Vertical Differentiation: Multiproduct Strategy to Face Entry?

In this paper, we consider an incumbent firm facing potential entry in a vertical differentiation model a la Mussa and Rosen where consumers differ only by their intensity of preference for quality. We ask whether the incumbent firm has the incentive to adopt a multi-product strategy in order to face entry, in a natural duopoly case. It turns out that this strategy is never profitable and that an incumbent always prefers to produce one quality. It appears that either a cost effect or an income dispersion are necessary to urge producers to offer several products.