Internal and External Cartel Stability: Numerical Solutions

The stability of imperfect cartel, i.e. a market with both cartel firms (k-firms) and noncartel or fringe firms (j-firms), has a long research history. Selten (1973) was the first to argue that four firms in a market are a few while six are too many. Ten years later, D'Aspremont et al (1983), modelled the behavior of j-firms as price-takers. Donsimoni et al (1986) and Daskin (1989), based on D'Aspremont et al (1983), showed that stable cartels exist with a small number of firms. On the other hand, when the number of firms in the market increase, the relative size of cartel firms decreases. Shaffer (1995), assuming constant marginal costs, was the first to introduce the Cournot behavior of the j-firms as followers, with the k-firms being the price-leaders. The conditions she proposed for internally stable cartels for nn ≥ 4, are kk ≤ nn 2 + 1, for even n, respectively kk ≤ nn+1 2 + 1, for odd n, and reverse bounds for externally stable cartels. It is noteworthy, as Shaffer argues, that the values of parameters do not influence these conditions. A similar model was used by Escrihuela-Villar (2008), arguing that, in stable cartels, the number of cartel firms with nn ≥ 4 is independent from the parameters and is determined, either by 1 4 (1 + 3nn − �(nn − 2)nn − 7, or by the same condition plus one. Konishi & Lin (1999) modified Shaffer’s model by assuming quadratic cost function, without a fixed part, and setting the slope of demand curve equal to unit. Due to algebra complexity, they conducted numerical simulations. Zu et al (2012), followed Konishi & Lin and provided an analytic approach to determine the size of stable cartels. Their estimates are slightly larger, compared to Zu et al (2012). In addition to that, they questioned the uniqueness of the stable cartel, argued by Konishi & Lin.