Thoughts on Social Welfare Functions

The way that most economists visualize the objective function of economic policy is in terms of the maximization of a social welfare function. This is a mathematical expression that summarizes the weight which society places on the preferences expressed by each member of society. A typical social welfare function today would be:

W (Ui) where W’ > 0, W’’ < 0 and i = 1…n where n is the number of individuals considered to be members of society.

Historically, there have been three great controversies about the nature of the social welfare function. The one on which economists have got most excited is the issue of whether utility is cardinally measurable. In the wake of Pareto, it is generally assumed that it is not possible to measure welfare cardinally, but only ordinally.

The second controversy is about an issue that makes sense only if utility is cardinally measurable and interpersonal comparisons are possible. The issue is inequality. Many of us feel it so important to recognize that equality should enter into the SWF that we take this as an important argument for abandoning the classical position that utility is only ordinal and that interpersonal comparisons are impossible. There is still the question of the weight to be assigned to inequality in comparison to an increase in total output. No one sensible (with due respect to Rawls) believes that only equality, or the income of the poorest, matters: the serious arguments are about trade-offs. The most satisfactory formalization of this is due to A.B. Atkinson (1970, 1975), who postulated that the marginal social value of the income of an individual i was equal to (Yi) where ε is a distribution parameter that varies from 0 (for those who subscribe to the economist’s de facto traditional view that distribution is unimportant) to ∞ (for those who subscribe to Rawls’ view that only the income of the poorest counts).

The third issue, which has been mainly debated in the political rather than the economic realm, concerns the domain of the welfare function: who are the i whose welfare is considered to matter? Initially society was organized in a way that implied that the only people who mattered were the king and maybe other aristocrats. The franchise was gradually broadened in England in the nineteenth century: first to the whole middle class, then to the urban working class, then to the rural working class, and finally (in the twentieth century) to women. And there it has remained, for almost a century. It would be unthinkable to deny economic rights (a role in the welfare function) to those accorded political rights, which implies that the i today are: all citizens.

Two categories of being are still denied a place in the social welfare function: foreigners, and animals. The exclusion of foreigners does not automatically mean that they are denied consideration: presumably they enter into the SWF of their home country, and then it is presumed that the political authorities of their respective countries enter into a bargaining process with the aim of maximizing the joint welfare of the two countries. The ideal end result may be to place individual foreigners in a comparable situation to nationals. (Consider ITO, or the arrangements for mutual provision of benefits under national health systems.)

There is no such reassurance for animals. They receive consideration only insofar as some of the people included in the SWF care about the continued existence of animals, or the lack of cruelty imposed on them. In Paul Collier’s book The Plundered Planet: Why We Must—and How We Can—Manage Nature for Global Prosperity, despite its main title, the analysis is all about the benefits that humans may enjoy from the policies recommended; there is no consideration of the consequences for nature per se. One may seriously doubt whether this provides a basis for meeting environmentalists halfway, as he seeks. Organized religion has long taught that animals are placed here for humanity’s benefit; we have no obligation to consider their interests. Even conservationists emphasize concepts like “existence value” to justify expenditure of resources on animal preservation.

It is high time for this arrogance, speciesism, to give way to a broad-based concern for the animal kingdom. Consider the following thought-experiment. Suppose that the human race ceases, for whatever reason, to exist. Would we want a SWF that declares itself indifferent between an incinerated world in which no advanced life survives and a world with an abundance of life-forms similar to that which existed prior to mankind’s advent? (In neither event are there any human beings, so that a SWF restricted to human preferences would have nothing to say about which state is preferable.) Another thought-experiment: would we really consider the crash of a couple of jumbo jets with total loss of life a greater tragedy than the extinction of, say, the mountain gorilla? To an economist, the implication of such thought-experiments is the necessity of placing the interests of animals in the SWF.

How to do this raises all the problems of interpersonal comparisons to the nth degree. But it can be argued that it is preferable to be forced to make such judgments explicitly rather than having them made implicitly, which typically involves completely ignoring animal interests (except maybe for pets), with the consequence that farm animals are treated quite atrociously (particularly in the United States). Personally I would give an equal weighting to humanity versus the millions of other life-forms. I would also be prepared to place a value on human life (which is how we behave). Presumably the death of any other life-form would not be weighed comparably heavily, though the extinction of a species (or subspecies) would carry a major cost. How large a cost would surely depend upon the uniqueness of the species or subspecies. It would be interesting to see if these judgments are widely shared.

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