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Most quality-then-price decision models under vertical product differentiation consider a predetermined market configuration. We endogenize market configuration considering quality-dependent marginal costs and conclude that a strictly interior full coverage duopoly holds for some parameter values, unveiling the relevance of this commonly assumed market structure. Moreover, we show that a monopoly never arises in equilibrium, and (i) there are multiple equilibria at the frontier between interior and corner full coverage duopoly, (ii) the market is fully (partially) covered when relative tastes' heterogeneity is low (high), and (iii) there is a discontinuity in the transition from partial coverage to full coverage duopoly.