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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?

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    $\begingroup$ 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. $\endgroup$ Dec 12, 2022 at 7:10
  • $\begingroup$ 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. $\endgroup$ Dec 12, 2022 at 7:14
  • $\begingroup$ 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. $\endgroup$ Dec 12, 2022 at 15:55
  • $\begingroup$ @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.) $\endgroup$
    – VARulle
    Dec 13, 2022 at 10:21
  • $\begingroup$ 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. $\endgroup$ Dec 13, 2022 at 11:19

3 Answers 3

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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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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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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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