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Social welfare function

This term has been used in economics in two senses. 1. As defined by A. Bergson and P. A. Samuleson, it is a mathematical relationship which associates a number with any allocation of commodities among consumers. The relationship will reflect same particular set of value judgements concerning the well-being of different consumers, and underlying it is a preference ordering over alternative allocations. The higher a particular allocation is in the ordering, the higher the number it will be assigned by the social welfare function. Different sets of value judgements will of course imply different social welfare functions in this sense. The concept of the social welfare function was proposed to enable economists to conduct analysis of the desirability of particular resource allocations in as objective a way as possible. Befare ‘desirability’ can be discussed it is necessary to have a set of value judgements to define what should be regarded as good or bad. By expressing these in terms of a social welfare function the economist can examine the consequences of different kinds of value judgements without himself being committed to any of them. This is then likely to be a more apen and healthy analytical procedure than one by which an economist begins with a commitment to one specific set of value judgements and makes policy prescriptions accordingly. Economists should then stand ready to analyse the implications of anybody’s value judgements by representing them as a social welfare function. 2. In a fundamental analysis of the possibility of the existence of a social welfare function of the Bergson-Samuelson type just described, or rather more generally of the preference ordering over resource allocations which underlies it, K. J. Arrow in fact introduced a second meaning of the term. Arrow defined a social welfare function as a sel of procedures for devising a social preference ordering, given the preference orderings passessed by the individuals in society. Arrow’s concept of a social welfare function is therefore akin to a ‘constitution’, i.e. a set of rules for transform ing the opinions and desires of the members of a social group into concrete choices made from the alternatives which confront that group. The degree of abstraction with which Arrow pursued his analysis was such that his conclusions are applicable to all social groups, from the executive board of a large company to H.M. government or the Security Council of the United Nations, and to all types of constitution, from simple majority voting to complex voting schemes. Arrow posed the following question. Suppose we require of our constitution that it possess certain general properties, e.g. that it should be responsive to the wishes of every member of the group and that no member of the group plays the role of dictator, that if everyone in the group prefers alternative A to B then A should exceed B in the social ranking, and a number of other technical properties. Is it then possible to design a constitution with these properties which will produce a complete, consistent ranking of the alternative social states? Arrow’s famous impossibility theorem shows that no such constitution is possible. It is in this sense that Arrow proved the impossibility of a social welfare function. An entire area of research has developed from this fundamental contribution, and most attention has been directed at the reasonableness of the properties which must be possessed by the constitution. On further analysis, some are less reasonable than might at first be supposed, and a weakening of the requirements can reverse the impossibility result. The area has become a fruitful field for mathematical logicians. Bergson’s and Arrow’s concepts of the social welfare function are related, in that Arrow’s social welfare function is the process by which Bergson’s social welfare function would be generated. Thus, Arrow’s analysis could be viewed as more fundamental than and logically prior to that of Bergson. However, it has been argued that Arrow’s work is of more significance for political theory than for economics, since in the economic analysis of public policy economists are concerned with the nature of the given social welfare function in Bergson’s sense, rather than with the properties of the political process which brings it about.

Reference: The Penguin Dictionary of Economics, 3rd edt.