Vertical externalities revisited: new results with public inputs and unit taxation.

This paper studies the provision of public inputs in a federal system with unit taxation on labor. We use a model with vertical tax and expenditure externalities to analyze the e¢ ciency of equilibria under di¤erent settings, particularly Nash and Stackelberg equilibria. Our results discuss some ?ndings from the previous literature. First, both vertical externalities are interrelated each other. Second, the condition for production e¢ ciency in the public sector becomes irrelevant to assess optimality. And third, the replication of the second-best outcome by the federal government behaving as Stackelberg leader crucially depends on the states?reaction function.