This paper compares the equilibrium outcomes under simultaneous and sequential output setting in a mixed duopoly in a vertically differentiated market. When the timing of the output game is determined endogenously, it is shown that simultaneous play in the quality stage with the public firm acting as the high-quality producer and simultaneous play in the second period in the output stage turn out to be the unique subgame perfect Nash equilibrium, which contrasts with the endogenous timing in a purely private duopoly.