Welfare reducing vertical integration in a bilateral monopoly under Nash bargaining

We consider a bilateral monopoly where a linear input price is determined by Nash bargaining. We show, with an increasing marginal cost of input production, that vertical integration reduces consumer surplus and welfare compared with bilateral monopoly if the bargaining power of the input supplier is low. This result is important for competition policies as it questions the common wisdom suggesting vertical integration increases welfare by eliminating the problem of double marginalization. Overproduction under bilateral monopoly compared with vertical integration is the reason for our result. Interestingly, consumer surplus and welfare can be higher under a linear input price compared with a two-part tariff input price.