Multiproduct Multimarket Price Competition under Capacity Constraints

Capacity constraints are central to oligopoly competition in many industries, yet existing multiproduct Bertrand theory does not characterize equilibrium when capacity binds across markets. We establish existence and uniqueness of Bertrand–Nash equilibrium in a multimarket, multiproduct oligopoly under multinomial logit demand, with both linear and convex costs. Capacity creates cross-market spillovers: pricing in one market affects the shadow value of capacity in others. Methodologically, we extend the aggregative-games framework to a nested fixed-point structure separating across-market capacity allocation from within-market pricing, using tools from nonsmooth analysis to handle kinks from binding constraints. The framework yields new insights for merger analysis: binding capacity dampens merger-induced price increases through shadow-cost relief, while post-merger reallocation of scarce capacity can raise consumer surplus. However, the merged firm's privately optimal reallocation generally differs from the social optimum, creating a role for merger remedies.