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Revisiting the Mathematical Difficulties in Patinkin Cartel Model and Joint Profit Maximization

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We discuss the mathematical difficulties encountered in Patinkin's classical cartel model. It may be impossible to derive Patinkin's cartel by finding the reciprocal marginal cost functions: it could be impossible for cartel members to compute a solution, unless certain assumptions are made to simplify the problem, such as quasi-linear marginal costs or constant marginal costs. The total cost function is incoherent with respect to the sum of members' total cost. The model cannot handle constant marginal costs but we remind that de Mesnard's (2009, 2011) model of cartel with exogenous market shares allow solving the problem. We conclude that the Patinkin model of cartel is not so self-evident. This is annoying because it remains the theoretical basis of cartel theory, even if it is in the background when game theory is used.