There A’int No Such Thing As A Social Welfare Function [TANSTAASWF]

People learn economics for two reasons. The first reason is to understand how the world works. The second reason is to try to improve the world in a logical manner and avoid common pitfalls. Occasionally, these two reasons lead to a major tension, and the concept of a social welfare function is one of these points.

We hear a lot about how some policy is in “the common interest” or is “socially beneficial.” An economist should instantly take a critical eye to such phrases based on one simple fact: society and community do not have benefits or interests because society and community are abstract nouns representing a series of relations and not acting individuals. So instead of starting with “the common good,” we have to start with the true actors and benefactors: individuals.

All economists who are not total cranks agree that individuals have preferences over available goods and rank these preferences in some way. Mainstream economists [of which I am, incidentally, not one] assume that these preferences follow certain rules: completeness (i.e., you can make up a big list of all comparisons of one good to another: redheads versus blondes, redheads versus brunettes, redheads versus V-8 engines), transitivity (if you prefer redheads to blondes and blondes to brunettes, you prefer redheads to brunettes), and convexity. Given these, you can formulate a continuous utility function, so that you can map the value of different bundles of goods. So each individual has an associated utility function. This utility function provides a very simple numerical representation of a bundle of goods to allow for quick comparisons. It also allows you to see the effects of a policy on a person, and whether this person will be harmed, helped, or neither.

“So, then, let’s just add up all of these utilities and pick which policies maximize this,” you might be saying. And, in fact, this was the conclusion of a lot of classical economists in the post-Bentham era (as well as folks in the neoclassical era). There were some discussions about how, exactly, to aggregate these values. You can take a strict summation of utilities, or you can weight certain utilities extra (like, for example, giving a higher weight to the poor than to the rich). You can even be like John Nash and propose that you multiply all of the utilities together, which gives even more weight to those lower-utility folks. So there’s a lot of different methods to choose from.

It’s just that there’s one problem: this assumes that utility functions are cardinal. In fact, they are ordinal. If I have a utility of 6 from one set of goods and a utility of 3 from another set of goods, this does not mean that the first set of goods provides me with twice as much utility; rather, it means only that the first set of goods provides me with more utility. It does not signify how much. The reason for this is that utility functions are just mappings of preferences, and as mapping, they can take on any value so long as they represent the preferences perfectly. Here’s a numerical example: suppose that two goods exist, cheese and bread. Let’s say that I prefer 3 cheese and 2 bread units to 2 cheese and 3 bread units, and the latter to 4 bread and 1 cheese unit. A way to represent these preferences would be the utility function
u(cheese, bread) = 5*cheese + bread

In the first case, my utility is 17 (5*3 + 2). In the second case, my utility is 13, and the last case has a utility of 9. If these values were cardinal, then I would prefer the first to the third by almost a factor of 2. But wait! What about:
u(cheese, bread) = 5*cheese + (bread + 15)?

Well, my first utility is 32. My second utility is 28. My third utility is 24. Now, instead of preferring the first option to the third option by a factor of two, I prefer the first to the third by a mere 33%. But both utility functions perfectly correspond to the underlying preferences, so both are equally valid. So this means that utility functions must be ordinal and not cardinal. Here’s another way to think about this difference: when classical economists thought of “utility,” they were thinking of a value given in return for consuming a product. And you could compare products by comparing the amount of “utility” received for each product. So you can have a function that looks like 1 car ride / 1 util or 2 horseback ride / 1 util, and thus get that 1 car ride = 2 horseback rides. The problem is, what’s a util? It was a fake measure thought up to try to explain that people receive value; it cannot be calculated. So without this calculation, you don’t have a denominator, and without the denominator, you can’t actually get a cardinal utility (or number of utils).

