# Inequality Aversion Utility Functions: Would $1000 Mean More to a Poorer Family than$4000 to One Twice as Rich?

Economists use utility functions to represent many aspects of people’s preferences. Even when an economic model has been simplified to (in some sense) have only one good–let’s call it “consumption”–the curved, concave shape of a utility function like the one above can be used to represent any of the following:

1. Risk aversion (either in a risky investment situation or an insurance situation)
2. Resistance to intertemporal substitution
3. Resistance to substituting between one’s own consumption and the consumption (at some ratio to one’s own consumption) of a child, parent, friend, or stranger one cares about
4.  A good part of how the value of a statistical human life varies with the income level of a society
5. How people feel about inequality–that is, how they feel about the situation of the poor and the rich.

If there are two goods–lets call them “consumption” and “leisure,” the curved, concave shape of the part of the utility function that depends on consumption can be used to represent how the need to work to be able to afford more consumption changes as the amount of consumption one is doing already increases–whether that increase in consumption occurs from the passage of time or because of luck.  I mention the many things that concave utility functions are used to represent because it is not at all clear that the utility function one should use to represent one of these things should look the same as the utility function one should use to represent another. In this post, I want to focus on just one thing a concave utility function can be used to represent: how people feel about inequality.

I want to emphasize that finding a good utility function to represent how people feel about inequality requires asking about people how they feel about the situation of the rich and the poor. There is no guarantee, for example, that you could ask about someone’s attitudes toward risk and get a good read on how they feel about inequality.

Yoshiro Tsutsui, Fumio Ohtake (both of the University of Osaka) and I arranged to collect data on a rider to the February, 2005 University of Michigan Survey of Consumers that asked directly about people’s feelings about the situation of the rich and the poor. The sample was the same sample as that used for the University of Michigan Consumer Confidence numbers–a sample intended to be representative of the adult U.S. population. This post gives a preview of some of the results from an academic paper we are working on, ably assisted by Daniel Reck and Fudong Zhang. It follows up on what I said about the poor and the rich in my first post “What is a Supply-Side Liberal?” In this post, I am taking the overall philosophical perspective is that of Utilitarianism, as developed by modern welfare economics using a social welfare function.

Yoshiro, Fumio and I wanted to ask questions that got at the key issues while minimizing reactions based in a shallow way on political ideology. To the extent these questions are about redistribution, the intent is to get at only the benefits of redistribution, as distinct from the costs of redistribution (say through tax distortions).

We began by asking

It is often said that one thousand dollars is worth more to a poor family than to a rich family. Do you agree?

90% of all respondents agreed.  Then we went on to ask questions to probe how much more $1000 is worth to a poor family than a rich family. I won’t give the whole sequence of questions here. Let me just choose two questions that are especially revealing about what the typical adult American thinks. When we asked Think of two families like yours, one with half the income of your family, the other with the same income as your family. Which would make a bigger difference, one thousand dollars to the family with half your family’s income or four thousand dollars to the family with an income like yours? 66% of all respondents thought the$1000 to the poorer family with half the income would make a bigger difference than $4000 to the richer family. (Everyone who had disagreed from the outset with the idea that$1000 is worth more to a poor family than to a rich family was counted as thinking the $4000 to the rich family would make a bigger difference.) When we asked Think of two families like yours, one with half the income of your family, the other with the same income as your family. Which would make a bigger difference, one thousand dollars to the family with half your family’s income or eight thousand dollars to the family with an income like yours? 66% of all respondents though the$8000 to the richer family would make a bigger difference than $1000 to the poorer family with half the income. Focusing on the middle opinion, I read this evidence as saying that the median adult American thinks that$1000 to a poorer family with half the income would have about the same impact on that family’s life as an amount of money somewhere between $4000 and$8000 to the richer family. Stretching the interpretation a little more, I am going to take the utilitarian perspective and talk about this median view as “inequality aversion” and as an indication that most people think there would be some benefit to redistribution, though the costs might sometimes–or even often–outweigh the benefits. I do think that view represents the views of those in the middle of the political spectrum.

