Internal and External Cartel Stability: Numerical Solutions

The size of imperfect cartels, i.e. with participating (k-firms) and non-participating or fringe (j-firms), has been investigated by some researchers since early 1980s. Most papers, based on simple cost and demand functions, have proposed various conditions to find the optimal number of k-firms. Recently, other researchers have conducted numerical simulations or used an analytic approach to determine the size of stable cartels. In this paper, I investigate two standard models, with additional parameters in demand and cost functions. In the first model the cartel faces j competitive firms, while in the second model the cartel is the Stackelberg leader and the j-firms are Cournot followers. In the second model, the entry to and exit from cartel are formulated as integer non-linear programming, using some parameter bounds. Contrary to other studies who relied on simplified functions and found exact algebraic conditions, Mathematica fails to provide such conditions. It does provide though global numerical solutions. In the first model, irrespectively of the number of firms, the optimal number of k-firms is always three. In the second model, the number of k-firms is moderate higher, but lower than other studies find.