We model a duopoly with a private and a public firm under the hypothesis of vertical product differentiation. Firms choose their quality levels first and then prices. We ask which firm will choose to serve the higher (lower) segment of the market. When firms act simultaneously in each stage, there are two subgame perfect Nash equilibria entailing opposite rankings between the quality levels. If the State-owned firm has a move advantage, then there is a unique Stackelberg equilibrium in which the public firm serves the upper segment of the market.