<p>Does quasi-concave utility function imply convex indifference curve?</p>

<p>No that is not true. Consider <span class="math-container">$u(x, y) = -x^2 - y^2$</span> defined on <span class="math-container">$\mathbb{R}^2_+$</span>. Since <span class="math-container">$u$</span> is concave it is quasiconcave. Observing the graph of the indifference curves, we see that ICs of <span class="math-container">$u$</span> are not "convex".</p>

<p><a href="https://i.sstatic.net/eNOPK.png" rel="noreferrer"><img src="https://i.sstatic.net/eNOPK.png" alt="enter image description here"></a></p>

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2019-11-04T13:22:16.663Z
The answer linked in the question does say that monotonicity is required. You are right that this information is not contained in the body of this question.
Giskard
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2019-11-04T07:09:38.503Z
2019-11-04T13:03:15.127Z
2019-11-04T13:03:15.127Z
Amit
<p>It is well-known that convex indifference curve (i.e. the function is convex)/ preference would imply quasi-concave utility function. But does quasi-concave utility function imply convex indifference curve?</p>

<p>It seems that <a href="https://economics.stackexchange.com/questions/26498/if-a-utility-function-is-quasi-concave-can-we-say-that-the-ic-curve-associated">this answer</a> give a brief sketch of the proof, but how can I show it in a more formal way?</p>

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2019-11-03T13:49:36.227Z
Seems like that answer had little effect, because I tried to point out in it that the indifference curve is usually not "convex".
Giskard
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2019-11-03T13:50:44.590Z
Your question seems fairly straightforward. Have you tried directly applying the definitions of convex preferences and quasi-concave utility functions?
Giskard
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2019-11-03T15:48:12.860Z
You said that "What you probably mean is that the IC curve implicitly defines a convex function f where f(x)=y." That is what I meant. I am not sure what do you mean by " IC curve is not convex in the usual meaning of the word convex when applied to sets".
Aqqqq
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2019-11-03T16:10:30.467Z
I know how to go from quasi-concave utility function to convex preference now. But what about from convex preference to convex indifference curve (formally)?
Aqqqq
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2019-11-03T17:10:19.463Z
Would you mind typing in your quasi-concave utility function to convex preference proof (edit your question, do not add more comments), so we can see why it is not reversible?
Giskard
CC BY-SA 4.0
2019-11-03T17:11:47.630Z
On convexity: Most curves are not convex sets. If you pick two points from the curve you can usually find a convex combination of them that is not on the curve.
Giskard
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2019-11-03T17:37:40.200Z
I never said that it is not reversible. I just wonder how to prove that convex preference leads to convex indifference curve.
Aqqqq
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2019-11-03T20:36:54.457Z
Would you please give us your definition of a quasi-concave function? In some cases, the claim is true by definition.
Bertrand
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2019-11-03T22:16:18.513Z
The definition is the same as the one here: https://en.wikipedia.org/wiki/Quasiconvex_function
Aqqqq
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2019-11-04T07:11:37.060Z
"Convex" indifference curves does not imply quasiconcave utility function. Consider $u = -xy$, $u$ has convex ICs but it is not quasiconcave.
Amit
<microeconomics><utility>
Does quasi-concave utility function imply convex indifference curve?
Aqqqq
# Does quasi-concave utility function imply convex indifference curve?

(Asked by: username_0 on 2019-11-03)

It is well-known that convex indifference curve (i.e. the function is convex)/ preference would imply quasi-concave utility function. But does quasi-concave utility function imply convex indifference curve?

It seems that [this answer](https://economics.stackexchange.com/questions/26498/if-a-utility-function-is-quasi-concave-can-we-say-that-the-ic-curve-associated) give a brief sketch of the proof, but how can I show it in a more formal way?

**Tags:** microeconomics, utility
> **Comments:**
> **username_1**: Seems like that answer had little effect, because I tried to point out in it that the indifference curve is usually not "convex".  
> **username_1**: Your question seems fairly straightforward. Have you tried directly applying the definitions of convex preferences and quasi-concave utility functions?  
> **username_0**: You said that "What you probably mean is that the IC curve implicitly defines a convex function f where f(x)=y." That is what I meant. I am not sure what do you mean by " IC curve is not convex in the usual meaning of the word convex when applied to sets".  
> **username_0**: I know how to go from quasi-concave utility function to convex preference now. But what about from convex preference to convex indifference curve (formally)?  
> **username_1**: Would you mind typing in your quasi-concave utility function to convex preference proof (edit your question, do not add more comments), so we can see why it is not reversible?  
> **username_1**: On convexity: Most curves are not convex sets. If you pick two points from the curve you can usually find a convex combination of them that is not on the curve.  
> **username_0**: I never said that it is not reversible. I just wonder how to prove that convex preference leads to convex indifference curve.  
> **username_2**: Would you please give us your definition of a quasi-concave function? In some cases, the claim is true by definition.  
> **username_0**: The definition is the same as the one here: https://en.wikipedia.org/wiki/Quasiconvex_function  
> **username_3**: "Convex" indifference curves does not imply quasiconcave utility function. Consider $u = -xy$, $u$ has convex ICs but it is not quasiconcave.  

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## 1 answers

### Answer by username_3 on 2019-11-04. Score: 6 | Accepted answer

Does quasi-concave utility function imply convex indifference curve?

No that is not true. Consider $u(x, y) = -x^2 - y^2$ defined on $\mathbb{R}^2\_+$. Since $u$ is concave it is quasiconcave. Observing the graph of the indifference curves, we see that ICs of $u$ are not "convex".

[![enter image description here](https://i.sstatic.net/eNOPK.png)](https://i.sstatic.net/eNOPK.png)

> **Comments:**
> **username_1**: The answer linked in the question does say that monotonicity is required. You are right that this information is not contained in the body of this question.  

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