<p>You can make literally any set into an indifference curve of a &quot;well-behaved&quot; (complete, transitive, reflexive) preference relation.</p>
<p>Let <span class="math-container">$H \subset \mathbb{R}^n$</span>.</p>
<p>Assume that the preference relation <span class="math-container">$\preceq$</span> is such that <br>
<em>(i)</em> for all <span class="math-container">$x,y \in H$</span> we have <span class="math-container">$x \sim y$</span> <br>
<em>(ii)</em> for all <span class="math-container">$x \in H$</span>, <span class="math-container">$y \notin H$</span>, we have <span class="math-container">$x \not\sim y$</span> (meaning <span class="math-container">$x \succ y$</span> or <span class="math-container">$x \prec y$</span>, the same for all <span class="math-container">$y \notin H$</span>). <br>
<em>(iii)</em> <span class="math-container">$\preceq$</span> is well-behaved on <span class="math-container">$\mathbb{R}^n \setminus H$</span>. <br>
A simple way to guarantee <em>(iii)</em> is to assume for all <span class="math-container">$x,y \notin H$</span> we have <span class="math-container">$x \sim y$</span>.</p>
<p>Then <span class="math-container">$H$</span> (whatever shape it may have) is an indifference curve of <span class="math-container">$\preceq$</span>.</p>
<p>Such a <span class="math-container">$\preceq$</span> may also be represented by a utility function <span class="math-container">$u$</span>: <br>
<span class="math-container">$u(x) = 0, \ \forall x \in H$</span> <br>
<span class="math-container">$u(x) = 1, \ \forall x \notin H$</span>.</p>
<p>Note that this <span class="math-container">$u$</span> is not continuous, but you did not require that. For closed sets <span class="math-container">$H$</span> it can be made continuous, by setting <br>
<span class="math-container">$u(x) = \text{distance}(x, H)$</span>.</p>

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2021-06-22T23:03:40.520Z
Right, so if we have a consumption bundle of order $2$ (that is, two different goods) and we represent it in the 2D plane, then [this](https://imgur.com/a/6EGkDtn) can be an example of an indifference curve as well right? (If yes, then do such cases exist in reality or we tend to work with curves with more distinct properties?)
not tdm's twin
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2021-06-22T23:34:07.107Z
Well, since 2D is $\mathbb{R}^2$ and your curve is a subset of $\mathbb{R}^2$, my answer applies.
Giskard
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2021-06-22T23:34:43.833Z
"*do such cases exist in reality*" I don't think anyone has ever observed an indifference curve in a non-laboratory setting, but you can post this as a new question.
Giskard
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2021-06-22T22:16:58.210Z
2021-06-23T07:25:11.417Z
2021-06-23T07:25:11.417Z
Giskard
<p>Does the indifference set have to be in the form of a curve, or of a form that is well-known? If it is not necessary to be a curve, how would the set look like? Can I get some examples?</p>
<p>If we follow the usual definition of the indifference curve, I don't see any reason why it can not be a function that's not a curve. The Wikipedia article does not have a particular description of the function; it just gives a <a href="https://en.wikipedia.org/wiki/Indifference_curve#Map_and_properties" rel="nofollow noreferrer">list of the possible forms</a>.</p>

CC BY-SA 4.0
2021-06-22T21:24:39.650Z
@Giskard It does not, because it assumes convexity. I am trying to figure out if an indifference set can have a form that is not well-known. I am not assuming monotonicity, or convexity. Let's assume the axioms and nothing more.
not tdm's twin
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2021-06-22T22:12:00.213Z
What are "the axioms"?
Giskard
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2021-06-22T22:15:17.793Z
@Giskard The completeness axiom, the reflexivity axiom and the transitive axiom.
not tdm's twin
<microeconomics>
Can the indifference curve (set) not be a curve at all?
not tdm's twin
# Can the indifference curve (set) not be a curve at all?

(Asked by: username_0 on 2021-06-22)

Does the indifference set have to be in the form of a curve, or of a form that is well-known? If it is not necessary to be a curve, how would the set look like? Can I get some examples?

If we follow the usual definition of the indifference curve, I don't see any reason why it can not be a function that's not a curve. The Wikipedia article does not have a particular description of the function; it just gives a [list of the possible forms](https://en.wikipedia.org/wiki/Indifference_curve#Map_and_properties).

**Tags:** microeconomics

> **Comments:**
> **username_0**: @username_1 It does not, because it assumes convexity. I am trying to figure out if an indifference set can have a form that is not well-known. I am not assuming monotonicity, or convexity. Let's assume the axioms and nothing more.  
> **username_1**: What are "the axioms"?  
> **username_0**: @username_1 The completeness axiom, the reflexivity axiom and the transitive axiom.  

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## 1 answers

### Answer by username_1 on 2021-06-22. Score: 2 | Accepted answer

You can make literally any set into an indifference curve of a "well-behaved" (complete, transitive, reflexive) preference relation.

Let $H \subset \mathbb{R}^n$.

Assume that the preference relation $\preceq$ is such that   
*(i)* for all $x,y \in H$ we have $x \sim y$   
*(ii)* for all $x \in H$, $y \notin H$, we have $x \not\sim y$ (meaning $x \succ y$ or $x \prec y$, the same for all $y \notin H$).   
*(iii)* $\preceq$ is well-behaved on $\mathbb{R}^n \setminus H$.   
A simple way to guarantee *(iii)* is to assume for all $x,y \notin H$ we have $x \sim y$.

Then $H$ (whatever shape it may have) is an indifference curve of $\preceq$.

Such a $\preceq$ may also be represented by a utility function $u$:   
$u(x) = 0, \ \forall x \in H$   
$u(x) = 1, \ \forall x \notin H$.

Note that this $u$ is not continuous, but you did not require that. For closed sets $H$ it can be made continuous, by setting   
$u(x) = \text{distance}(x, H)$.

> **Comments:**
> **username_0**: Right, so if we have a consumption bundle of order $2$ (that is, two different goods) and we represent it in the 2D plane, then [this](https://imgur.com/a/6EGkDtn) can be an example of an indifference curve as well right? (If yes, then do such cases exist in reality or we tend to work with curves with more distinct properties?)  
> **username_1**: Well, since 2D is $\mathbb{R}^2$ and your curve is a subset of $\mathbb{R}^2$, my answer applies.  
> **username_1**: "*do such cases exist in reality*" I don't think anyone has ever observed an indifference curve in a non-laboratory setting, but you can post this as a new question.  

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