# The mathematical proof of a monotonic utility transformation does not restrict the use of strictly decreasing monotonic functions. Why bar them?

I understand from an intuitive sense that decreasing monotonic transformations will skew the choices and ordinality.

But mathematically the $$F'(U(x,y))$$ just cancels out each other out in numerator and denominator when we take the MRS and thus gives us the same MRS as our old utility function. But this cancelling out can also happen with decreasing monotonic transformations. Then why bar them from use ?

## 2 Answers

The single most important feature that we care about a utility function is that if a bundle $$X$$ is preferred to a bundle $$Y$$ then $$U(X)>U(Y)$$ (with weak inequalities whenever appropriate). This is how we define a utility function. You can immediately see that if a function has that property, any monotonic increasing transformation of it will also have that property. However, if the transformation is decreasing, call it $$V(X)=f(U(X))$$ with $$f'<0$$, then $$V(X) i.e. now $$X$$ gives less utility. Therefore, we see that $$V$$ cannot represent the same preferences.

What about the MRS invariance property? Well, once you understand that utility representations are not unique, (or unique up to monotonic increasing transformations), you can ask what is constant across all functions that represent the same preferences? The answer is: they have the same MRS.

You can clearly see that it is necessary for two functions to have the same MRS if they represent the same preferences, but it is not sufficient. As you pointed out, decreasing transformations also have the same MRS. Note however, that the interpretations of the MRS slightly changes when considering monotonic decreasing transformations.

Ordinal levels of measurement (of which ordinal utility is but one example) are often defined in terms of their properties with respect to monotonic transformations. They show up in voting theory, economics, nonparametric statistics, and a variety of other places. In short, they're used for a lot more than just the Marginal Rate of Substitution. So while using a monotone decreasing mapping may not impact some calculations (where a double negative will cancel out, essentially) it will certainly impact others.

• Ordinal measurements should preserve the order, but a decreasing mapping reverses it. – Regio Jun 17 '19 at 5:40
• Yes, but the OP was asking about their use in a specific calculation in which there are two reversals, which end up cancelling each other out. I’m just trying to clarify why the rule about order-preserving transformations exists — not all calculations involving ordinal data will have a double reversal. We’re both saying the same thing; we’re just explaining it differently. – Bill Clark Jun 17 '19 at 14:23