<p>I agree with you that the answer should be: <span class="math-container">$(D)$</span> <em>All the above</em>.</p>
<p>All the given tranformations in the exercise are monotonic transformations of the utility fuction <span class="math-container">$u(x,y)=\sqrt (xy)$</span>. <span class="math-container">$(A)$</span> and <span class="math-container">$(B)$</span> are respectively the fourth power and the square of your utility function.</p>
<p>I think that there is sometimes a misunderstanding about this matter of monotonic transformation, as sometimes I read that odd  power are monotonic transformations, seeming to imply that even powers are not.
But this is not necessarily true for utility functions, both odd and even powers can be in this case monotonic transformations.</p>
<p>The misunderstanding comes from the fact that even powers are not monotonic tranformation on <span class="math-container">$\mathbb{R}$</span>, of course, but they are if we restrict to positive values: actually, if <span class="math-container">$f(x)$</span> is an even power,  for <span class="math-container">$x&gt;0$</span> we have <span class="math-container">$f'(x) &gt;0$</span>.</p>
<p>A problem could exist if we have a utility function with negative values, in this case <span class="math-container">$f'(x) &lt;0$</span>, and the order of preferences changes, is inverted.</p>
<p>But in your case you have a square root, and a square root can't be negative.</p>

CC BY-SA 4.0
2022-12-12T08:16:41.703Z
2022-12-12T08:16:41.703Z
BakerStreet
<p>The answer depends on the context. If this is meant to be an ordinal utility function, (D) should be correct, as all transformations are positive in the first quadrant. But if this is meant to be a von-Neumann-Morgenstern utility function, then only positive affine linear transformations lead to equivalent utility functions and (C) is correct.</p>

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2022-12-12T11:29:20.863Z
2022-12-13T10:09:15.793Z
2022-12-13T10:09:15.793Z
VARulle
<p>C) is the correct answer if you work with cardinal preferences, D) is the  correct answer if you work with ordinal preferences.</p>
<p>I think that your teacher assumed that the preferences are cardinal and not made a mistake but I agree the question should be better worded.</p>

CC BY-SA 4.0
2022-12-12T19:59:53.947Z
2022-12-12T21:10:01.877Z
2022-12-12T21:10:01.877Z
csilvia
<p>I'm confused on how to determine if utility functions represent the same preferences. The question is as follows:</p>
<p>Which of these utility functions represent the same preferences as <span class="math-container">$u(x,y) = \sqrt(xy)$</span>?</p>
<p>(A) <span class="math-container">$u(x,y) = x^2y^2$</span>
(B) <span class="math-container">$u(x,y) = xy$</span>
(C) <span class="math-container">$u(x,y) = 10\sqrt(xy)$</span>
(D) All of the above</p>
<p>Why would the answer be (C) in this case? Why isn't D the answer?</p>

CC BY-SA 4.0
2022-12-12T07:10:33.067Z
What are the domains of the functions? We cannot have negative $xy$ in the original function and in c, in the rest we can have. But nothing's stopping us from restricting the domains of the other functions to non-negative $xy$ too. So the question needs to be clearer imo.
Ishan Kashyap Hazarika
CC BY-SA 4.0
2022-12-12T07:14:39.923Z
Another possibility that comes to mind is: are these utilities ordinal or interval? For example, if these are von Neumann- Morgenstern utilities, then they are unique only upto an affine transformation. In that case also, only c is the correct answer.
Ishan Kashyap Hazarika
CC BY-SA 4.0
2022-12-12T15:55:42.713Z
I think, however, that the utility function of the exercise isn’ t a Morgenstern-von Neumann utility function. That should be linear, that is it should have the ‘expected utility form’ $U(\sum \alpha_hL_k)=\sum \alpha_kU(L_k)$ ( see for instance Mas Colell, *Microeconomic Theory*  or Varian, *Microeconomic Analysis*.)  The utility function in the exercise is actually a Cobb-Douglas with $\alpha=1/2$, and hasn’t the expected utility property, unless you take a logarithmic transformation of it.
BakerStreet
CC BY-SA 4.0
2022-12-13T10:21:00.930Z
@BakerStreet, a von-Neumann-Morgenstern utility function has to be linear in *probabilities* if it takes a lottery as its argument, but if the $(x,y)$ are consumption bundles, then $u(x,y)=(xy)^{1/2}$ defines a von-Neumann-Morgenstern utility function on lotteries over consumption bundles by the usual extension to expected utilities. (MCWG would probably call $u(x,y)$ the associated Bernoulli utility function, but the terminology for this varies.)
VARulle
CC BY-SA 4.0
2022-12-13T11:19:26.403Z
Maybe you are right, Varian says explicitly that a Cobb-Douglas isn't a expected utlitiy fnction, unless you take a logaritmic transformation, probably it is a matter of variable terminology. I will read what say Varian and Mas Colell again, thank you for the observation.
BakerStreet
<microeconomics><utility>
Confused on finding same preferences
neoncat
# Confused on finding same preferences

