Monday, February 14, 2022

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.

11 Comments:

At 10:53 PM, February 14, 2022, Anonymous Arqiduka said...

Probably some flavor of silliness, but does it matter that you can "fit" many, many more choices between "burned house" and "chock ice cream" than you can between the latter and "vanilla ice cream", even in an ordinal framework?

 
At 11:44 AM, February 15, 2022, Blogger Carl Edman said...

Agree entirely. Would just add that a temperature scale with steps like Fahrenheit but zero at absolute zero exists: https://en.wikipedia.org/wiki/Rankine_scale?wprov=sfti1

I think they all got it wrong and we should use a measure based on the inverse of temperature as currently conceived. It would make it easier to understand that temperatures “below” absolute zero do exist but are actually “hotter” than the highest conventional temperatures.

 
At 12:07 PM, February 15, 2022, Blogger Outside The Box said...

As has been true for decades David, I find myself nodding along with both the way you approach thinking about this and thus your conclusions... it is so important to realize that we are talking about models here, and that models are just tools in that they are better at some tasks than others - just like a screwdriver and a hammer - and different models have different strengths and weaknesses. In general, the ordinal model's strength is that because it makes so few assumptions, it applies to a huge number of situations; but the flipside is that it says *less* about those scenarios, precisely because so few assumptions went into it. Other models like the cardinal model can tell us more about some scenarios (as you detail), but they require a bit more information and so aren't so helpful when we don't have that information. Neither model is "right" any more than either a hammer or a screwdriver is "better": it depends on what questions you are asking.

And to that point, there can be more models than these and sometimes they also tell us useful things. The trick is in knowing which to use and when, and in understanding how they might be in error.

 
At 7:51 AM, February 17, 2022, Blogger Modern Mugwump said...

Very convincingly arugued (to this non-economist). I wonder if there are different types of cardinal utility (i.e. log vs. linear). Most (maybe all) human sensory perception is log - that seems obviously more useful.

 
At 6:40 PM, February 17, 2022, Anonymous Anonymous said...

《everything that was being done in economics by treating utility as a number》

Is the goal to prove free markets produce fair, non-arbitrary prices? But if preferences can be intransitive, neither cardinal nor ordinal utility will help you rule out arbitrary prices?

So what do you do with the old AA homile, "one is too many and a thousand is not enough"?

Also, how does assessing preferences by paying people work in a society like on North Sentinel Island, or on an individual like the North Pond hermit, both of whom had no use for your money?

Thirdly, why did my brother commit suicide despite having been indoctrinated in utility maximization theory by his economics teachers at Berkeley, leading to a successful career in accounting, unless the zero point of utility is potentially around $100k/year? Was my brother afraid to give up his lucrative salary to pursue spiritual fulfillment, because economists taught him that would be irrational? Did he need $1 million/year, because "1000 is not enough"?

Fourth, is the idea of budget constraints relaxed by finance? With an implicit Fed put, are "too big to fail" agents really subject to any budget constraints, and what does that say about your ability to prove prices aren't arbitrary?

 
At 4:47 AM, February 18, 2022, Anonymous Arqiduka said...

Another thought that occurred to me earlier. Your reasoning in this post is a prime example of the methodological differences between Austrians and various others, to the degree I understand them: to an Austrian "cardinal explains as much as ordinal, and a bit more, so we assume it" is just wrong. Cardinal v ordinal is a question of which you can derive from the logical framework, model simplicity or predictive power be damned.

 
At 9:10 PM, February 20, 2022, Anonymous Anonymous said...

"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."

Except that money is just a proxy for the goods that are also subjectively valued against ice cream flavors.

It's not the money he's valuing, it's the goods he will buy that he's valuing.

The offer of money tells you that John thinks he can satisfy a higher-ranked end with X amount of money, but money units aren't a measure of units of satisfaction.

If, at any given time, John prefers a pizza to a certain flavor of ice cream, that preference doesn't change with the price of a pizza - if the pizza was a million bucks, he'd still prefer it to a particular ice cream flavor.

"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."

What if John only wants a bigger bribe to switch to chocolate because then he can now afford to give a special lady *her* preferred flavor of ice cream and now also go buy himself some vanilla and go enjoy the ice cream with his lady friend?

The bigger bribe would tell you nothing about how much more John values vanilla over chocolate than he does 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."

For any desired end there's going to be amounts of stuff that's required to satisfy that end.

But the quantity of stuff required to do so is only meaningful because of the subjective ends that are desired.

It's not the quantity that determines how useful something is, rather it's the desired utility that determines the quantity of stuff that is required to attain that utility.

And that's why utility is ordinal, not cardinal.

 
At 1:24 PM, February 21, 2022, Blogger Adam Ruth said...

I tried to comment, but it was deleted on the submit. I’m trying one more time.

 
At 1:24 PM, February 21, 2022, Blogger Adam Ruth said...

Goodbye.

 
At 1:53 PM, February 21, 2022, Blogger Adam Ruth said...

My comment was that ordinal utility between chocolate and pistachio is meaningless without the difference in utility between vanilla and chocolate.

Is the vanilla option unavailable because of a shortage of vanilla or a reaction to the banning of vanilla? It matters, and the hypothetical is incapable of compensating for it. Hypotheticals, such as this, that don’t account for environmental effects are, by their own nature, invalid an uninstructive.

Think of the trolley car problem, it’s all BS without a full systemic analysis.

 
At 9:06 PM, February 21, 2022, Anonymous Anonymous said...

"It matters, and the hypothetical is incapable of compensating for it."

Ordinal utility just means that someone has a preference for one thing over another. There is always a ranking of choices, so an absense of vanilla would not render an ordinal ranking of chocolate and pistachio impossible.

"Hypotheticals, such as this, that don’t account for environmental effects are, by their own nature, invalid an uninstructive.

"Think of the trolley car problem, it’s all BS without a full systemic analysis."

Henry Hazlitt had a great response to that argument:

Failure of the New Economics
Chapter VII: "Statics" vs. "Dynamics"
by Henry Hazlitt
https://mises.org/library/failure-new-economics-0

"It is a mistake to believe that we can skip over all “static” assumptions for the superficial reason that such assumptions are “unreal.” This would be as foolish as it would be for a ballistic-missile designer to skip over all preliminary calculations of the probable flight or parabola of his missile through a frictionless medium, on the ground that no actual medium is every really frictionless.

"In order to understand the consequences of dynamic assumptions we must first of all understand the consequences of static assumptions. The method of science is that of experimental or (when that is impossible) “hypothetical isolation.” 6 It is the method of “successive approximations.” 7 It is to study one change, force, or tendency at a time, whenever that is possible, even when it usually, or perhaps always, acts in combination with other forces, and then to study later the combinations, interrelations, and mutual influences of all the main changes, forces, or tendencies at work.

"The belief that we can skip over all these tedious preliminaries, and surprise the secrets of the actual economy in one great leap by the use of simultaneous differential equations, is a double delusion. It disdains a method that is indispensable in order to embrace a method that is inappropriate and illegitimate."

 

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