In economics, there are two types of numbers that we use. Cardinal numbers express amounts. For example, “one”, “two”, “three”, etc. are all cardinal numbers. You can add them, subtract them, or even take them to an exponent.
Money prices are cardinal, which is why you can calculate precise profits and loss.
On the other hand, ordinal numbers express ranks. For example “first, “second”, “third”, etc. are all ordinal numbers. It doesn’t really make sense to talk about adding (or subtracting or exponentiating) ranks.
Almost all economists believe that utility is ordinal. This means your preferences are ranked: first most preferred, second most preferred, and so on. Here is a made up value scale:
1st. Having a slice of pizza
2nd. Having $2 in cash
3rd. Having a cyanide pill
Someone with the above preferences would give $2 in cash in order to get a slice of pizza. But would rather keep their $2 than to have a cyanide pill. By the same principle, they would also prefer to have a slice of pizza to a cyanide pill.
This is in contrast to cardinal utility, which requires the existence of something like “utils”. It’s just as nonsensical to say that “Sally gets twice as many utils from her first preferred good than the next best thing,” as it is to say “I like my first best friend twice as much as my second best friend.”
Usually, this is where most discussions of ordinality as it applies to economics end. But I believe I have a new extension of this concept that affect utility theory.
A New Perspective on Ordinal Preferences
Some people are dissatisfied with the ordinal approach to utility. “Sure, I prefer pizza over cyanide,” they’ll say, “but I really, really prefer pizza. You can’t show this intensity of preferences ordinally!” In other words, they believe the ordinal approach is lacking something real that a cardinal approach could approximate.
Well, it’s true that in a specific moment when I observe you choosing pizza over cyanide, I can’t really tell “how much” you preferred it.
But one way I can model it is that in your mind, you have a value scale of all things you wanted in that moment. And that the thing that you “really, really” wanted is ranked “much, much” higher relative to the other thing.
Let’s say pizza was first on your value scale, and cyanide was 1000th. So while it’s wrong to say you preferred pizza “one thousand times” as much as cyanide, it would be correct to say you would have preferred 999 other things to cyanide.
In other words, you would rather have any one of these 999 other things instead of instead cyanide—with pizza being chief among them. This is the sense in which you “really, really” prefer pizza to cyanide. We’ve been able to express the “intensity” sentiment without resorting to cardinal numbers.
Let’s extend the example. If your choice was between pizza and sushi, and sushi was your 2nd ranked good, then we can say several equivalent things: (i) you’re closer to indifference (i.e., viewing them as the same good) between pizza and sushi than pizza and cyanide; (ii) your preference for pizza over cyanide is stronger than your preference for pizza over sushi; (iii) you prefer pizza less intensely to sushi than to cyanide; and (iv) it’s easier for you to choose between pizza and cyanide than it is to choose between pizza and sushi.
Of course, we don’t walk around with an exhaustive list of all the goods we could possibly want at any time. This fact may make it virtually impossible to empirically test this account of psychology. But this way of thinking about “intensity of preferences” is at least consistent with ordinal preferences, meaning we can better understand this phenomena using mental tools we’re already familiar with.
I believe this way of thinking is also useful in interpreting “hard choices”. Everyone is familiar with being in a situation where you don’t know what to choose between to seemingly attractive alternatives. An easily relatable example might be choosing a drink at the self-serve cola machines that are in most fast food restaurants now. You might really like both Vanilla Coke and Cherry Coke, but you can only choose one. Because you feel as though you like both of them equally, this is what makes it a hard choice.
In other words, I’m proposing the reason that this is a hard choice is because these two goods are positioned very close together on your value scale. So close, in fact, that you have a difficult time determining which outranks the other. This is what makes the choice hard.
Applications of this framework
You might object. “This might be fun to think about, it might even be a contribution to psychology. But what implication does this have for economics—that is, the science of human action?”
My answer is this: when presented with a difficult choice, a person will choose to wait instead of choosing instantly. Waiting allows them to collect more information, deliberate more, and consider other options.
(More formally: a person facing a choice between two goods that are indistinguishably close on their value scale will not choose either good in the present period; instead, they will postpone the choice to a future period where they expect to have a larger information set.)
How can we apply this framework to the real world? Before we begin, note well that hesitation is itself an action. And as an action, it has a place on the individual’s value scale. For an entrepreneur, this has at least two implications.
First, the entrepreneur’s consumers may be facing hesitation because they can’t choose between the goods on sale. Think back to the cola examples. It’s possible that you’re in such a rush that the hesitation is not worth your time. The consumer may choose not to buy cola at all.
Second, I believe this approach can shed new light on the issue of so-called “transfer pricing”. While transfer pricing typically is used in the context of tax ramifications to a firm that is trying to buy or sell assets from a subsidiary, we can generalize the concept by considering how a either a very large firm that has “horizontally integrated” by buying and selling inputs for its final product from itself, either because it has grown so large that it’s merged with all its competitors and suppliers, or it has a government monopoly where no other firm is allowed to produce that input. In short, if a firm has monopolized the production of inputs to the extent where no market prices exist for them, how should he calculate his own costs (and therefore profits)?
Murray Rothbard was the first to observe that in effect a firm that has grown so large where this is a problem has become a socialist economy. And just as how a socialist economy can’t produce efficiently without market prices, neither can this hypothetical firm. (For more on this point, see pp. 659-660 of Murray Rothbard’s Man, Economy, and State with Power and Market.)
And so while the standard story of why a socialistic economy can’t rationally calculate profits and losses is based on the cardinal notions of money—without money prices, you literally cannot subtract costs from revenues—I approach from a different angle. Namely, action becomes “harder” because the lack of a market does not allow the firm (or socialist government) to observe its own ordinal rankings for its inputs. And so, the firm faces a “hard choice” in how to optimize its own production schedule.
Firms and governments also demonstrate that they’re engaging in hard choices, by establishing bureaucracies. Firms hire “transfer pricing specialists”, governments set up councils that “determine” prices, and so on. This is analogous to an indecisive person hiring a consultant to help them pick between Vanilla Coke and Cherry Coke.
In summary, a “hard choice” is when a person cannot distinguish where two heterogeneous goods rank on their own value scale. This can be demonstrated through hesitation and/or deliberation. The harder it is to establish a market price for goods, the more hesitation and deliberation will be required.
I believe this approach, if correct, can be insightful for both entrepreneurs and research. A new question that is raised is, “how do individuals, firms, and governments respond under different circumstances when faced with a hard choice?”
In a future post, I hope to extend this framework of ambiguous ordinal rankings to probabilities as well. In the meantime, I look forward to any feedback on this post.