I provide a full characterization of the quality choice in duopolies with vertical differentiation, without assuming ex-ante if the market is fully covered or not. This will allow to show that covered or uncovered market configurations are endogenous outcomes of firms' strategic interaction. To this purpose, I assume that firms are characterized by quadratic fixed costs of quality improvements and check whether pure-strategy subgame perfect equilibria with a corner solution always exist. Finally, my results are compared to the quality choice that maximizes the total surplus of the economy. I show that the welfaremaximizing choice of qualities does not have to be found in the same market configuration of the corresponding market equilibrium.