Based on my economics book, monotonic transformations for a utility function can look something like this:

$f(u) = u + 17 $ or even like this: $f(u) = u^3$

That being said what it purpose in the case of the utility function, as far as I know, the number of the utility doesn't matter to us (the only thing that matters is if it's bigger or smaller than the utility of another bundle)?

The book explains it in the following way:

monotonic transformation is a way of transforming one set of numbers into another set of numbers in a way that preserves the order of the numbers.

Though it the order stays the same why would we do it in the first place?


Utility is subjective, and as you wrote, "it doesn't matter to us", but only to the person who orders her choices. We are not able to quantify a subjective concept like the utility of another person, but each person can do this for herself. This is the reason why utility is ordinal, which means that only the order of the numbers matters, and not the way we are measuring these numbers: if 10 > 3.123456, then the first bundle is preferred to the second one, by the person who orders her choices. And this ordering does not change if you change the units of measurement, and measure utility with square roots, exp, logs, (positive) power functions, etc. You can try to measure this in dollars if you want, you will never be able to measure utility.


Certain calculations using utility require utility functions to have some particular properties. The point of bringing up the invariance of preferences represented by different utility functions that are just monotonic transformations of each other is to show that these utility-based calculations are not entirely dependent on the utility functions that you happen to choose. The same results should at least hold up to order-preserving transformations.


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