How can we represent these views in an inequality-aversion utility function? To make the numbers a little easier, let me lowball the degree of inequality aversion a little, and act as if $1000 to the poorer family with half the income had exactly the same life-impact as$4000 to the richer family. Let me also simplify by assuming that those ratios hold regardless of the initial income level.  With those simplifications, some moderately advanced mathematics implies that the utility function must be of the form

U© = A - B/C

where A and B are some positive numbers and C is the level of consumption spending of an individual or family of a given size. The reason A and B are not determined is that we need some yardstick. It is easy to forget, but almost all measurement requires the choice of some arbitrary yardstick. The exact length of an Earth day is an accident of how our solar system formed and the geological era we are in, but we used it to develop units of time.  Similarly, a kilometer was originally intended to by a 1/40,000 of the circumference of the Earth. In addition to the size of units of measurement, we also often need arbitrary starting points. Our measures of longitude start at 0 at Greenwich, England,  which has to do with historical accidents of geopolitical and scientific power and influence at the time the system of latitude and longitude was chosen.

Fortunately, no economic logic depends on the values of A and B. The value of A doesn’t matter because for economic decisions because in any decision it is the comparison of how well-off one is under two or more possible situations that matters. When comparing any pair of options,  the difference in utility between those two choices will leave “A” cancelled out. This is analogous to the fact that the path from Ann Arbor to the Detroit Metro Airport would be the same even if, in an egocentric change, Ann Arbor were the starting point for longitude instead of Greenwich, England. And the path from Ann Arbor to the Detroit Metro Airport would also be the same if Kabul, Afghanistan were the starting point for longitude. The value of B doesn’t matter because using one value of B rather than another is like the choice to measure distances in miles rather than kilometers, or in inches rather than yards. The real-world answers are going to come out the same.

For convenience–and only for convenience, since it doesn’t matter–I have chosen A=0 and B=1 for the graph at the top of this post. It may seem odd that utility is then always negative for this functional form, but utility being represented by a negative number is literally meaningless except in relation to what 0 utility means. With A=0, a utility of zero is material bliss–the maximum utility possible. So negative utility simply means that one has fallen short of material bliss.

Marginal utility is the slope of the utility function. It tells how much extra utility there is from a little more consumption. Even before choosing A=0 and B = 1, we can say that marginal utility here is

Marginal Utility = U’© = B/[C squared]

Notice how A has already dropped out. After choosing B=1, marginal utility becomes

Marginal Utility = U’© = 1/[C squared]

This means that doubling consumption C will reduce marginal utility to one quarter of what it would have been at the lower level of consumption, so $4000 at that higher level of consumption means only as much as$1000 at a consumption level half as big. What this shows is only that the utility function (with its associated slope, marginal utility) is doing OK at representing what we designed the utility function to represent: $1000 to a poorer family with half the income meaning the same as$4000 to a richer family.

It is my contention that bringing the discipline of mathematics to discussions of redistribution is useful in informing the debate about redistribution. Let me give just one example. Looking at the utility function at the top of the post, the slope shows how much a little extra money means to someone at each level of consumption. The difference between the slope at different levels of consumption shows how much benefit there is from redistributing from a richer to a poorer individual or family–a benefit that then needs to be weighed against the costs–for example costs to freedom from the compulsion of taxes, or costs from people’s efforts to evade and avoid taxes. If one thinks of a consumption of 1 as representing the middle class, a consumption of 4 as representing the rich and a consumption of ¼ as representing the poor, one can see that there is a much bigger difference in the slope for the poor minus the slope for the middle class than the difference in the slope for the middle class minus the slope for the rich. So with a utility function that has the slope depend inversely on the square of consumption as here, there are much bigger gains from redistributing dollars from the middle class to the poor than there are from redistributing dollars from the rich to the middle class.