Social Choice with Analytic Preferences

A social welfare function is a mapping from a set of profiles of individual preference orderings to the set of social orderings of a universal set of alternatives. A social choice correspondence specifies a nonempty subset of the agenda for each admissible preference profile and each admissible agenda. We provide examples of economic and political preference domains for which the Arrow social welfare function axioms are inconsistent, but whose choice-theoretic counterparts (with nondictatorship strengthened to anonymity) yield a social choice correspondence possibility theorem when combined with a natural agenda domain. In both examples, agendas are compact subsets of the nonnegative orthant of a multidimensional Euclidean space. In our first possibility theorem, we consider the standard Euclidean spatial model used in many political models. An agenda can be interpreted as being the feasible vectors of public goods given the resource constraints faced by a legislature. Preferences are restricted to be Euclidean spatial preferences. Our second possibility theorem is for economic domains. Alternatives are interpreted as being vectors of public goods. Preferences are monotone and representable by an analytic utility function with no critical points. Convexity of preferences can also be assumed. Many of the utility functions used in economic models, such as Cobb-Douglas and CES, are analytic. Further, the set of monotone, convex, and analytic preference orderings is dense in the set of continuous, monotone, convex preference orderings. Thus, our preference domain is a large subset of the classical domain of economic preferences. An agenda can be interpreted as the set of feasible allocations given an initial resource endowment and the firms' production technologies. To establish this theorem, an ordinal version of the Analytic Continuation Principle is developed.