<p>Seems to me <span class="math-container">$dU = 0$</span> is true by definition unless <span class="math-container">$dU$</span> is not defined</p>
<ol>
<li>because the function <span class="math-container">$U$</span> is not differentiable in some variable.
E.g. <span class="math-container">$$U(x,y) = |x| + y$$</span></li>
<li>because there are a countably infinite number of dimensions but the partial differentials are not always absolute convergent and hence cannot be summed up. E.g.: <span class="math-container">$$U(x_1,x_2,x_3,x_4...) = x_1 - x_2 + x_3 -x_4...$$</span></li>
<li>because the dimensions have a larger than countably infinite cardinality. I have never seen this in micro but it is theoretically possible. E.g. expected utility over continuum states.</li>
</ol>
<p>Edit:
<a href="https://web.archive.org/web/20151023205733/http://people.hss.caltech.edu/%7Ekcb/Notes/Demand4-Integrability.pdf" rel="nofollow noreferrer">This</a> (2014Nov. <a href="https://web.archive.org/web/20210507010811/http://www.its.caltech.edu/%7Ekcborder/Notes/Demand4-Integrability.pdf" rel="nofollow noreferrer">Here</a>'s a 2016Nov update) seems to offer an extremely thorough walkthrough of the integrability problem in <span class="math-container">$n$</span> dimensions.</p>

