<p>If you are using the definition of indifference curve on <a href="https://en.wikipedia.org/wiki/Indifference_curve" rel="nofollow noreferrer">wikipedia</a> 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 <span class="math-container">$x+y=const$</span> which is of finite length (in the positive quadrant). </p>

<p>A necessary criteria for infinite length would be that having <span class="math-container">$0$</span> 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.</p>

<p><strong>Edit in response to comment:</strong> Indifference curves come in families, one can consider the family of indifference curves <span class="math-container">$x\cdot y = const.$</span>, they satisfy the definition, are of infinite length and if either <span class="math-container">$x$</span> or <span class="math-container">$y$</span> is equal to zero, then the utility is equal to <span class="math-container">$0$</span> as well.</p>

CC BY-SA 4.0
2019-12-24T04:05:19.623Z
But then the indifference curve at $u = 0$ would just be a point, not an infinitely long curve?
Art
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2019-12-24T11:56:43.610Z
@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
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2019-12-20T15:03:27.123Z
2019-12-22T09:30:17.140Z
2019-12-22T09:30:17.140Z
quarague
<p>Indifference curves are often of infinite length.</p>

<p>Is this implied by monotonicity or non-satiation?</p>

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

<hr>

<p>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?</p>

CC BY-SA 4.0
2019-12-19T02:23:17.710Z
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
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2019-12-19T02:41:51.393Z
@Art How to prove the general case?
High GPA
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2019-12-19T02:57:02.150Z
The point is you should have an idea first which property is required... then we can discuss proving :)
Art
<microeconomics><preferences>
Why are indifference curves (often) of infinite length?
High GPA
# Why are indifference curves (often) of infinite length?

(Asked by: username_0 on 2019-12-18)

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?

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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?

**Tags:** microeconomics, preferences

> **Comments:**
> **username_1**: 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.  
> **username_0**: @username_1 How to prove the general case?  
> **username_1**: The point is you should have an idea first which property is required... then we can discuss proving :)  

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## 1 answers

### Answer by username_2 on 2019-12-20. Score: 3 | Accepted answer

If you are using the definition of indifference curve on [wikipedia](https://en.wikipedia.org/wiki/Indifference_curve) 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.

> **Comments:**
> **username_1**: But then the indifference curve at $u = 0$ would just be a point, not an infinitely long curve?  
> **username_2**: @username_1 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.  

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