<p>May be, all you need is in the videos on Youtube Amit quoted above, but I would add some concise observations, mainly through examples, that I hope could be useful to you and other contributors.</p>
<p>Let's begin from Point 3 of your question.</p>
<p><strong>Point 3.</strong></p>
<p>To have a simple and intuitive grasp of the concept of quasi-concavity, it could be convenient to see the symmetric concept of quasi-convexity  for <strong>functions from  <span class="math-container">$\mathbb{R}$</span> to <span class="math-container">$\mathbb{R}$</span></strong>.</p>
<p>(Using quasi concavity would be the same, as quasi-concavity is quasi-convexity with the minus sign before. I use quasi-convexity because I think that pictures and formulas are clearer).</p>
<p>The <strong>formal definition</strong> is: a function <span class="math-container">$f$</span> from <span class="math-container">$\mathbb{R}$</span> to <span class="math-container">$\mathbb{R}$</span>  is <em>quasi-convex</em> if, <span class="math-container">$\forall \alpha \in \mathbb{R}$</span>, the set</p>
<p><span class="math-container">$$\{x\in D: f(x)&lt;\alpha\},$$</span></p>
<p>where <span class="math-container">$D$</span> is the domani of <span class="math-container">$f$</span>, is <em>convex</em>.</p>
<p>In the picture below there is an example of a non quasi-convex function and of a quasi-convex function which is not convex.</p>
<p><a href="https://i.sstatic.net/FMOBB.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/FMOBB.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/z3Qvv.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/z3Qvv.png" alt="enter image description here" /></a></p>
<p>In the first picture we have a non quasi-convex function, <span class="math-container">$f(x)= x^2(x^2-2)$</span>: the set of points for which <span class="math-container">$f(x)$</span> is under the green line is the union of the two red intervals, and is not convex.</p>
<p>The second picture shows the function  <span class="math-container">$f(x)=\sqrt|x|$</span>, which is quasi-convex, but not convex.</p>
<p><strong>Point 1.</strong></p>
<blockquote>
<p>how can the function be quasi-concave and the plotted curve be convex?</p>
</blockquote>
<p>Remember that indifference curves are level curves of a utility function.
A quasi-concave function <strong>can</strong> ,of course,  have convex level curves.</p>
<p>Let's see an example, of a function from <span class="math-container">$\mathbb{R^2}$</span>  to <span class="math-container">$\mathbb{R}$</span>.</p>
<p>Consider a <strong>Cobb-Douglas utility function</strong>, <span class="math-container">$U(x,y)= kx^\alpha  y ^{1-\alpha}$</span>, <span class="math-container">$x,y\geq0$</span>, <span class="math-container">$k \in R_+$</span> and  <span class="math-container">$\alpha \in (0,1)$</span>.</p>
<p>The Cobb Douglas is strictly concave, so it is concave, so it is quasi-concave. The indifference curves are convex.</p>
<p>You can have an intuitive idea  from the following picture, where the graphs of the Cobb Douglas in three dimensions, and of its level curves (indifference curves) are represented.</p>
<p><a href="https://i.sstatic.net/9NVyl.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9NVyl.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/XOveS.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XOveS.png" alt="enter image description here" /></a></p>
<p>If you calculate analytically the level curves of the Cobb-Douglas, you can see that they are convex.</p>
<p>For example, for <span class="math-container">$k=1$</span> and <span class="math-container">$\alpha=1/2$</span>, we have the hyperbolas</p>
<p><span class="math-container">$$ y=c^2/x,$$</span> <span class="math-container">$c\in R$</span>, for <span class="math-container">$x&gt;0$</span>.</p>
<p><strong>Point 2.</strong></p>
<blockquote>
<p>Quasi-concavity implies concavity so I can just prove utility is
concave to prove ID curves are convex? Concavity can be proven through
a Hessian matrix of second derivatives right?</p>
</blockquote>
<p>I think you made a typo, the reverse of what you wrote is true: concavity implies  quasi-concavity (quasi-concavity is a weaker property), so it is sufficient to prove concavity.</p>
<p>Concavity can be proved through second derivatives for functions from <span class="math-container">$\mathbb{R}$</span>  to <span class="math-container">$\mathbb{R}$</span>, or through the Hessian matrix, for functions from <span class="math-container">$\mathbb{R^n}$</span> to <span class="math-container">$\mathbb{R}$</span>, but provided that the utility function is differentiable  as needed.</p>
<p>For example, one can’t prove the convexity of <span class="math-container">$f(x))= |x|$</span> using the derivatives, because this function is not differentiable in all <span class="math-container">$\mathbb{R}$</span> ( it is not differentiable in <span class="math-container">$0$</span>).</p>