CC BY-SA 3.0
2016-10-21T05:44:47.017Z
2 and 3 --> thanks! 1 --> this case is probably excluded because we are supposed to have $dU=0$ for 2D. So I guess we must assume U is differentiable in all variables. Any idea what any of this has to do with integrability?
BCLC
CC BY-SA 3.0
2016-10-21T05:50:46.187Z
3 --> could this happen in finance? Like $x_t$ is like a stock bought at time $t$? Seems like treating the same stock as a different good depending on $t$. Or maybe option to buy stock at time $t$? So, uncountably infinite options?
BCLC
CC BY-SA 3.0
2016-10-21T05:52:56.700Z
Edit to earlier comment: so something not integrable may make $dU$ undefined?
BCLC
CC BY-SA 3.0
2016-10-21T06:31:53.997Z
$U$ not being differentiable would make it undefined. E.g. $|x|$ is not differentiable w.r.t $x$ but you can calculate the integral just fine. An integrability problem may arise if you try to calculate the indifference curve from its slope, i.e. $MRS(x,y)$. The slope is the derivative of the function, integrating returns the function ($+c$) from the slope. If the function was not differentiable to begin with $MRS(x,y)$ does not exist.
Giskard
CC BY-SA 3.0
2016-10-21T06:38:02.973Z
On continuum states: The consumption bundle may be a stock, this will have different values in different world states $i$. The actual good is not the stock though, it is unit value in the different world states. As you say these may be thought of as options.
Giskard
CC BY-SA 3.0
2016-10-21T07:07:54.273Z
denesp, but how about the converse (integrability prob --> dU is not defined) ?
BCLC
CC BY-SA 3.0
2016-10-21T12:49:31.577Z
@BCLC for detailed definition of the problem and conclusions please see my edit.
Giskard
CC BY-SA 3.0
2016-10-22T08:28:00.220Z
Thanks but TL;DR, denesp. Where's non-integrability there?
BCLC
CC BY-SA 3.0
2016-10-22T14:39:53.577Z
@BCLC The second paragraph was too far down...? You can also search the document for the phrase 'integrability'.
Giskard
CC BY-SA 4.0
2022-11-14T20:20:01.953Z
Giskard where is the $dU > 0$ or $dU$ is not defined there? I see an integrability theorem...Is that what you mean? So ok if the assumptions are not justified then maybe $dU > 0$ or $dU$ is not defined you mean?
BCLC
CC BY-SA 4.0
2022-11-15T21:16:08.547Z
It seems that someone deleted my comment? Please don't do so again (at least let me know why). I am reposting it: "Let me get back to you in 5-10 years."
Giskard
CC BY-SA 4.0
2022-11-23T13:51:20.177Z
@Giskard the original comment was deleted because there were flags by multiple users flagging it as unfriendly or unkind. Now, while the comment is snarky, I do not believe the comment itself is as offensive as to violate site policy, but at the same time it is completely pointless. Comments are supposed to be used either to *constructively* critique answer, add some minor addition to an answer, or to request clarification. Comments should not be used for things like just saying "thank you" or snarky remarks. While we mods won't go out of our way and delete every thank you comment
1muflon1
CC BY-SA 4.0
2022-11-23T13:52:06.173Z
or every snarky remark if multiple people complain about a comment and the comment literally does not add anything, I do not see reason not to just delete it.
1muflon1
CC BY-SA 4.0
2022-11-23T16:57:23.643Z
It is absurd to continue a technical conversation 6 years after it ended.
Giskard
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2022-11-23T17:01:02.303Z
@1muflon1 As for multiple people flagging the comment - I have been feeling for a while that I am a poor fit for the site, and will probably stop visiting. It was a bad habit for me anyway. Thank you for taking the time to respond.
Giskard
CC BY-SA 4.0
2022-11-23T17:10:09.767Z
@Giskard just because some people flagged your comment does not mean you are not welcomed here, in any case why do you respond to comments like that at all? Just ignore it if you think its not worth to follow up on old question
1muflon1
CC BY-SA 4.0
2022-11-25T19:57:05.730Z
@Giskard 'after it ended' - did it end?
BCLC
CC BY-SA 4.0
2022-11-28T18:15:31.760Z
@1muflon1 Thanks again. I don't think I was unwelcome to most users; I simply see no benefit to coming any longer, it was just the result of a habit. The failures of this site reflect what I consider to be the failures of the profession. It is taxing my mental health to see these, and I don't have to, since I have already left the profession. Thus my decision to leave this SE. Thanks for moderating these past few years. I hope you find joy in using the site, as otherwise the amount of hours put in is simply mental.
Giskard
CC BY-SA 4.0
2022-11-28T20:17:47.797Z
@Giskard sorry to hear that, you will be missed. Of course, health comes first. If you would ever change your mind you will always be welcome here though. I wish you all well I always enjoyed discussing things with you even at times when I was disagreeing with you. Sorry about your breakup with the econ profession. I hope you will have good luck and success in whatever you are doing now or will do next
1muflon1
CC BY-SA 4.0
2016-10-21T04:40:25.837Z
2022-11-15T07:22:40.207Z
2022-11-15T07:22:40.207Z
Giskard
<p>In two dimensions, we have on an indifference curve that <span class="math-container">$dU=0$</span>.</p>
<p>Does this apply to indifference objects in higher dimensions?</p>
<p>I was thinking that if <span class="math-container">$dU &gt; 0$</span>, then one is moving to a higher indifference curve/object</p>
<p>My Macroeconomics professor, in the first week of class, gave a microeconomics review. I asked if <span class="math-container">$dU = 0$</span> in higher dimensions. He said he would answer the question next time, and during the next class, he said it may not hold true in higher dimensions because of the problem of integrability.</p>
<p>I asked another economics professor in the department who said that <span class="math-container">$dU=0$</span> holds by definition of indifference curves/objects.</p>
<p>What is this problem of integrability, and what does it have to do with indifference curves?</p>
<p>Is it <a href="https://books.google.com/books?id=Z6Z8AgAAQBAJ&amp;pg=PA192&amp;lpg=PA192&amp;dq=total%20differential%20utility%20function%20problem%20integrability&amp;source=bl&amp;ots=ANphWOglG5&amp;sig=WVDJaYCE3r2I7lYK2qbi9TvYR8s&amp;hl=en&amp;sa=X&amp;ved=0ahUKEwjirJuMzurPAhUHKo8KHaQlDhsQ6AEIKDAD#v=onepage&amp;q=total%20differential%20utility%20function%20problem%20integrability&amp;f=false" rel="nofollow noreferrer">this one</a>?</p>
<p>I've taken 2 courses of real analysis, 2 on probability theory and 2 on stochastic calculus (and 4 on statistics), but the mathematics I've encountered in economics is only up to basic linear algebra and separable ordinary differential equations (Actually the linear algebra's mainly just matrices. I hardly recall learning concepts like 'kernel' or 'eigenvalue' in economics, though I have learned uses for both in finance).</p>
<hr />
<p>Giskard says</p>
<blockquote>
<p>Seems to me <span class="math-container">$dU = 0$</span> is true by definition unless <span class="math-container">$dU$</span> is not defined</p>
</blockquote>
<p>and gives the ff link: <a href="https://web.archive.org/web/20151023205733/http://people.hss.caltech.edu/%7Ekcb/Notes/Demand4-Integrability.pdf" rel="nofollow noreferrer">This</a> (2014Nov. <a href="https://web.archive.org/web/20210507010811/http://www.its.caltech.edu/%7Ekcborder/Notes/Demand4-Integrability.pdf" rel="nofollow noreferrer">Here</a>'s a 2016Nov update)</p>
<p>Where is the <span class="math-container">$dU &gt; 0$</span> or <span class="math-container">$dU$</span> is not defined there? I see an integrability theorem...if this is what is meant, then the assumptions are not justified then maybe <span class="math-container">$dU &gt; 0$</span> or <span class="math-container">$dU$</span> is not defined?</p>
<hr />
<p>BUT ANYWAY the bottom line is yeah there's no reason a defined <span class="math-container">$dU$</span> should not be <span class="math-container">$0$</span> right? It's an INDIFFERENCE curve??!?!! If your <span class="math-container">$dU &gt; 0$</span>, then you area already on a higher curve or something right?</p>

