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This paper studies supply chain competition and vertical coordination in a differential linear-quadratic game setting. In this setting, supply chains produce complement goods (in the sense of Cournot: that is, consumers need one unit of both products provided by the supply chains). Each supply chain is made up of a single supplier or manufacturer and a single seller or retailer who coordinate their decisions through a revenue-sharing contract and who play a Stackelberg game where the supplier is the leader and the retailer is the follower. Competition occurs at both levels of the supply chain. Retailers play Nash and compete in price; manufacturers also play Nash but they compete in choosing their production capacities. We consider both cases where investments are reversible and irreversible. Manufacturers provide any level of production required by the retailers (up to their production capacity). Moreover, as leader of a Stackelberg game in their respective supply chain, both manufacturers exploit the equilibrium price decisions made by the retailers. In keeping with the literature on dynamic competition between supply chains, we focus on the case where manufacturers rely on open-loop strategies. We show that when investment is reversible there are no equilibria. Where investment is irreversible, equilibria exist only when the production capacities of the manufacturers are equal--- in that case it turns out that all strategies are time consistent. The non-existence of equilibrium stems from the fact that manufacturers' instant profits are discontinuous functions of their production capacities.