<p>The single most important feature that we care about a utility function is that if a bundle <span class="math-container">$X$</span> is preferred to a bundle <span class="math-container">$Y$</span> then <span class="math-container">$U(X)&gt;U(Y)$</span> (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 <span class="math-container">$V(X)=f(U(X))$</span> with <span class="math-container">$f'&lt;0$</span>, then <span class="math-container">$V(X)&lt;V(Y)$</span> i.e. now <span class="math-container">$X$</span> gives less utility. Therefore, we see that <span class="math-container">$V$</span> cannot represent the same preferences. </p>

<p>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. </p>

<p>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. </p>

CC BY-SA 4.0
2019-06-14T17:15:42.637Z
2019-06-14T17:15:42.637Z
Regio
<p>Ordinal levels of measurement (of which ordinal utility is but one example) are often <em>defined</em> 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 <em>decreasing</em> mapping may not impact <em>some</em> calculations (where a double negative will cancel out, essentially) it will certainly impact others.</p>

CC BY-SA 4.0
2019-06-17T05:40:48.020Z
Ordinal measurements should preserve the order, but a decreasing mapping reverses it.
Regio
CC BY-SA 4.0
2019-06-17T14:23:32.877Z
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
CC BY-SA 4.0
2019-06-12T13:35:45.000Z
2019-06-12T13:35:45.000Z
Bill Clark
<p>I understand from an intuitive sense that decreasing monotonic transformations will skew the choices and ordinality.</p>

<p>But mathematically the <span class="math-container">$F'(U(x,y))$</span> 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 ?</p>

<utility><consumer-theory><preferences><choice-theory>
The mathematical proof of a monotonic utility transformation does not restrict the use of strictly decreasing monotonic functions. Why bar them?
metrics24
# The mathematical proof of a monotonic utility transformation does not restrict the use of strictly decreasing monotonic functions. Why bar them?

(Asked by: username_0 on 2019-06-12)

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 ?

**Tags:** utility, consumer-theory, preferences, choice-theory

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## 2 answers

### Answer by username_1 on 2019-06-14. Score: 1

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.


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### Answer by username_2 on 2019-06-12. Score: 0

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.

> **Comments:**
> **username_1**: Ordinal measurements should preserve the order, but a decreasing mapping reverses it.  
> **username_2**: 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.  

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