The Idea of Social Choice by David J. Mayston (auth.)
By David J. Mayston (auth.)
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The scope for relaxing the UD condition again appears a limited one. In order for MDR to qualify as a SDF, the necessaryl and sufficient condition for a set of individual orderings over a finite A to be in the corresponding domain is that every triple of alternatives in A must satisfy at least one of the conditions VR, ER and LA [47], with Kramer's contribution [30] again relevant here. Thus if one wants to achieve the consistency properties implicit in a SDF with quasi-transitivity but not sacrifice UD, P and WD, then, as with the parallel conditions in the context of achieving a SWF with full transitivity, the logic of the impossibility result is for a re-examination of Arrow's independence condition.
1 The changes will be the same only if the relevant MRE's are unity. We can now indeed formulate a 'neo-Marshallian' Law of Diminishing Marginal Utility along indifference curves. 2 If good k is a normal good, a partial increase in its price relative to pI = 1 (as in Fig. 10) associated with an increase in income of good one to compensate for the price rise, will imply a fall in the marginal utility of good one. 3 Given these changes in Ai along each indifference curve, the amount of money which is required to achieve a movement between two indifference curves will very naturally vary with the prices at which it is paid, and hence with the path of income variation taken.
This reasoning is, though, slightly misleading. As previously stressed, Arrow's SWF corresponds to the process by which a social ordering is derived given anyone combination of individual orderings from amongst the admissible domain of those individual preferences. Bergson's swf is then a numerical representation of the resulting social ordering. e. the SWF, to the condition that it operate for this given domain. If a SWF is impossible for the given domain (under stated conditions), we will not be able to find any Bergson swf's for at least some of the combinations of individual preferences in this domain.