<p>No. Actually, standard preferences restricted to a budget line will not have this property.</p>
<p>For concreteness, take the preferences on <span class="math-container">$\mathbb{R}^2_{+}$</span> defined by the utility function given by <span class="math-container">$u(x_1,x_2)=\sqrt{x_1}+\sqrt{x_2}$</span>. Clearly, these preferences have all the desired properties. Now let <span class="math-container">$$X=\{(x_1,x_2)\in \mathbb{R}^2_{+}\mid x_1+x_2=1\}.$$</span>
Then the indifference curve corresponding to <span class="math-container">$(1,0)$</span> when restricted to <span class="math-container">$X$</span> is <span class="math-container">$\{(1,0),(0,1)\}$</span>. Indeed, the preferences are strictly convex, and any other point on the line must be, as a proper convex combination, strictly better and, therefore, not indifferent.</p>

CC BY-SA 4.0
2023-07-08T09:21:13.427Z
Thanks a lot for this counter example ! It answers perfectly to my question.
Peter
CC BY-SA 4.0
2023-07-08T15:31:44.853Z
May I ask if you are aware of additional conditions that would imply this connectedness (apart from convexity of indifference sets obviously) ?
Peter
CC BY-SA 4.0
2023-07-08T16:15:29.423Z
I don't have a proof, but I think that $X$ contains with every points all larger points might work. Note that monotonicity has absolutely no bite on the $X$ in my example, no point there is comparable according to size.
Michael Greinecker
CC BY-SA 4.0
2023-07-08T17:27:21.297Z
By “X contains with every points all larger points”, you mean “if x belongs to X, then all points dominating x also belong to X” am I right ? In that case, the proof consisting in constructing the homeomorphism between each indifference class and the simplex that I mentioned in the original proof would work, but, obviously, X would not be compact. I would be looking for conditions on the relation defined on a convex compact. Anyway, thanks a lot for your answers !
Peter
CC BY-SA 4.0
2023-07-08T17:50:45.067Z
Do you have any meaningful example on a convex compact for which it works?
Michael Greinecker
CC BY-SA 4.0
2023-07-08T18:17:58.707Z
I have the example of a relation that can be represented by a function which is the restriction to X of a linear function. But this satisfies the stronger property of having convex, and not simply connected, indifference sets.
Peter
CC BY-SA 4.0
2023-07-08T08:50:49.137Z
2023-07-10T08:27:52.127Z
2023-07-10T08:27:52.127Z
Michael Greinecker
<p>Let <span class="math-container">$\succsim$</span> a complete, reflexive and transitive binary relation defined on <span class="math-container">$X$</span>, a non-degenerated (i.e not identical to a singleton) convex compact subset of <span class="math-container">$\mathbb{R}^n_{++}$</span> (the set of n-dimensional vectors with positive components). Suppose <span class="math-container">$\succsim$</span> is continuous for the order topology and strictly monotonic (i.e, it preserves the usual partial order of <span class="math-container">$\mathbb{R}^n_{++}$</span>).</p>
<p>Are the associated indifference sets connected ?</p>
<p>For a preference relation defined on the whole convex cone <span class="math-container">$\mathbb{R}^n_{++}$</span> the properties I introduced are indeed sufficient, as shown in this post <a href="https://economics.stackexchange.com/questions/45752/are-indifference-curve-graphs-continuous-given-the-preferential-definition-of-co">Are Indifference Curve graphs continuous given the preferential definition of continuity?</a></p>
<p>Can you think of a counter-example or, on the contrary, provide the proof of the sufficiency of these conditions ?</p>
<p>Thank you very much !</p>

<microeconomics><mathematical-economics><utility><preferences>
Sufficient conditions for connectedness of indifference sets of a preference relation defined on a compact and convex set only
Peter
# Sufficient conditions for connectedness of indifference sets of a preference relation defined on a compact and convex set only

(Asked by: username_0 on 2023-07-07)

Let $\succsim$ a complete, reflexive and transitive binary relation defined on $X$, a non-degenerated (i.e not identical to a singleton) convex compact subset of $\mathbb{R}^n\_{++}$ (the set of n-dimensional vectors with positive components). Suppose $\succsim$ is continuous for the order topology and strictly monotonic (i.e, it preserves the usual partial order of $\mathbb{R}^n\_{++}$).

Are the associated indifference sets connected ?

For a preference relation defined on the whole convex cone $\mathbb{R}^n\_{++}$ the properties I introduced are indeed sufficient, as shown in this post [Are Indifference Curve graphs continuous given the preferential definition of continuity?](https://economics.stackexchange.com/questions/45752/are-indifference-curve-graphs-continuous-given-the-preferential-definition-of-co)

Can you think of a counter-example or, on the contrary, provide the proof of the sufficiency of these conditions ?

Thank you very much !

**Tags:** microeconomics, mathematical-economics, utility, preferences


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## 1 answers

### Answer by username_1 on 2023-07-08. Score: 5 | Accepted answer

No. Actually, standard preferences restricted to a budget line will not have this property.

For concreteness, take the preferences on $\mathbb{R}^2\_{+}$ defined by the utility function given by $u(x\_1,x\_2)=\sqrt{x\_1}+\sqrt{x\_2}$. Clearly, these preferences have all the desired properties. Now let $$X=\{(x\_1,x\_2)\in \mathbb{R}^2\_{+}\mid x\_1+x\_2=1\}.$$
Then the indifference curve corresponding to $(1,0)$ when restricted to $X$ is $\{(1,0),(0,1)\}$. Indeed, the preferences are strictly convex, and any other point on the line must be, as a proper convex combination, strictly better and, therefore, not indifferent.

> **Comments:**
> **username_0**: Thanks a lot for this counter example ! It answers perfectly to my question.  
> **username_0**: May I ask if you are aware of additional conditions that would imply this connectedness (apart from convexity of indifference sets obviously) ?  
> **username_1**: I don't have a proof, but I think that $X$ contains with every points all larger points might work. Note that monotonicity has absolutely no bite on the $X$ in my example, no point there is comparable according to size.  
> **username_0**: By “X contains with every points all larger points”, you mean “if x belongs to X, then all points dominating x also belong to X” am I right ? In that case, the proof consisting in constructing the homeomorphism between each indifference class and the simplex that I mentioned in the original proof would work, but, obviously, X would not be compact. I would be looking for conditions on the relation defined on a convex compact. Anyway, thanks a lot for your answers !  
> **username_1**: Do you have any meaningful example on a convex compact for which it works?  
> **username_0**: I have the example of a relation that can be represented by a function which is the restriction to X of a linear function. But this satisfies the stronger property of having convex, and not simply connected, indifference sets.  

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