# Indifference curve - Does $dU = 0$ hold in higher dimensions? / Problem of integrability

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?

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).

Giskard says

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

and gives the ff link: This (2014Nov. Here'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?

• $dU$ seems ill-defined? Commented Oct 17, 2016 at 9:59
• @denesp ?
– BCLC
Commented Oct 21, 2016 at 0:05
• Okay, so its the full differential. Commented Oct 21, 2016 at 4:32
• 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). Commented Oct 21, 2016 at 9:09
• @Ubiquitous Thanks! Maybe there are some conditions where Not integrable --> Not differentiable or something?
– BCLC
Commented Oct 21, 2016 at 11:52

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 (2014Nov. Here's a 2016Nov update) seems to offer an extremely thorough walkthrough of the integrability problem in $$n$$ dimensions.

• $U$ 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. Commented Oct 21, 2016 at 6:31
• 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. Commented Oct 21, 2016 at 6:38
• @BCLC for detailed definition of the problem and conclusions please see my edit. Commented Oct 21, 2016 at 12:49
• Thanks but TL;DR, denesp. Where's non-integrability there?
– BCLC
Commented Oct 22, 2016 at 8:28
• 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." Commented Nov 15, 2022 at 21:16