Arrovian theorems for economic domains. The case where there are simultaneously private and public goods
After Arrow's famous impossibility result the question arose whether his result also applies to economic domains, i.e. situations in which the set of social alternatives is a set of public goods or a set of collections of private goods, or a combination of both public and private goods. For public goods, \textit{E. Kalai} and \textit{E. Muller} [J. Econ. Theory 16, 457-469 (1977; Zbl 0407.90005)] showed that Arrow's theorem holds, for private goods where the individuals are assumed to be interested solely in their own commodity bundle, \textit{E. Kalai} and \textit{Z. Ritz} [ibid. 22, 23-36 (1980; Zbl 0429.90006)] and the authors [ibid. 47, No.2, 257- 281 (1989; Zbl 0677.90003)] among others, have proved that Arrow's result still holds. For the case of both public and private goods, the present paper provides an analogous answer. It is proved that for standard domains where there are infinitely many public and private goods and where the individuals have continuous, strictly increasing and convex complete preorderings, every weakly Paretian Arrow social welfare function is dictatorial. An Arrow function is one which by definition satisfies the condition of independence of irrelevant alternatives. The model which the authors are working with considers a set of social alternatives such that each alternative is composed of n private components, one for each person, and a public component. The proof is established for so-called ultrasaturating domains which are related both to saturating domains, a requirement used in the case of purely public goods, and to hypersaturating domains, used in the case of private alternatives only. It is shown that a standard domain is ultrasaturating. Finally, the authors show that a version of \textit{A. Mas-Colell} and \textit{H. Sonnenschein}'s theorem [Rev. Econ. Studies (1972)] also holds for standard domains in the case of public and private goods. For both results, the requirement of an ultrasaturating domain is a sufficient condition. On the other hand, the authors' approach admits the indifference within the individuals' preference relations.
scientific article; zbMATH DE number 4141756