(Asked by: username_0 on 2022-12-12)

I'm confused on how to determine if utility functions represent the same preferences. The question is as follows:

Which of these utility functions represent the same preferences as $u(x,y) = \sqrt(xy)$?

(A) $u(x,y) = x^2y^2$
(B) $u(x,y) = xy$
(C) $u(x,y) = 10\sqrt(xy)$
(D) All of the above

Why would the answer be (C) in this case? Why isn't D the answer?

**Tags:** microeconomics, utility

> **Comments:**
> **username_1**: What are the domains of the functions? We cannot have negative $xy$ in the original function and in c, in the rest we can have. But nothing's stopping us from restricting the domains of the other functions to non-negative $xy$ too. So the question needs to be clearer imo.  
> **username_1**: Another possibility that comes to mind is: are these utilities ordinal or interval? For example, if these are von Neumann- Morgenstern utilities, then they are unique only upto an affine transformation. In that case also, only c is the correct answer.  
> **username_2**: I think, however, that the utility function of the exercise isn’ t a Morgenstern-von Neumann utility function. That should be linear, that is it should have the ‘expected utility form’ $U(\sum \alpha_hL_k)=\sum \alpha_kU(L_k)$ ( see for instance Mas Colell, *Microeconomic Theory*  or Varian, *Microeconomic Analysis*.)  The utility function in the exercise is actually a Cobb-Douglas with $\alpha=1/2$, and hasn’t the expected utility property, unless you take a logarithmic transformation of it.  
> **username_3**: @username_2, a von-Neumann-Morgenstern utility function has to be linear in *probabilities* if it takes a lottery as its argument, but if the $(x,y)$ are consumption bundles, then $u(x,y)=(xy)^{1/2}$ defines a von-Neumann-Morgenstern utility function on lotteries over consumption bundles by the usual extension to expected utilities. (MCWG would probably call $u(x,y)$ the associated Bernoulli utility function, but the terminology for this varies.)  
> **username_2**: Maybe you are right, Varian says explicitly that a Cobb-Douglas isn't a expected utlitiy fnction, unless you take a logaritmic transformation, probably it is a matter of variable terminology. I will read what say Varian and Mas Colell again, thank you for the observation.  

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## 3 answers

### Answer by username_2 on 2022-12-12. Score: 3

I agree with you that the answer should be: $(D)$ *All the above*.

All the given tranformations in the exercise are monotonic transformations of the utility fuction $u(x,y)=\sqrt (xy)$. $(A)$ and $(B)$ are respectively the fourth power and the square of your utility function.

I think that there is sometimes a misunderstanding about this matter of monotonic transformation, as sometimes I read that odd power are monotonic transformations, seeming to imply that even powers are not.
But this is not necessarily true for utility functions, both odd and even powers can be in this case monotonic transformations.

The misunderstanding comes from the fact that even powers are not monotonic tranformation on $\mathbb{R}$, of course, but they are if we restrict to positive values: actually, if $f(x)$ is an even power, for $x>0$ we have $f'(x) >0$.

A problem could exist if we have a utility function with negative values, in this case $f'(x) <0$, and the order of preferences changes, is inverted.

But in your case you have a square root, and a square root can't be negative.


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### Answer by username_3 on 2022-12-12. Score: 2

The answer depends on the context. If this is meant to be an ordinal utility function, (D) should be correct, as all transformations are positive in the first quadrant. But if this is meant to be a von-Neumann-Morgenstern utility function, then only positive affine linear transformations lead to equivalent utility functions and (C) is correct.


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### Answer by username_4 on 2022-12-12. Score: 0

C) is the correct answer if you work with cardinal preferences, D) is the correct answer if you work with ordinal preferences.

I think that your teacher assumed that the preferences are cardinal and not made a mistake but I agree the question should be better worded.


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