What that means for our social welfare function is that just adding up these different welfares isn’t such a good idea. Let’s say that Dan, Tony, and I make up society, and there is some type of policy. Here are our utility functions:
Dan – 3*cheese + 2*bread
Tony – 9*cheese + bread
Kevin – 4*cheese + 4*bread

We implement the Cheese Renewal Act of 2007, in which we work together to increase cheese production and make it so that there is an additional sum of 6 cheese units in our society. Let us also say that, for political reasons, each of us gets 1 cheese unit automatically, so that there are now 3 cheese units left to distribute. If we just use a social welfare function in which we add up the utility levels, a policy to give the last 3 units of cheese to Tony would net us 27 utility points. To give it to Kevin would give us 12. To give it to Dan would give us 9. And to give one to each would give us 16 points. So thanks to Tony’s intense love of cheese (and that we don’t experience any diminishing marginal demand for cheese, thanks to the easy utility function), giving him all of the cheese would be a policy winner.

But as I said before, utility functions are ordinal, and this means that they can survive any monotonic transformation. I showed you a case in which I added some constant to the utility, but the other monotonic transformation is to multiply the entire thing by a constant. So let’s multiply Kevin’s utility function by 10. This is still an ordinal utility function and covers exactly the same preferences, so it’s a valid thing to do:
Dan – 3*cheese + 2*bread
Tony – 9*cheese + bread
Kevin – 40*cheese + 40*bread

Now, giving Kevin the 3 additional cheese units nets 120 points of utility, while giving Tony the 3 only gets us 27. And you can do the same thing for Dan so that he “should” get the three cheese units. The reason for our problem is that we do not have a consistent denominator here, again due to the fact that these utility functions are all ordinal. And without a consitent denominator, we can aggregate these utilities however we would like. If Kevin happens to be the brother of the guy coming up with the social welfare function, then we could see Kevin have the 40*cheese + 40*bread. But if Tony bribes the guy, he might just pick 4*cheese + 4*bread. And if the fellow happens to be a Catholic, perhaps Dan could get a cardinal to talk to the guy and maybe make it so that Dan has a utility function of 300*cheese + 200*bread. So taking all of these individual utilities and add them or multiply them or whatever doesn’t really solve our problem; or, better put, it doesn’t give us a unique solution, but rather an infinite number of solutions.

But wait, there’s more! See, I’m not the first guy who thought of this. In fact, some rather smart people understood that you can’t have a social welfare function like this, as you can’t add up ordinal measures to get a cardinal measure. So instead, they asked themselves, “Self, what should a social welfare function look like if we have ordinal individual welfare functions?” The answer is that such a function should have four properties: completeness, transitivity, irrelevance of alternative options, and non-dictatorship. So a function should be complete and transitive, just like the individual functions. And the third option, in which outside alternatives are irrelevant, means that if I offer you the choice between cheese and bread and you choose bread, you should still choose bread over cheese if I offer you cheese, bread, and crackers. So adding in new options should not change the social choice. If you have these three, you can map the social preferences to a numerical social utility function, which gets rid of the problems associated with social utility functions above, and lets you choose the policy with the highest social welfare. The final option means that a social welfare function should not correspond exactly with any one individual’s welfare function, as we want some notion of “fairness” here. Well, add all of this up and you get Arrow’s Impossibility Theorem.

The upshot of Arrow’s Impossibility Theorem is that you can’t have 1, 2, and 3 hold true and still have 4 hold true. So if you do not want a dictatorial social welfare function, you have to relax either condition 1 or condition 3. But if you relax condition 3 (the irrelevance of alternative options), you can’t have a numerical social utility function, which defeats the entire purpose of trying to develop a social welfare function. And if you relax condition 1, you are really just evading the problem.

So in the end, there still isn’t a viable social welfare function, even with mainstream theory. Any aggregation of utilities can be tampered with, and an ordinal social welfare function leads to dictatorship, in which one person gets the best of the deal in every single case. And it is for this reason that Pareto optimality is seen as the most beneficial condition for a social policy. If no alternative policy could make somebody better off without harming anybody else, then the policy in question is Pareto optimal.

Thus, if you are to speak of “the common good” or “society’s best interest,” the only way you can do so is by talking in terms of Pareto optimality. But what if you have two policies which are Pareto optimal? Well, then economics doesn’t really have much to say. This is because you can’t compare Pareto optimal policies in terms of some type of welfare function—because There A’int No Such Thing As A Social Welfare Function.

4 thoughts on “There A’int No Such Thing As A Social Welfare Function [TANSTAASWF]

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