# Why are indifference curves (often) of infinite length?

Indifference curves are often of infinite length.

Is this implied by monotonicity or non-satiation?

If not, what is/are some condition(s) that are sufficient for indifference curves to have infinite length?

More generally, indifference curves are almost always manifolds without boundary. Which means that the curves don't have endpoints. What property of the utility function guarentees this?

• You could come up with utility function whose indifference curve is not infinite (e.g. $u(x, y) = x^2 + y^2$) and see what property it violates. – Art Dec 19 '19 at 2:23
• @Art How to prove the general case? – High GPA Dec 19 '19 at 2:41
• The point is you should have an idea first which property is required... then we can discuss proving :) – Art Dec 19 '19 at 2:57

If you are using the definition of indifference curve on wikipedia then they can be of finite or infinite length. For example if you are completely indifferent between the two goods, the indiffernce curve would be of the form $$x+y=const$$ which is of finite length (in the positive quadrant).
A necessary criteria for infinite length would be that having $$0$$ of one good provides you with zero utility, so that any combination of a little bit of both has a higher utility than an arbitrary quantity of one good and zero of the other good.
Edit in response to comment: Indifference curves come in families, one can consider the family of indifference curves $$x\cdot y = const.$$, they satisfy the definition, are of infinite length and if either $$x$$ or $$y$$ is equal to zero, then the utility is equal to $$0$$ as well.
• "A necessary criteria for infinite length would be that having $0$ of one good provides you with zero utility" -- I'm not sure if this is correct. Consider $U(x,y)=xy+1$ -- the indifference curves are of infinite length but it is not true that "having $0$ of one good provides you with zero utility". – Kenny LJ Dec 21 '19 at 3:31
• But then the indifference curve at $u = 0$ would just be a point, not an infinitely long curve? – Art Dec 24 '19 at 4:05
• @Art No, the $0$ utility curve would correspond to all allocations where you have $0$ of one good, and some quantity of the other. So it is an infinitely long curve as well. It is just not smooth because it has a kink at the point where $x$ and $y$ both zero. – quarague Dec 24 '19 at 11:56