I analyze the effects of downstream competition when there is bargaining between downstream firms and upstream agents (firms or unions). When bargaining is over a uniform input price, a decrease in the intensity of competition (or a merger) between downstream firms may raise consumer surplus and overall welfare. When bargaining is over a two-part tariff, a decrease in the intensity of competition reduces downstream profits and upstream utility and raises consumer surplus and overall welfare. Standard welfare results of oligopoly theory can be reversed: less competition can be unprofitable for firms and/or beneficial for consumers and society as a whole.