Though utility is as inherently personal as joy or shame, their [mainstream economists] textbooks suggests that ‘units of utility’ can be measured and added. (Butler 2010: 98).As utility is only assumed to be ordinal, and not cardinal, I have never understood this criticism. The only thing that matters with utility functions is that if A is preferred to B then a utility function will append a higher number to A than B. That is, if u is a utility function representing the above preference relation then u(A)=10 and u(B)=9 will do. But a utility function v where v(A)=1 and v(B)=0.1 will also do the job. Now if we assume u and v are the utility functions for two different people, the sum of u+v makes no sense at all. Thus I don’t get the point Butler is trying to make.

## Sunday, 5 September 2010

### A point I have never understood

Previously I made mention of Eamonn Butler’s new book Austrian Economics: A Primer. At one point Butler raises an issue I have seen raised before in some Austrian writings. He says,

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## 5 comments:

isn't he saying what you are saying - that textbooks are erroneously promoting the idea that apples and oranges are to be measured in the same way?

"As utility is only assumed to be ordinal, and not cardinal"

Not always true. Von Neumann utility, used to analyze choice under uncertainty, is cardinal.

That doesn't solve the problem of interpersonal comparisons of utility, but it does mean that "U(B)-U(A) = 2 [U(C)-U(A)], or in English "I prefer B to A by twice as much as I prefer C to A," is a meaningful statement.

It means that you are indifferent between a certainty of C and a coin flip that gives you .5 probability of A and .5 probability of B.

I get that for choice under certainty the utility functions that represent the underlying preference relation need only be ordinal and that a lot of results in micro theory (e.g. existence of general equilibrium) do not require cardinal utility.

But it seems to me that most textbooks in fact switch to cardinal utility fairly quickly. In most you'll see statements like "the consumer's optimal consumption plan satisfies marginal rate of substitution = price ratio", which - unless I am badly mistaken - only makes sense if the utility function of which you are taking the partial derivatives is cardinal.

David: You're right but there is no mention of risk in Butler's discussion.

Tobias: The slope of the indifference curve equals the price ratio but as utility is constant along the indifference curve does the ordinality/cardinality matter? The slope of the difference curve just says how much of good one I will give up to get enough good 2 to remain at a constant level of utility.

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