Local public goods and average cost pricing
An economy with local public goods is defined as a specified data \({\mathcal E}:=\{(T,{\mathcal T},v)\), \((X_ p(t)\), \(X_ g(t)\), e(t), \(\succ_ t\), \(\theta^ j(t)\), \((Y^ j)\), \((Y^ i)\}\), where (T,\({\mathcal T},v)\) is a measure space of consumers, for each \(t\in T\) \(X_ p(t)\) \((\subset R^{\ell})\) is the consumption set for private goods, \(X_ g(t)=\{e^ 1,...,e^ m\}\) is the consumption set consisting of m local public goods \(e^ i\) being supplied by community i, \(\succ_ t\) is the preference relation on \(X_ p(t)\times X_ g(t)\), \(\theta^ j(t)\) is the relative share of t in private firm j \((=1,...,u)\), \(Y^ j\) \((\subset R^{\ell})\) is the production set of private firm j, and \(Y^ i\) \((\subset (-R_+^{\ell})\times R_+)\) is the production set of local government i to produce its public good. Each consumer t chooses a maximal element (with respect to \(\succ_ t)\) of his budget constraint, private firm j maximizes its profit, and local government i passively charges the necessary cost (of producing its public goods) evenly to its residents as tax. Consumer's choice determines a partition \((n^ i)\) of T into the residents of community i. Thus a Tiebout equilibrium of \({\mathcal E}\) is ((x(t)), \((y^ j)\), \((y^ i)\), \((p_ g)\), p)\(\in \prod_{t}(X_ p(t)\times X_ g(t))\times \prod^{u}_{j=1}Y^ j\times \prod^{m}_{i=1}Y^ i\times R^ m\times \Delta^{\ell -1}\) such that x(t) is a maximal element (with respect to \(\succ_ t)\) of \(\{\) \(x|\) \(p\cdot x_ p+p_ g\cdot x_ g\leq p\cdot e(t)+\sum_{j}\theta^ j(t)\max p\cdot Y^ j\}\), \(p\cdot y^ j=\max p\cdot Y^ j\), \(p^ i_ g=\) per capita cost for producing local public good in community i (i.e., tax), and the market clearance condition is satisfied. A somewhat weaker notion of Tiebout equilibrium is proposed and its existence theorem is provided. A logical analogy of this result with an existence theorem for an economy with natural monopolies and contestable markets is also pointed out.
scientific article; zbMATH DE number 3896613