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Social Welfare Function and Social Indifference Surfaces

Ken-ichi Inada

Abstract

The purpose of this paper is to prove an impossibility theorem for the existence of a social welfare function. Our treatment of the social welfare function differs from that of Arrow insofar as the social choice function generated through the social welfare function from the individual choice functions is defined in the disaggregated social state in his case, while in our case it is defined in the aggregated social state. We assume that every individual's choice function is defined in an n-dimensional commodity space. Then, the social choice function is defined in Arrow's case in an (m x n)-dimensional commodity space while it is defined in our case in an n-dimensional space. With our modification, the requirement insuring the existence of the social welfare function yielding social indifference surfaces becomes weaker than in Arrow's case. Nevertheless, we still obtain an impossibility.