Usually, market models analyse competition between firms with either quantity or price as decision’s variables. This paper considers mixed duopoly competitions in which a state-owned public firm and a private firm produce complementary goods. We analyse, separately, the model in which the state-owned public firm sets the price and the private firm chooses the output production; and the model in which the state-owned public firm sets the output production and the private firm chooses the price. By considering domestic and international models, we analyse the effects of opening the market to a foreign firm. We also compare the results obtained with different types of decision variables. This paper contributes to the understanding of the implications of firms’ decisions on social welfare. As a result, the paper shows that, in the domestic competition, social welfare is higher when the state-owned public firm sets price and the domestic private firm sets production outputs than in other competitions. Furthermore, when the market is opened to foreign firms, social welfare is higher when both firms set prices than in other competitions.