There has been a flareup in free-market economist circles over the issue of “interpersonal utility comparisons.” First Tyler Cowen wrote a post that took it for granted that a rich man got less utility from an extra dollar than a poor man did (although Cowen didn’t think that justified government redistribution by itself). In response, David R. Henderson expressed puzzlement that Cowen overlooked the fact that such an interpersonal utility comparison (IUC) was nonsense, because utility is an ordinal concept. Then a bunch of people, including David Friedman, jumped into the fray, claiming that Henderson was outdated because von Neumann and Morgenstern had shown how we could in fact construct cardinal, not ordinal, utility functions.

In the present blog post I’ll hit the key points in this dispute. To cut to the chase, I agree with David R. Henderson: The way economists use the term, “utility” is an ordinal concept, which expresses a subjective ranking, not an objective measurement. Therefore it makes no sense to say Jim gets more or fewer utils than Sally. Furthermore, the work of von Neumann and Morgenstern does not alter this basic fact: Whether we “believe in” cardinal utility has nothing to do with their demonstrations.

*Brother Can You Spare a Dime?*

For a full treatment see my earlier blog post here, when Dan Klein (like Tyler Cowen) also stated matter-of-factly that rich people had a lower “marginal utility” of money than poor people. But for our purposes, let me give a brief summary of the main argument:

Economic theory uses the concept of diminishing marginal utility (DMU), which says that people attach less utility to subsequent units of a good. This concept explains why a shopper in the grocery store doesn’t spend all of his money on the first type of item he puts into his cart. For example, suppose a guy starts with $20 in his wallet and sees that soda is on sale for $1 per bottle. The guy puts the first bottle in his cart because he values the 1st soda bottle more than his 20th dollar bill. He puts another in his cart, because he values the 2nd soda bottle more than his 19th dollar bill. But (let us suppose) he stops there, because he values the 18th dollar bill more than the 3rd bottle of soda. The marginal utility of the soda diminishes with each acquired unit, while the marginal utility of money increases as the man (mentally) spends away his units of money.

Without being able to “think on the margin,” it would be hard to explain why people only make limited purchases: If the man thought the first bottle of soda was “worth it” at $1, why would he stop? Only marginal thinking can explain everyday consumer behavior.

Now the problem comes in when people sometimes try to use the concept of DMU to justify government income redistribution. Specifically, the argument is that (say) the billionth dollar to Bill Gates has hardly any marginal utility, while the 10th dollar to a homeless man carries enormous marginal utility. So clearly–the argument goes–taking a dollar from Bill Gates and giving it to a homeless man raises “total social utility.”

There are several serious problems with this type of claim. Most obvious, even if we thought it made sense to attribute units of utility to individuals, there is no reason to suppose we could compare them across individuals. For example, even if we thought a rich man had units of utility–akin to the units of his body temperature–and that the units declined with more money, and likewise for a poor person, nonetheless we have no way of placing the two types of units on the same scale.

*Ordinal vs. Cardinal*

However, a deeper problem is that utility itself–in the way it has traditionally been used in consumer theory by economists–is an ordinal concept. To explain these types, note that the ordinal numbers are things like 1st, 2nd, 15th, and so on. In contrast, the cardinal numbers fall on the real number line; they are things like 2, 3.78, and 604.2. Note that it makes sense to perform arithmetical operations on cardinal numbers; we can say that 12.4 is twice as much as 6.2. But we can’t perform operations on ordinal numbers, except to rank them. 1st is better or higher than 2nd, but it’s not “twice” as good. Furthermore, we can’t say that the gap between 1st and 2nd is smaller than the gap between 2nd and 18th. Such statements are meaningless, if all we have is the ordinal ranking itself.

Even though modern mathematical economics uses utility functions that spit out cardinal numbers, strictly speaking the underlying theory is ordinal. (This is one of the points Bryan Caplan made in his essay criticizing Austrians.) The building block of modern preference theory is a “preference relation,” which takes any two bundles of goods and ranks them–either one is strictly better than the other, or the consumer is indifferent between them. There are theorems showing that if the preference relation for a consumer obeys certain properties, then we can construct a cardinal utility function to “represent” the preferences. But there’s nothing magical about the “utils” such a function shoots out. All we care about is that the utility function assigns a greater number of utils to a preferred bundle. So if the individual prefers bundle A to bundle B, then the utility function had better say bundle A yields more utils than bundle B.

Yet to repeat, if we understand what the utility function means in this context, there should be no confusion about the utils being “real” cardinal things. If a certain utility function represents the underlying ordinal preference relation, then we can transform it into a different utility function–say by squaring it–and that new utility function will also work, so long as our transformation is “monotonic,” which is just a fancy term meaning that the ranking is respected.

*Von Neumann-Morgenstern Utility*

So far so good. But as David Friedman and others pointed out, in 1947 John von Neumann and Oskar Morgenstern famously proved that under certain conditions, we can represent a person’s ordinal preference rankings over lotteries with a cardinal utility function such that the person makes choices as if he were maximizing the mathematical expectation of the utility of the underlying prizes. In order for the cardinal utility function to obey this special property, it could only be transformed in a linear way (the jargon term is “positive affine transformation”). This is why David Friedman and others think that vNM utility theory showed that economics had returned to a cardinal notion of utility.

To help the novice understand the two approaches, let’s work through two examples involving fruit. Suppose John has the following ordinal preference ranking:

1st: Apple

2nd: Banana

3rd: Orange

We could say, “John gets more utility from an Apple than from a Banana,” and we could conclude that if John can afford only a Banana or Orange (because an Apple is too expensive), then John will buy the Banana.

