Finiteness property with vertical and horizontal differentiation: does it really matter?

Abstract

The paper proves the consistency of vertical and horizontal differentiation models regarding the finiteness property. In the case of constant returns, it turns out that the attitude of consumers towards products sold at marginal costs is crucial for the finiteness property to hold or not. We first prove concerning the finiteness property results with horizontal preferences consistent with results already obtained with vertical preferences. Second we prove with vertical and horizontal preferences, a new type of result relative to the existing literature: the existence of an indifferent consumer and a population unanimous w.r.t. their most preferred variant when all variants are sold at marginal cost, is sufficient to allow to an unbounded number of firms to be active. In the case of non-constant returns, it is the cost structure in a neighborhood of the zero output that drives the market structure. Results consistent with results already proved with vertical preferences, are proved with horizontal preferences. Finally, existence results with horizontal preferences are provided to show that the demonstrated properties have cases of application.