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Given some irrational preferences, that can be represented by a function. If the function does not satisfy rationality (transitivity, completeness), does this imply it is not a utility function.

I know rationality over $\preccurlyeq$ does not imply a utility function. But rationality and continuity over $\preccurlyeq$ implies a utility function. But what about the reverse direction?

For example, $u(x) = sin(x) + 1$, is not rational, but is continuous, is it a utility function?

In my books I see a lot about the requirements needed to make a utility function, but given a function, what are the requirements for it to be a valid utility function?

My Answer A utility function is the representation of a preference relation $\preccurlyeq$. All preference relations are by assumption (or definition), rational. Given a function, if there does not exist any rational preference relation, then it must not be a utilty function.

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    $\begingroup$ Does this answer your question: economics.stackexchange.com/a/18234/42? $\endgroup$ – Herr K. Oct 29 '20 at 21:03
  • $\begingroup$ @HerrK. Not exactly, they are considering the existence of utility functions based on non-continuous preference relations. I am slightly asking about the existence of utility functions based on non-rational preferences. More directly I am asking, given a function with a non-rational preferences relation, can it still be a utility function? $\endgroup$ – Tsangares Oct 29 '20 at 23:14
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    $\begingroup$ As my linked answer and the answer by @WalrasianAcutioneer suggest, any real-valued function can represent some rational preference. I think you might be confusing rationality of preferences with rationality of functions. A rational function is one that can be expressed as a ratio of two polynomial functions, so the word "rational" here is used as an adjective of "ratio". In contrast, a rational preference is one that embeds some intuitive sense of reasonableness. Although the same word is used, but it has different meanings in the two cases. $\endgroup$ – Herr K. Oct 30 '20 at 0:40
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I don't particularly understand the question.

Start with any function $f:X \rightarrow \mathbb{R}$.

Define $x \succeq y$ if $f(x) \geq f(y)$.

We get a rational preference over $X$.

By the way $\sin(x) + 1$ is a perfectly valid utility representation.

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