Now if we want, we can assign a cardinal utility function to the elements in the set of fruits. For example, let’s say that U(Apple) = 5, U(Banana) = 3, and U(Orange) = 1. Then it works to say that John chooses fruit in order to maximize his utility function U, if John is only picking one fruit. But, we shouldn’t say anything silly like “The Apple gives 5 times as much happiness to John as the Orange,” because we just as well could square the original function by assigning utility scores of 25, 9, 1 and get the same result, namely a function U() that represents the underlying ordinal preference ranking, in the sense that it yields the correct behavior from John. But with this new, redefined U(), the utils of the Apple are 25 times higher than the utils of the Orange. So it should be clear–if we actually understand what we’re doing–that using the cardinal utility function doesn’t imbue any cardinality to the person’s subjective preferences.

Ah, but along come von Neumann and Morgenstern. They want to deal with choices made under conditions of uncertainty. In order to model that, they don’t have the individual rank the various outcomes, but instead the individual ranks the set of all possible lotteries over the underlying outcomes.

Back to our friend John, he now has to give us a lot more information. It’s not enough to know how he ranks the Apple, Banana, and Orange. Now we want to know how John ranks things like the following two choices:

Lottery1 with {10% Apple, 40% Banana, 50% Orange}

versus

Lottery2 with {20% Apple, 20% Banana, 60% Orange}

Just knowing the rankings of Apple, Banana, and Orange, it’s not obvious how John should rank the two lotteries above. Let’s suppose for the sake of argument that he prefers Lottery2.

Now IF our friend John catalogs his ordinal preference ranking between any two such lotteries, each taken from the set of all possible lotteries, and furthermore IF these ordinal preference rankings obey certain properties, THEN von Neumann and Morgenstern can create a little function u() that gives utils for the Apple, Banana, and Orange, such that you can create a big function U() that generates a number by the mathematical expectation of the probabilities of a fruit times the u() for that fruit.

For example, if u(Apple) = 5, u(Banana) = 3, and u(Orange) = 2, then Lottery1 above would yield overall utility of U(L1) = 0.1×5 + 0.4×3 + 0.5×2 = 0.5 + 1.2 + 1 = 2.7, whereas Lottery2 would yield overall utility of 0.2×5 + 0.2×3 + 0.6×2 = 1.0 + 0.6 + 1.2 = 2.8. Therefore, John would prefer Lottery2 to Lottery1. We could say that John was an expected utility maximizer, and that the expected utility from Lottery2 was higher.

To be clear, the choice of the little u() is still arbitrary, even here. If we, say, doubled all the values and added 2, that wouldn’t mess up the implied rankings of the lotteries, and so we wouldn’t be losing any information; we could still say we had modeled John’s underlying ordinal preference rankings with the cardinal utility function. However, we can’t do any old monotonic transformation. We can’t square the numbers in u(), for example, because then we’d lose our ability to claim John was maximizing the expectation (using the probabilities in the lottery) of the little u() function.

**Wait, Huh?**

So did we just prove that people have cardinal utility, after all? The analogy proponents often invoke is temperature. It’s true, they will admit, any temperature scale is arbitrary. Three popular ones are Fahrenheit, Celsius, and Kelvin. It would be silly for someone to say that an object that’s 100 degrees F is twice as hot as an object that is only 50 degrees F, because the numbers wouldn’t work out like that in Celsius or Kelvin.

Even so, we still think that temperature is a cardinal, measurable, objective property of the world. And notice that the transformation between the three popular temperature scales is an affine one: You can change the size of a unit proportionally, and on top of that you can slide the scale up or down a fixed number of units, but that’s it. You can’t, say, square the Fahrenheit scale to come up with an equivalent scale that works just as well to measure temperature. [Note for purists: Just focus on the positive portion of the temperature scales. I’m making a point about the unit sizes being distorted, not about, say, negative 4 degrees F turning into positive 16 degrees if we square it.]

I’m not sure that this popular move really works. For one thing, I think the reason we reject a “Fahrenheit Squared” temperature scale is that we just know that heat really *is* a cardinal, measurable quantity. If all we wanted to do was rank objects in terms of their heat from hottest to coldest, then a “Fahrenheit Squared” scale would work. But that’s not really what we want out of a temperature scale. We want an increase of one degree to correspond to the same increment in heat intensity, wherever we already are on the scale. And we want this in a scale because we know that heat is cardinal.

Do we have a similar situation with subjective preferences? I don’t think so. Most obvious, people in the real world do not obey the axioms that von Neumann and Morgenstern need for their theorem. And this isn’t so much about people being “irrational” and needing an economist to help them think through the situation; I think many people have no problem with their choices in the famous Allais paradox. It’s only a “paradox” if we assume that people ought to be maximizing expected utility.

*Conclusion*

Ultimately we have to be clear when we ask, “Do people have cardinal utility?” If all we mean is, “We can make certain assumptions in a model,” then yes I’m OK with economists claiming that they have discovered cardinal utility. But if other economists still want to cling to the idea that modern consumer theory only relies on ordinal preference rankings, they’re on solid ground too. (For a technical discussion, see William Baumol’s article.) Furthermore, empirically, people do not obey the requirements of vNM expected utility theory.

In any event, this is all a moot point regarding the original question of interpersonal utility comparisons. Even if we thought individuals had cardinal utilities, it wouldn’t follow that redistribution would raise total social utility.

Even if we retreat to the everyday usage of terms, it still doesn’t follow as a general rule that rich people get less happiness from a marginal dollar than a poor person. There are many people, especially in the financial sector, whose self-esteem is directly tied to their earnings. And as the photo indicates, Scrooge McDuck really seems to enjoy money. Taking gold coins from Scrooge and giving them to a poor monk would not necessarily increase happiness, even in the everyday psychological sense.

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