"What about the theorem - that Rothbard dismissed - which claims that utility-maximizing individuals equalize the marginal utilities of goods consumed divided by their prices? Doesn't this show that neoclassicals believe in cardinal utility? No, it does not; statements made in technical jargon often sound absurd if you forget the underlying definitions. A utility function just uses numbers to summarize ordinal rankings; it doesn't commit us to belief in cardinal utility. Deriving the marginal utility of individual goods from this function commits us to nothing extra.[10]"

Deriving the marginal utility from ordinal utility functions commits us to nothing extra. Is that true? Is cardinality defined by being non-ordinal? ie. is marginal operations, taking derivatives (for the moment ignoring discrete vs continious and just focusing on ordinal vs cardinal) possible in ordinal rankings? can you derive a derivative, but then continue to say yes yes this derivative is from an ordinal ranking. Is he trying to have his cake and eat it too?

Could someone point me to a mathematics text (introductory as possible) that explains the requirements of a scale before these operations can be conducted?

That's the beauty of math. If you map them onto the seven dwarfs, manipulate the dwarfs system under logically admissable rules, and then invert the map back from dwarf space to the original space, the result is true in the original space, even though the intermediate dwarfish dance has no meaning in the original space. As I'm sure you know.

So that if economists claim that embedding the ordinal relations into cardinals and thern taking derivatives etc in the cardinal space yields a result that can be interpreted in terms of ordinals only, then that result is true for the ordinals. Which they claim they have done.

The paper linked to above laughs at the economists for using dwarfs, which do not exist in the real world etc. I am pointing out that such an argument is specious.

>The paper linked to above laughs at the economists for using dwarfs, which do not exist in the real world etc. I am pointing out that such an argument is specious.

Could someone point me to a mathematics text (introductory as possible) that explains the requirements of a scale before these operations can be conducted?

Bootm line, a derivative, no matter how you slice it, is a linear approximation. Which means, at the very least, that you need numbers involved. Cardinal rankings use numbers, ordinal rankings do not. So no derivative is possible using ordinal rankings.

Deriving the marginal utility from ordinal utility functions commits us to nothing extra. Is that true?

He didn't say that. He said deriving the marginal utility from cardinal functions adds nothing extra. I don't know if that's true or not.

Is cardinality defined by being non-ordinal?

Ordinal means order. There is an order to preferences. I like apples more than oranges is an ordinal statement. Cardinal means numbers. I like apples 5 units worth of liking, and oranges 3 units worth of liking, is a cardinal statement.

ie. is marginal operations, taking derivatives (for the moment ignoring discrete vs continious and just focusing on ordinal vs cardinal) possible in ordinal rankings?

As above, no. But see next answer.

can you derive a derivative, but then continue to say yes yes this derivative is from an ordinal ranking.

What you can do is start with an ordinal ranking [I like apples more than oranges, oranges more than potatoes]. Then you can make a cardinal ranking from that, by saying let apples have 3 units of liking, oranges 2, and potatoes one. You then take that cardinal ranking, find some differentiable function that has the same values as the cardinal rankings, and take derivatives of it. You then define the derivative of that very first ordinal ranking to be the derivative of that final differentiable cardinal function.

Of course you then have to prove that the final number you assign to the ordinal function is independent of the cardinal functions you chose in the intermediate steps [which can be done in infinitely many ways]. Caplan claims there is such a proof.

Bob Murphy claims that the result Caplan got using cardinals can be derived using ordinals alone, but didn't provide a link to back up that statement.

I've seen a paper by McCullogh that [I think] says that there exist simple ordinal rankings [of four objects] than can be proven to be impossible to make cardinal.

An ordinal ranking that satisfies some basic and intuitive criteria has a continuous cardinal representation. As for differentiability, this is often not neccessary for many economic models. Therefore, even if you think derivatives are suspicious, many mainstream results would still stand.

As for differentiability, this is often not neccessary for many economic models. Therefore, even if you think derivatives are suspicious, many mainstream results would still stand.

But what is the point, then, of using a cardinal ordering, something mainstream economists concede is disconnected from reality [=nobody has a cardinal ordering in his head of anything]?

I've read McCulloch's work before; thanks for reminding me of it. It deserves more attention, since it's method of utility analysis is very unique.

McCulloch's counterexample is not really a counter example. He is imposing very strict and unusual assumptions about the utility function, which most theorists do not impose.

Cardinality makes things much easier to work with, regardless of differentiability. I won't get into the details, but many theorems have been discovered that allow one to get the same results as differentiable functions without this feature.

McCulloch's counterexample is not really a counter example. He is imposing very strict and unusual assumptions about the utility function, which most theorists do not impose.

Couls you be more specific, please? What strict and unusual aasumptions does he impose?

Well, he directly states his assumption at the begining of the section. He wants a function that ranks subsets using the relative rankings of their elements. That's not exactly standard. Most utility functions are not measures. It may be plausible in some cases, but it does not serve as a universal counterexample.

He wants a function that ranks subsets using the relative rankings of their elements. That's not exactly standard.

But it seems quite realistic, something that might happen constantly in a real life situation. So are you saying that the standard just doesn't bother with those ordinals because they are inconvenient to deal with? Sort of like a doctor who doesn't treat broken bones because it's too messy and not amenable to pills?

...it does not serve as a universal counterexample.

Not sure what you mean by a universal counterexample. Something is either a counterexample or it isn't.

My point was just that it's a very strict assumption, which is clearly not true for many situations in life. If you read the paper he cites by Pratt and Kraft, you can see more clearly that this is a strange assumption, since, in essence, he is assuming that preference representations look like measures. Why is this realistic?

What I originally said still stands. His example is not a counterexample. It is only a counterexample to his own question, as posed at the beginning of that section.

My point was just that it's a very strict assumption, which is clearly not true for many situations in life.

But it is true in many other situations in life.

In any case, we aren't looking for something that true in all situations in life, or even many situations in life. We are looking for a counterexample to the statement that all the interesting and realistic ordinal rankings can be turned into cardinal rankings, and he has found one [in fact, two. Table 4 and table 16 of his paper].

Why is this realistic?

Because it quite possible that four real life objects, in the minds of real life people, have the ordinal ranking he assigns them. AKA realistic. What is unrealistic about it?

Perhaps you could provide a link to the paper by Pratt and Kraft.

What I originally said still stands. His example is not a counterexample.

That's not what you said earlier. You said it is indeed a couterexample, but not a "universal" one. But you did not explain what a universal counterex is.

It is only a counterexample to his own question, as posed at the beginning of that section.

But his question, as posed in the beginning of the section, is whether one can find an ordinal ranking that cannot be made cardinal. Which is exactly what we are looking for, and that he displayed.

If you don't think I'm correct, I encourage you to email McCulloch. I've corresponded with him before, and he answers emails.

That said, I'll try to explain this again. McCulloch is asking whether a particular kind of cardinal representation exists for a given ordinal ranking. This is similar to asking whether a differentiable representation exists. In either case, the answer may be in the negative, as McCulloch shows.

That does not mean that theredoes not exist a cardinal representation of some kind. Therefore, my original comment is correct.

OK, I finally see what you are getting at. You are saying McCullough is placing a restriction on the cardinal function, not on the ordinal function [as I thought you meant until now].