An Impossibility Theorem for Fixed Preferences: A Dictatorial Bergson- Samuelson Welfare Function

Abstract

Arrow's impossibility theorem (Arrow [1]) shows that a social welfare function (which specifies a transitive complete ordering on social alternatives for each m-tuple of individual preferences) is necessarily dictatorial if it has an unlimited domain (all possible m-tuples of preferences), is independent of irrelevant alternatives, Pareto consistent and there are at least three alternatives. It is shown here that such a social welfare function must be dictatorial even if there is only one m-tuple of preferences in the domain if it is neutral, Pareto consistent and there are enough alternatives so that in one m-tuple of preferences there is diversity of preference among the alternatives. Although Arrow did not require neutrality, it is a logical consequence of his conditions. In the second section we argue that this result shows that the Bergson-Samuelson welfare function is dictatorial. Samuelson [7] and Plott [5] argue that the Arrow theorem really has nothing to do with the Bergson problem since Bergson assumes that the preferences of the society are fixed. In fact, Samuelson argues that there are as many Bergson functions as there are m-tuples of preferences in the domain of Arrow's function, and hence the impossibility theorem has nothing to do with Bergson's welfare function. The result here, though, is shown for a fixed m-tuple of preferences and so can be applied to the Bergson concept.