CC BY-SA 4.0
2023-02-14T17:49:35.980Z
Honestly 50% of the time I read a great answer on this site, I scroll to the end and it's you @BakerStreet! I have a question related to this one here if you're interested: https://economics.stackexchange.com/questions/54438/quasiconvex-constraints-in-maximisation
CormJack
CC BY-SA 4.0
2023-02-14T17:50:40.957Z
Also in your first diagram surely its the humped part of the curve above the green line, between the two read lines that fails to be convex and is actually a concave interval?
CormJack
CC BY-SA 4.0
2023-02-14T18:56:54.833Z
Thank you very much @CormJack! You are my advertising agent :-). You are very kind. If you consider that part alone, it is a part of the function that is concave. This  part of the function is concave, not the interval on the abscissa, of course.  An interval is a convex set.
BakerStreet
CC BY-SA 4.0
2023-02-14T19:11:21.173Z
Hahhaha yeah we'll start a campaign to vote for you as a moderator! So just to clarify then, the hump above the green line between 1/2 and -1/2 is concave...And then the function from -1/2 to - infinity, and 1/2 to infinity is convex? I only ask because I think you wrote the opposite in your post, or am I misreading your quote: "In the first picture we have a non quasi-convex function....the set of points for which ()...is under the green line is the union of the two red intervals, and is not convex."
CormJack
CC BY-SA 4.0
2023-02-14T19:54:42.000Z
You have to considre the function as a whole, to say if it is convex and so on. If you take a single part it is different. In the phrase you quote I say that the *union of the two red intervals* is not convex, meaning that it is not a *convex set*. Don't confuse *convex set* with *convex function*, they are different, even if related, concepts.
BakerStreet
CC BY-SA 4.0
2023-02-27T15:11:50.313Z
Ah I understand now, yes the union of those two sets you define (above the function but below the red line) is indeed not a convex set. However given that convexity/concavity can be considered as non-global properties (vs quasi concavity/convexity - which are not defined at points). Could we say that the function defined over the interval on the LHS red line to negative infinity, is convex over this interval. And the same for the function defined on the interval given by RHS red line to positive infinity? (Just to remind you, we were talking about the first graph with the hump btw)
CormJack
CC BY-SA 4.0
2022-10-15T16:23:19.540Z
2022-10-15T18:54:44.950Z
2022-10-15T18:54:44.950Z
BakerStreet
<p>So my professor told that quasi-concave utilities lead to convex preferences/indifference curves. I have some conceptual problems understanding this statement.</p>
<ol>
<li>Indifference curves are plotted from utility functions and they are convex, so how can the function be quasi-concave and the plotted curve be convex?</li>
<li>Quasi-concavity implies concavity so I can just prove utility is concave to prove ID curves are convex? Concavity can be proven through a Hessian matrix of second derivatives right?</li>
<li>I understand the difference between concavity vs strictly concave. What is the meaning of quasi-concave in a simpler language?</li>
</ol>
<p>Thank you so much. I understand that these doubts are lame but I want to understand the concept correctly.</p>

CC BY-SA 4.0
2022-10-13T11:34:47.827Z
You can watch this playlist: https://youtube.com/playlist?list=PLUJGfL_499TLYEd-9IO1DmQKwm6Mtdt4u
Amit
CC BY-SA 4.0
2022-10-13T20:02:45.847Z
Thank you @Amit, your videos are a hidden gem. I hope more people discover them and benefit from your amazing content. I started watching 8 hours ago and have constantly been thinking about them.
EconNoob
<microeconomics><utility><preferences>
Shape of indifference curves and quasi-concavity of utility
EconNoob
# Shape of indifference curves and quasi-concavity of utility

(Asked by: username_0 on 2022-10-13)

So my professor told that quasi-concave utilities lead to convex preferences/indifference curves. I have some conceptual problems understanding this statement.

1. Indifference curves are plotted from utility functions and they are convex, so how can the function be quasi-concave and the plotted curve be convex?
2. Quasi-concavity implies concavity so I can just prove utility is concave to prove ID curves are convex? Concavity can be proven through a Hessian matrix of second derivatives right?
3. I understand the difference between concavity vs strictly concave. What is the meaning of quasi-concave in a simpler language?

Thank you so much. I understand that these doubts are lame but I want to understand the concept correctly.

**Tags:** microeconomics, utility, preferences

> **Comments:**
> **username_1**: You can watch this playlist: https://youtube.com/playlist?list=PLUJGfL_499TLYEd-9IO1DmQKwm6Mtdt4u  
> **username_0**: Thank you @username_1, your videos are a hidden gem. I hope more people discover them and benefit from your amazing content. I started watching 8 hours ago and have constantly been thinking about them.  

---

## 1 answers

### Answer by username_2 on 2022-10-15. Score: 2 | Accepted answer

May be, all you need is in the videos on Youtube username_1 quoted above, but I would add some concise observations, mainly through examples, that I hope could be useful to you and other contributors.

Let's begin from Point 3 of your question.

