Social Welfare Functions and Income Distributions

Old welfare economics, Pigou (1920), considered social welfare as a cardinal notion, while new welfare economics,1 Little (1950) and Graaff (1957), consider social welfare as an ordinal notion. An in depth introduction to welfare economics and a discussion of the transition from old to new welfare economics, is expounded quite well by Samuelson (1947, Chap. 8, pp. 203–219, and pp. 249–252). For the purpose of our discourse, the ordinal notion will be adopted. To be clear, when speaking of individual welfare we refer to the utility associated to a single consumer or economic agent; when considering social welfare, or simply welfare, the notion is associated to society as a whole. It is well known that, given a set of alternatives, a binary choice relation, or preference relation, endowed with some mathematical properties, can be selected as the basis to define a welfare function,2 since, from a binary choice relation, generally, one can build3 a welfare function which represents that binary choice relation. But, at least when the set of alternatives is very large, working with a welfare function is mathematically more operative than working with a preference relation. Hence, in this monograph we shall start directly by considering welfare functions.