Methods of demand theory are applied to the problem of the existence of a social welfare function under specific public choice algorithms. Integrability conditions, necessary for the derivation of social demand functions from utility maximization, are used. The social choice function, which chooses the mean of all voters demand for public goods to be the public provision, is analysed in detail. Necessary conditions for the existence of a social utility function, and by implication, a transative social ordering, are derived for this case. These conditions imply restrictions on individual preferences.
Aggregating individual preferences to achieve consistent social orderings is a central issue of public choice theory. Arrow (1963) showed the nonexistence of mappings which obey a reasonable set of conditions from individual preferences to consistent social preferences. Later Inada (1964) using Bergson (1938)welfare functions shows that mappings from individual marginal rates of substitution to social marginal rates of substitution will not, in general, be consistent either. In other words, the Arrow problem cannbt be solved by defining a social welfare function on the distribution of individual utilities. From the point of view of this present paper, Inada's method of analysis is particularly interesting.
Inada used conditions traditionally associated with revealed preference theory. Namely (integrability) conditions that are necessary for the consistent solution of the set of differential equations implied by the market