CC BY-SA 3.0
2016-10-17T09:59:32.200Z
$dU$ seems ill-defined?
Giskard
CC BY-SA 3.0
2016-10-21T00:05:10.563Z
@denesp [?](https://wikimedia.org/api/rest_v1/media/math/render/svg/60c3df75aab218f7c26647cb0418c300b19aa0f4)
BCLC
CC BY-SA 3.0
2016-10-21T04:32:55.583Z
Okay, so its the full differential.
Giskard
CC BY-SA 3.0
2016-10-21T09:09:41.403Z
It's not clear what your professor means by the problem of integrability. Usually, when we talk about "the integrability problem" we mean the problem of recovering preferences from a demand function (MWG have a section on this, starting on p75). But this is not obviously related to the question of whether utility is differentiable (I would agree with you that, if $U$ is differentiable, then $dU=0$ by definition of an indifference hypersurface).
Ubiquitous
CC BY-SA 3.0
2016-10-21T11:52:23.377Z
@Ubiquitous Thanks! Maybe there are some conditions where Not integrable --> Not differentiable or something?
BCLC
<macroeconomics><microeconomics><mathematical-economics><utility><measuring-utility>
Indifference curve - Does $dU = 0$ hold in higher dimensions? / Problem of integrability
BCLC
# Indifference curve - Does $dU = 0$ hold in higher dimensions? / Problem of integrability

(Asked by: username_0 on 2016-10-17)

In two dimensions, we have on an indifference curve that $dU=0$.

Does this apply to indifference objects in higher dimensions?

I was thinking that if $dU > 0$, then one is moving to a higher indifference curve/object

My Macroeconomics professor, in the first week of class, gave a microeconomics review. I asked if $dU = 0$ in higher dimensions. He said he would answer the question next time, and during the next class, he said it may not hold true in higher dimensions because of the problem of integrability.

I asked another economics professor in the department who said that $dU=0$ holds by definition of indifference curves/objects.

What is this problem of integrability, and what does it have to do with indifference curves?

Is it [this one](https://books.google.com/books?id=Z6Z8AgAAQBAJ&pg=PA192&lpg=PA192&dq=total%20differential%20utility%20function%20problem%20integrability&source=bl&ots=ANphWOglG5&sig=WVDJaYCE3r2I7lYK2qbi9TvYR8s&hl=en&sa=X&ved=0ahUKEwjirJuMzurPAhUHKo8KHaQlDhsQ6AEIKDAD#v=onepage&q=total%20differential%20utility%20function%20problem%20integrability&f=false)?

I've taken 2 courses of real analysis, 2 on probability theory and 2 on stochastic calculus (and 4 on statistics), but the mathematics I've encountered in economics is only up to basic linear algebra and separable ordinary differential equations (Actually the linear algebra's mainly just matrices. I hardly recall learning concepts like 'kernel' or 'eigenvalue' in economics, though I have learned uses for both in finance).

---

username_1 says

> Seems to me $dU = 0$ is true by definition unless $dU$ is not defined

and gives the ff link: [This](https://web.archive.org/web/20151023205733/http://people.hss.caltech.edu/%7Ekcb/Notes/Demand4-Integrability.pdf) (2014Nov. [Here](https://web.archive.org/web/20210507010811/http://www.its.caltech.edu/%7Ekcborder/Notes/Demand4-Integrability.pdf)'s a 2016Nov update)

Where is the $dU > 0$ or $dU$ is not defined there? I see an integrability theorem...if this is what is meant, then the assumptions are not justified then maybe $dU > 0$ or $dU$ is not defined?

---

BUT ANYWAY the bottom line is yeah there's no reason a defined $dU$ should not be $0$ right? It's an INDIFFERENCE curve??!?!! If your $dU > 0$, then you area already on a higher curve or something right?

**Tags:** macroeconomics, microeconomics, mathematical-economics, utility, measuring-utility

> **Comments:**
> **username_1**: $dU$ seems ill-defined?  
> **username_0**: @denesp [?](https://wikimedia.org/api/rest_v1/media/math/render/svg/60c3df75aab218f7c26647cb0418c300b19aa0f4)  
> **username_1**: Okay, so its the full differential.  
> **username_2**: It's not clear what your professor means by the problem of integrability. Usually, when we talk about "the integrability problem" we mean the problem of recovering preferences from a demand function (MWG have a section on this, starting on p75). But this is not obviously related to the question of whether utility is differentiable (I would agree with you that, if $U$ is differentiable, then $dU=0$ by definition of an indifference hypersurface).  
> **username_0**: @username_2 Thanks! Maybe there are some conditions where Not integrable --> Not differentiable or something?  

