Paolo Morganti

“Towards a General Theory of Vertical Differentiation”

Abstract
We develop a general approach to models of vertical differentiation that admits a wide variety of distributions of consumer preferences. We find that the traditional sufficiency conditions for equilibrium existence (Caplin and Nalebuff (1991)) impose qualitative restrictions in vertical differentiation settings, in particular on the high-quality firm’s behavior. By relaxing them, we are able to extend the class of distributions allowed. Interestingly, we observe only three possible behaviors concerning the price competition stage of the game. In the Full Coverage setting, we identify a novel set of necessary and sufficient conditions for existence and uniqueness of a pure strategy equilibrium in prices. Furthermore, we show that the Principle of Maximum Differentiation is the only equilibrium outcome, even when the subgames are played in mixed strategies. In the Partial Coverage setting, regardless of the distribution, the first-stage best response for the low-quality firm is always linear in the quality of the rival. Moreover, if a subgame perfect equilibrium exists, the high-quality firm chooses the highest possible quality. Curiously, we notice that any issue of existence is related to the level of dispersion of the distribution. Finally, ranking families of distributions according to their Gini index, our simulations show novel comparative statics on prices and profits.