Nash behaviour and public good

In this note we analyse the provision of a pure public good with non constant production cost in the context of a federation of jurisdictions with two tiers of Government: the central and the local. The central government aims at welfare maximization but this objective is constrained to the use of lump sum transfer. Local governments aim at their own utility maximization and they behave according to the Nash rule. The production cost for the public good is affected by the jurisdiction's type (high or low) and by the quantity of the good that is produced. It is shown that a social welfare improvement might take place, in some circumstances, even without any central government intervention. On the other hand a first best is unreachable under the hypothesis of Nash behaviour and lump sum transfer among jurisdictions