**Point 3.**

To have a simple and intuitive grasp of the concept of quasi-concavity, it could be convenient to see the symmetric concept of quasi-convexity for **functions from $\mathbb{R}$ to $\mathbb{R}$**.

(Using quasi concavity would be the same, as quasi-concavity is quasi-convexity with the minus sign before. I use quasi-convexity because I think that pictures and formulas are clearer).

The **formal definition** is: a function $f$ from $\mathbb{R}$ to $\mathbb{R}$ is *quasi-convex* if, $\forall \alpha \in \mathbb{R}$, the set

$$\{x\in D: f(x)<\alpha\},$$

where $D$ is the domani of $f$, is *convex*.

In the picture below there is an example of a non quasi-convex function and of a quasi-convex function which is not convex.

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

[![enter image description here](https://i.sstatic.net/z3Qvv.png)](https://i.sstatic.net/z3Qvv.png)

In the first picture we have a non quasi-convex function, $f(x)= x^2(x^2-2)$: the set of points for which $f(x)$ is under the green line is the union of the two red intervals, and is not convex.

The second picture shows the function $f(x)=\sqrt|x|$, which is quasi-convex, but not convex.

**Point 1.**

> how can the function be quasi-concave and the plotted curve be convex?

Remember that indifference curves are level curves of a utility function.
A quasi-concave function **can** ,of course, have convex level curves.

Let's see an example, of a function from $\mathbb{R^2}$ to $\mathbb{R}$.

Consider a **Cobb-Douglas utility function**, $U(x,y)= kx^\alpha y ^{1-\alpha}$, $x,y\geq0$, $k \in R\_+$ and $\alpha \in (0,1)$.

The Cobb Douglas is strictly concave, so it is concave, so it is quasi-concave. The indifference curves are convex.

You can have an intuitive idea from the following picture, where the graphs of the Cobb Douglas in three dimensions, and of its level curves (indifference curves) are represented.

[![enter image description here](https://i.sstatic.net/9NVyl.png)](https://i.sstatic.net/9NVyl.png)

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

If you calculate analytically the level curves of the Cobb-Douglas, you can see that they are convex.

For example, for $k=1$ and $\alpha=1/2$, we have the hyperbolas

$$ y=c^2/x,$$ $c\in R$, for $x>0$.

**Point 2.**

> Quasi-concavity implies concavity so I can just prove utility is
> concave to prove ID curves are convex? Concavity can be proven through
> a Hessian matrix of second derivatives right?

I think you made a typo, the reverse of what you wrote is true: concavity implies quasi-concavity (quasi-concavity is a weaker property), so it is sufficient to prove concavity.

Concavity can be proved through second derivatives for functions from $\mathbb{R}$ to $\mathbb{R}$, or through the Hessian matrix, for functions from $\mathbb{R^n}$ to $\mathbb{R}$, but provided that the utility function is differentiable as needed.

For example, one can’t prove the convexity of $f(x))= |x|$ using the derivatives, because this function is not differentiable in all $\mathbb{R}$ ( it is not differentiable in $0$).

> **Comments:**
> **username_3**: Honestly 50% of the time I read a great answer on this site, I scroll to the end and it's you @username_2! I have a question related to this one here if you're interested: https://economics.stackexchange.com/questions/54438/quasiconvex-constraints-in-maximisation  
> **username_3**: Also in your first diagram surely its the humped part of the curve above the green line, between the two read lines that fails to be convex and is actually a concave interval?  
> **username_2**: Thank you very much @username_3! You are my advertising agent :-). You are very kind. If you consider that part alone, it is a part of the function that is concave. This  part of the function is concave, not the interval on the abscissa, of course.  An interval is a convex set.  
> **username_3**: Hahhaha yeah we'll start a campaign to vote for you as a moderator! So just to clarify then, the hump above the green line between 1/2 and -1/2 is concave...And then the function from -1/2 to - infinity, and 1/2 to infinity is convex? I only ask because I think you wrote the opposite in your post, or am I misreading your quote: "In the first picture we have a non quasi-convex function....the set of points for which ()...is under the green line is the union of the two red intervals, and is not convex."  
> **username_2**: You have to considre the function as a whole, to say if it is convex and so on. If you take a single part it is different. In the phrase you quote I say that the *union of the two red intervals* is not convex, meaning that it is not a *convex set*. Don't confuse *convex set* with *convex function*, they are different, even if related, concepts.  
> **username_3**: Ah I understand now, yes the union of those two sets you define (above the function but below the red line) is indeed not a convex set. However given that convexity/concavity can be considered as non-global properties (vs quasi concavity/convexity - which are not defined at points). Could we say that the function defined over the interval on the LHS red line to negative infinity, is convex over this interval. And the same for the function defined on the interval given by RHS red line to positive infinity? (Just to remind you, we were talking about the first graph with the hump btw)  

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