Core existence in vertically differentiated markets

Abstract

We prove that a su¢cient condition for the core existence in a n-Örm vertically di §erentiated market is that the qualities of Örmsí products are equally-spaced along the quality spectrum. This result contributes to see that a fully collusive agreement among Örms in such markets is more easily reachable when product qualities are not distributed too asymmetrically along the quality ladder.

Cartel formation with quality differentiation

“…This is because Örms that participate in a coalition create positive externalities by setting higher prices than when they operate as independent outsiders. 17 As will become clear in the following, this has the implication that if a cartel is stable under the aggregative rule it is a fortiori stable for any other coalition partition of outsiders. This includes the possibility of a fringe that exclusively consists of singletons, which e §ectively would be the outcome under the unanimity rule.…”

Collusive Agreements in Vertically Differentiated Markets

“…In this section we consider the incentive of …rms to form the whole industry cartel (grand coalition). Following Gabszewicz et al (2016b), a partition function game can be associated to the vertically di¤erentiated market introduced in Section 3 and, from this, it can be proved that the core of this game is nonempty when the qualities of the products sold by the …rms are equispaced along the quality spectrum. Moreover, it can be easily shown that, when this regularity condition does not hold, the core can be empty.…”

“…Following Gabszewicz et al (2016b) we adopt the concept of delta core by Hart and Kurz (1983), also denoted projection core in the recent axiomatization by Bloch and van den Nouweland (2014). Since for the case of vertically di¤erentiated markets the coalitional worth possesses positive coalition externalities, 13 the delta or projection-core is the smallest core and, 1 3 This means that every …rm is advantaged when rivals merge in coalitions 1 4 Gabszewicz et al (2016b) use this notion of core in order to provide the strongest core existence result.…”

“…However, the latter strategy appears usually more as a temporary than a permanent strategy, since a …ghting brand may risk to cannibalize the market of the merging …rms. Ultimately, a consolidated group can …nd more advantageous to re-brand its economy products rather than un-brand some of its In Gabszewicz et al (2015) we introduce a simple framework in which three ex ante heterogeneous …rms, initially producing three vertically di¤erentiated goods, low (…rm 1), medium (…rm 2) and high (…rm 3), enter a negotiation to decide whether to merge or not with some or all rival …rms and, once merged, optimally reshape the qualities and prices according to the new market structure.…”

“…Assume as in Gabszewicz et al (2015) a three-stage game where, at the …rst stage, every …rm expresses its willingness to form an alliance or, alternatively, to stay as singleton. Then, at the second and third stage each formed coalition can decide, in turn, the qualities and prices of its goods.…”