---

## 1 answers

### Answer by username_1 on 2016-10-21. Score: 7

Seems to me $dU = 0$ is true by definition unless $dU$ is not defined

1. because the function $U$ is not differentiable in some variable.
   E.g. $$U(x,y) = |x| + y$$
2. because there are a countably infinite number of dimensions but the partial differentials are not always absolute convergent and hence cannot be summed up. E.g.: $$U(x\_1,x\_2,x\_3,x\_4...) = x\_1 - x\_2 + x\_3 -x\_4...$$
3. because the dimensions have a larger than countably infinite cardinality. I have never seen this in micro but it is theoretically possible. E.g. expected utility over continuum states.

Edit:
[This](https://web.archive.org/web/20151023205733/http://people.hss.caltech.edu/%7Ekcb/Notes/Demand4-Integrability.pdf) (2014Nov. [Here](https://web.archive.org/web/20210507010811/http://www.its.caltech.edu/%7Ekcborder/Notes/Demand4-Integrability.pdf)'s a 2016Nov update) seems to offer an extremely thorough walkthrough of the integrability problem in $n$ dimensions.

> **Comments:**
> **username_0**: 2 and 3 --> thanks! 1 --> this case is probably excluded because we are supposed to have $dU=0$ for 2D. So I guess we must assume U is differentiable in all variables. Any idea what any of this has to do with integrability?  
> **username_0**: 3 --> could this happen in finance? Like $x_t$ is like a stock bought at time $t$? Seems like treating the same stock as a different good depending on $t$. Or maybe option to buy stock at time $t$? So, uncountably infinite options?  
> **username_0**: Edit to earlier comment: so something not integrable may make $dU$ undefined?  
> **username_1**: $U$ not being differentiable would make it undefined. E.g. $|x|$ is not differentiable w.r.t $x$ but you can calculate the integral just fine. An integrability problem may arise if you try to calculate the indifference curve from its slope, i.e. $MRS(x,y)$. The slope is the derivative of the function, integrating returns the function ($+c$) from the slope. If the function was not differentiable to begin with $MRS(x,y)$ does not exist.  
> **username_1**: On continuum states: The consumption bundle may be a stock, this will have different values in different world states $i$. The actual good is not the stock though, it is unit value in the different world states. As you say these may be thought of as options.  
> **username_0**: denesp, but how about the converse (integrability prob --> dU is not defined) ?  
> **username_1**: @username_0 for detailed definition of the problem and conclusions please see my edit.  
> **username_0**: Thanks but TL;DR, denesp. Where's non-integrability there?  
> **username_1**: @username_0 The second paragraph was too far down...? You can also search the document for the phrase 'integrability'.  
> **username_0**: username_1 where is the $dU > 0$ or $dU$ is not defined there? I see an integrability theorem...Is that what you mean? So ok if the assumptions are not justified then maybe $dU > 0$ or $dU$ is not defined you mean?  
> **username_1**: It seems that someone deleted my comment? Please don't do so again (at least let me know why). I am reposting it: "Let me get back to you in 5-10 years."  
> **username_3**: @username_1 the original comment was deleted because there were flags by multiple users flagging it as unfriendly or unkind. Now, while the comment is snarky, I do not believe the comment itself is as offensive as to violate site policy, but at the same time it is completely pointless. Comments are supposed to be used either to *constructively* critique answer, add some minor addition to an answer, or to request clarification. Comments should not be used for things like just saying "thank you" or snarky remarks. While we mods won't go out of our way and delete every thank you comment  
> **username_3**: or every snarky remark if multiple people complain about a comment and the comment literally does not add anything, I do not see reason not to just delete it.  
> **username_1**: It is absurd to continue a technical conversation 6 years after it ended.  
> **username_1**: @username_3 As for multiple people flagging the comment - I have been feeling for a while that I am a poor fit for the site, and will probably stop visiting. It was a bad habit for me anyway. Thank you for taking the time to respond.  
> **username_3**: @username_1 just because some people flagged your comment does not mean you are not welcomed here, in any case why do you respond to comments like that at all? Just ignore it if you think its not worth to follow up on old question  
> **username_0**: @username_1 'after it ended' - did it end?  
> **username_1**: @username_3 Thanks again. I don't think I was unwelcome to most users; I simply see no benefit to coming any longer, it was just the result of a habit. The failures of this site reflect what I consider to be the failures of the profession. It is taxing my mental health to see these, and I don't have to, since I have already left the profession. Thus my decision to leave this SE. Thanks for moderating these past few years. I hope you find joy in using the site, as otherwise the amount of hours put in is simply mental.  
> **username_3**: @username_1 sorry to hear that, you will be missed. Of course, health comes first. If you would ever change your mind you will always be welcome here though. I wish you all well I always enjoyed discussing things with you even at times when I was disagreeing with you. Sorry about your breakup with the econ profession. I hope you will have good luck and success in whatever you are doing now or will do next  

---

