In Defense of Cardinal Utility

The originators of the marginal revolution in economics treated utility as cardinal, able to be represented by numbers. Early in the Twentieth Century Hicks, expanding on an idea of Pareto, showed that everything that was being done in economics by treating utility as a number could be done by treating it as an ordering instead. About ten years later Von Neumann and Morgenstern found something cardinal utility could do and ordinal utility could not — model choice under uncertainty. Which approach is better in general remains an open question. 

Robert Murphy recently posted his arguments for ordinal utility on Mises Wire. Reading his arguments and thinking about the issue I concluded that cardinal utility was the better approach, not only for choice under uncertainty but for doing, and thinking about, economics in general.

Robert writes:

If we hypothetically knew that John would pick vanilla over chocolate, and chocolate over pistachio, then we know the first, second, and third items in his ranking of ice cream flavors. But we couldn’t say that John’s preference for vanilla over chocolate is bigger than his preference for chocolate over pistachio.

To find out whether John’s preference for vanilla over chocolate is bigger than his preference for chocolate over pistachio we find something else John values that is unlikely to have its value to him changed by what flavor of ice cream he is eating — say money. We offer to pay him to eat chocolate instead of vanilla and see how much it takes to make him switch. We repeat the experiment with chocolate and pistachio. If it takes a bigger bribe to get him to switch from vanilla to chocolate than from chocolate to pistachio that is evidence that his preference for vanilla over chocolate is greater than his preference for chocolate over pistachio. It is not proof — perhaps, for some bizarre reason, having chocolate ice cream instead of vanilla changes his value for money — but it is evidence. If the comparison was, as Robert believes, meaningless, there could not be evidence for it.

Robert writes:

Finally, the Austrian approach to utility definitely rules out interpersonal comparisons.

If utility is only revealed in choices, which I think is part of the Austrian approach, interpersonal comparison is ruled out for either cardinal or ordinal utility since there is no chooser standing above the parties to reveal it. This has nothing to do with the issue of ordinal vs cardinal, since the assumption that utility is ordinal does not rule out interpersonal comparisons; an ordinal ranking could be defined over everyone.

Robert writes:

Sometimes people—even other economists—are incredulous that the Austrians deny the possibility of interpersonal utility comparisons. “Do you really mean to tell me,” they exclaim, “that you don’t know if a starving man gets more utility from a sandwich than a sleeping man gets from rat poison?”

The problem here is that this approach uses the word “utility” in an everyday sense, rather than the formal sense Austrians use in economic theory.

The main reason to be interested in the issue of interpersonal utility is the utilitarian argument for wealth redistribution, that a dollar is worth more to the poor man than the rich.  For that argument it is the everyday sense of “utility,” more precisely the sense used in utilitarian philosophy, that is relevant. The same is true for other arguments, such as those around the concept of economic efficiency, that try to say something about the net effect of a change that affects multiple people.

Finally, Robert takes up the economic argument for cardinal utility due to Von Neumann — that it can be used to make sense of choices under uncertainty. His first argument against:

In the first place, the axioms necessary to satisfy their theorem are falsified in everyday experience.

Surely true. The question is not whether cardinal utility does a perfect job of describing human behavior under uncertainty but whether it does a better job than ordinal utility. VN utility gives us some, but not perfect, knowledge of how individuals will choose among uncertain outcomes. Ordinal utility gives us none. It is a mistake to reject a useful tool because it is not perfect, to make the best the enemy of the good.

Robert writes:

In the von Neumann and Morgenstern framework, they admit that the cardinal utility functions are unique only “up to a positive affine transformation,” so that should have nipped in the bud the notion that we were really grappling with underlying psychic quantities that governed human choices.

Temperature also is unique only up to a positive affine transformation — we can do physics in centigrade, Fahrenheit, or Kelvin. Robert, having heard that argument, responds by pointing to the existence of absolute zero. One can still do physics in Fahrenheit, you just have to remember that absolute zero is not 0 but -459.67°F. That aside, a temperature scale that starts at absolute zero but uses Fahrenheit degrees instead of centigrade degrees works just as well as Kelvin does. Why doesn’t that fact “nip in the bud” the notion that temperature is dealing with underlying physical quantities? Similarly for measuring length in feet or meters, weight in pounds or grams.

Robert asks:

In contrast, do we say that a dead man has zero utility? What about someone being tortured, does he have even fewer utils?

You do not need a zero of utility to do economics, but if you want one the obvious candidate is the suicide point, the level of utility at which someone is indifferent between living and dying. The man being tortured who would commit suicide if he could then has negative utils. There are numbers lower than zero.

None of Robert’s arguments, even if correct, would imply that cardinal utility does a worse job than ordinal of describing human action; they at most show that one can do economics without using cardinal utility, not that one must. Hicks proved that everything then being done by cardinal could be done by ordinal, but it is equally true that everything done by ordinal can be done by cardinal. The only argument for the superiority of ordinal is the one offered by Hicks: Occam’s Razor. Fewer assumptions are better.

Preferring ordinal utility in order to avoid unnecessary assumptions might make sense if there were no advantages to cardinal utility, but there are. To begin with, it better describes our subjective experience of choice. One advantage to studying humans over studying electrons is that, being ourselves humans and not electrons, we can get some relevant information by introspection. Doing so, I observe that my preference for chocolate ice cream over vanilla feels weaker than my preference for Baskin-Robbins’ Pralines n’ Cream over either. That preference, in turn, feels much weaker than my preference for not having my house burn down. Those are facts I directly observe about the contents of my head. I expect Robert observes similar facts about the contents of his.

A second and related reason is that cardinal utility makes it easier to intuit economics, which is why the early versions of neoclassical economics, both Austrian and Marshallian, were done that way. The marginalist revolution was about marginal utility, a concept not even meaningful if utility is only ordinal. That is why it is replaced, at least in Marshallian economics that treat utility as ordinal, by statements about the characteristics of indifference curves.

One of the problems with modern economics is the increasing emphasis on formal mathematics over economic understanding. I believe one reason for that development is the shift from thinking about economics the way Alfred Marshall did to thinking about it the way John Hicks did. The further your mental model is from what you understand, the more you have to rely on formal mathematics instead. It is easier to understand the idea of declining marginal utility than the convexity to the origin of an indifference curve.