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Often in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of the requirements), nor do I know why this is necessary (though I imagine the why will depend on each case).

Can someone explain it to me, and if there is an explanation of the why as well, I would be grateful. Thanks!

EDIT: To give one example where the term appears, see this Wikipedia entry for utility functions, which says at one point:

In order to simplify calculations, various assumptions have been made of utility functions.

CES (constant elasticity of substitution, or isoelastic) utility
Exponential utility
Quasilinear utility
Homothetic preferences

Most utility functions used in modeling or theory are well-behaved. They are usually monotonic, quasi-concave, continuous and globally non-satiated.

I might be wrong, but I don't think "well-behaved" means monotonic, quasi-concave, continuous and globally non-satiated. What about twice differentiable?

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@Harry Gindi: well, it is not my fault that economists use this term in their papers and books. How can you expect me to know that well behaved is the same as nonsingular, smooth, or whatever? If I knew all of that I wouldn't probably be asking the question! Instead of down-voting me and telling me off for not being "formal", why don't you write an answer and explain this in a nice way? Is that how you expect this website to get more users? By down-voting legitimate questions rather than questions written by someone who already knows the answer but only wants reputation points? –  Vivi Jul 21 '10 at 0:25
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Also, according to the FAQ, "Mathematics - Stack Exchange is for people studying math at any level & professionals in related fields." If you start turning down people with less knowledge of the field this website will be exactly the same as mathoverflow, and there will be no point in having two websites that are exactly the same. –  Vivi Jul 21 '10 at 0:37
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Vivi, I don't think your question is inappropriate, but Harry is right that "well-behaved" is context-dependent. Even in the world of economics, it may have different meanings when talking about different things. If you can provide specific examples of when the term was used, you will get better answers. –  Larry Wang Jul 21 '10 at 0:39
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There is a way to avoid offending Harry Gindi?! –  Larry Wang Jul 21 '10 at 1:14
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a well-behaved function is a function which behaves just like you need :-) (it's a circular definition, I know, but the concept is ill-defined) –  mau Jul 21 '10 at 8:56
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4 Answers

up vote 16 down vote accepted

In the sciences (as opposed to in mathematics) people are often a bit vague about exactly what assumptions they are making about how "well-behaved" things are. The reason for this is that ultimately these theories are made to be put to the test, so why bother worrying about exactly which properties you're assuming when what you care about is functions coming up in real life which are probably going to satisfy all of your assumptions.

This is particularly ubiquitous in physics where it is extremely common to make heuristic assumptions about well-behavior.

Even in mathematics we do this sometimes. When people say something is true for n sufficiently large, they often won't bother writing down exactly how large is sufficiently large as long as it's clear from context how to work it out. Similarly, in an economics paper you could read through the argument and figure out exactly what assumptions they need, but it makes it easier to read to just say "well-behaved."

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In practical application of mathematical analysis, the most useful "well-behavedness" condition on a function would be Lipshitz continuity. This ensures that the variations of a function are not too wild. The most important consequence is that differential equations will have a unique solution. All sorts of modelling of natural and physical situations uses differential equations, and it is valuable to know abstractly that they would have solutions under some tame conditions. Given such a theorem, one can concentrate exclusively on finding the solutions.

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In general, we think of well-behaved functions as simpler, somehow. In any field, we might want to limit ourselves to considering only well-behaved functions in order to avoid having to deal with nasty edge cases. And in each of these domains, the community is free to choose whatever definition of 'well-behaved' makes sense for them. A quick look at the wiki link that e.James posted will show you the diversity in ideas of what it means to be well-behaved. I am no economist, so I will take for granted that the definition you put forth in your question is the one in common use.

I can see twice-differentiable as a reasonable requirement for a utility function to be 'well-behaved' is because the derivative of the utility function is marginal utility, and economists often care about the derivative of marginal utility. For example, if the second derivative of utility is negative, this means that the marginal utility has a negative derivative in other words, additional quantity of the good or service does not add utility as quickly. Also commonly referred to as diminishing returns.

If we want to be able to take the derivative even once, of course, we need the function to be continuous. You probably don't need to worry about the formal definition here. Pencil test should work fine.

The requirement for utility to be monotonic means that it is either always increasing or always decreasing. In other words, that a particular good or service is either desirable or not. If 10 widgets were good, 20 must be better. Of course, as mentioned above, maybe not that much better.

Monotonic means that it will also be quasi-concave. (Except for weird stuff like a flat function) That is, they have at most one local maximum. We would prefer that functions be quasiconcave because we wish to avoid cases like the one below. It just makes it so much easier to optimize when you only have one possible maximum to worry about.
alt text

Globally non-satiated someone else can talk about. I don't know enough to be sure I won't just be misleading you further.

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thanks... that much I understand. I know that being twice-differentiable is one of the assumptions, but what about the rest? –  Vivi Jul 21 '10 at 1:29
    
So are all of those part of being "well-behaved" in the example above? –  Vivi Jul 21 '10 at 2:18
    
@Vivi: see my edits. –  Larry Wang Jul 21 '10 at 2:31
    
economists often care about the derivative of marginal utility - Sloppy, physics-envying economists, anyway. +1, very nice, pretty picture. –  Charles Stewart Jul 22 '10 at 9:38
    
Yeah, it was a nice answer, and thank you so much for that, @Kaestur :) –  Vivi Jul 22 '10 at 10:40
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The short answer is that there is no "exact" meaning. Ideally, additional axioms are introduced to ensure that a certain function (or any mathematical object, for that matter) is "well-behaved" which, in effect, makes analysis easier. So, the meaning of "well-behaved" should be derived from those specific additional axioms.

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+1 and here's a Wikpedia article to back it up: en.wikipedia.org/wiki/Well-behaved –  e.James Jul 21 '10 at 0:47
    
@e.James: That wiki article would actually a pretty good answer –  Casebash Jul 21 '10 at 0:50
    
I find all of that so vague and confusing! Maybe that is just a consequence of the term "well-behaved" also being vague, but I find that sometimes things can be explained in a way that even non-mathematicians can understand. Once I asked an econometrics professor about this, and he said he couldn't remember, but that twice differentiability and continuity were two of the requirements. That made me think that there was a right answer, or at least for how economists use the term. –  Vivi Jul 21 '10 at 0:56
    
@Casebash: Yes, but the fundamental point would be the same as this answer, so we would just be splitting up the votes. If I had the rep level, I would edit Yaser Sulaiman's answer to put in the link, but for now a comment will have to do :) –  e.James Jul 21 '10 at 0:57
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@Vivi: there might be a right answer specific to economics. I was surprised to find that the Wikipedia article didn't even mention economics as one of the fields in which it was used. –  e.James Jul 21 '10 at 0:58
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