Optimal punishments in linear duopoly supergames with product differentiation

We analyze optimal penal codes in both Bertrand and Cournot supergames with product differentiation. We prove that the relationship between optimal pu nishments and the security level (individually rational discounted profit s tream) depends critically on the degree of supermodularity in the stage gam e, using a linear duopoly supergame with product differentiation. The secur ity level in the punishment phase is reached only under extreme supermodula rity, i.e., when products are perfect substitutes and firms are price sette rs. Finally, we show that Abreu's rule cannot be implemented under Cournot behavior and strong demand complementarity between products.