The equivalence of the cost share equilibria and the core of a voting game in a public goods economy
The author shows the equivalence of the cost share equilibria and the core of a voting game in a public goods economy. For individual \(i\), a consumption vector is a pair \((x,y^ i)=(x_ 1,\dots,x_ m;y^ i)\), where \(x_ j\) \((j\in\{1,\dots,m\})\) is the person's consumption of public good \(j\) and \(y^ i\) is her consumption of the private good. Individual \(i\)'s initial endowment is \(I^ i\); \(i\)'s payment for the production of the public goods is specified by a cost share function \(T^ i\) \((i\in\{1,\dots,n\})\). Given \(T=(T^ 1,\dots,T^ n)\), an allocation \((x_ 0,y_ 0)\) is said to be a cost share equilibrium iff \(y^ i_ 0=I^ i-T^ i(x_ 0)\) for all \(i\in\{1,\dots,n\}\), and \(u^ i(x_ 0,y^ i_ 0)\geq u^ i(x,I^ i-T^ i(x))\) for all \(x\in X\) and for all \(i\). Given any \(T\), a voting game \(G(T,W)\) introduces a structure of winning coalitions \(W\), where the set of simple majority coalitions is an example of \(W\). The core of the voting game \(G(T,W)\) is the set of all allocations which cannot be improved upon by any feasible allocation, denoted by \(C(G(T,W))\). The author proves two theorems, viz. (a) that any cost share equilibrium with respect to a given \(T=(T^ 1,\dots,T^ n)\) belongs to \(C(G(T,W))\), (b) under the assumptions of differentiability and that \((x_ 0,y_ 0)\) is an interior solution, if \((x_ 0,y_ 0)\) belongs to \(C(G(T,W))\), then it is a cost share equilibrium. These results are a generalization of those of \textit{M. Kaneko} [J. Econ. Theory 16, 123-136 (1977; Zbl 0399.90019)] where the cost share functions were restricted to be proportional (ratio equilibrium) to the total cost of providing the public goods, \(C(x)\). \textit{M. Kaneko} [Econometrica 45, 1589-1594 (1977; Zbl 0367.90046)] introduced a game in which the ratio was determined endogenously. In the present paper, it is not explained how the cost share functions \(T^ 1,\dots,T^ n\) are chosen.
scientific article; zbMATH DE number 35338