In most countries, there are three levels of government: national, regional and local. In Europe, there is even a fourth one, the European Union. In many instances, higher level authorities provide funds to lower levels, either through block grants (allocations for a general purpose) or matching grants (allocations that requires matching funds from the grantee). How this should occur is not well studied, especially when one considers that these funds can be used to build local public capital.
Heng-Fu Zou makes an attempt at this, with a cascading Stackelberg structure from national to regional and local governments. I do not want to mention the conclusions, though, because I think the paper is fundamentally flawed. The most interesting aspects of the problem are bypassed here: first, there is a strong redistributive aspect to block grants, hence taxation need to be part of the model, but it is only modeled as a fix lump sum payment here. Second, the very reason why there are block grant for specific purposes instead on general grants is that lower governments may be tempted to put it all in public consumption. That variable is absent from the model, everything flows into public capital.
Third, the utility function is assumed to be log-linear in all public capital and expenditures individually. This implies that all of them are essential (a government can for example not take over responsibilities from another) and in particular that private consumption or investment is completely useless. As a consequence, it is always good to increases taxes, no matter their current level. Other results also derive directly from this assumption about the utility function. Fourth, the matching grant is so poorly set up that it allows the author to claim in all seriousness that capital can go instantly to infinity if the higher authority matches at 100% the local investment. Fifth, capital does not depreciate, which matters immensely when you write about the long run. Etc.
Interesting question, horrible execution.