The choice of the roles by firms in a vertically differenitated duopoly is analysed introducing a preplay stage where firms set the timing of moves, under the alternative assumptions of full or non full market coverage. Under the first, it turns out that the unique subgame perfect equilibrium entails simultaneous play in the quality stage, followed by sequential play in the price stage, where both firms would prefer to be price leader, contrarily to the results obtained by previous literature. Under partial market coverage, it is possible to analyse both price and quantity competition. In such a case, simultaneous play in both stages emerges as the optimal behaviour. Overall, contrarily to the conclusions reached by the previous literature, it is not possible to claim that the choice of the strategy space dominates the distribution of roles.
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Luca Lambertini (1996) studied this question.
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