality level of an otherwise homogenous good, and consumers who are heterogeneous in their valuation of the good's quality. If the marginal cost of production is convex with respect to quality, multiple firms coexist, and their equilibrium markups are determined by the degree of convexity and the density of quality-competition. To endogenize the latter, we nest this industry setup in a Schumpeterian model of endogenous growth. Each firm enters the industry as the technology leader and successively transits through the product cycle as it is superseded by further innovations. The intrinsic reason that innovation happens in our economy is not one of displacing the incumbent; rather, innovation is a means to differentiate oneself from existing firms and target new consumers. Aggregate growth arises if, on the one hand, increasingly wealthy consumers are willing to pay for higher quality and, on the other hand, private firms' innovation generates income growth by enlarging the set of available technologies. Because the frequency of innovation determines the toughness of product market competition, in our framework, the relation between growth and competition is reversed compared to the standard Schumpeterian framework. Our setup does not feature business stealing in the sense that already marginal innovations grant non-negligible profits. Rather, innovators sell to a set of consumers that was served relatively poorly by pre-existing firms. Nevertheless, "creative destruction" prevails as new entrants make the set of available goods more differentiated, thereby exerting a pro-competitive effect on the entire industry.