This paper examines a price-setting mixed duopoly model in which a state-owned public firm and a private firm produce complementary goods. There is no possibility of entry or exit. Each firm has one owner and can hire one manager to make its production decisions. The paper first analyzes the following four possible cases: neither firm hires a manager, only the private firm hires a manager, only the state-owned public firm hires a manager and both firms hire managers. It is shown that economic welfare is identical in all the four cases. Next, this paper presents the equilibrium of the model. The paper shows that there exist two equilibrium outcomes: only the public firm hires a manager and neither firm hires a manager. As a result, it is found that the equilibrium of the paper is contrast with that obtained under price-setting mixed duopoly competition with substitute goods, where both the public firm and the private firm hire managers.