<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>

CC BY-SA 4.0
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
CC BY-SA 4.0
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>

CC BY-SA 4.0
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
CC BY-SA 4.0
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
CC BY-SA 4.0
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
CC BY-SA 4.0
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
CC BY-SA 4.0
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
CC BY-SA 4.0
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
CC BY-SA 4.0
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
CC BY-SA 4.0
2019-11-03T22:16:18.513Z
The definition is the same as the one here: https://en.wikipedia.org/wiki/Quasiconvex_function
Aqqqq
CC BY-SA 4.0
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.  

---

## 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.  

---

