<p>Take <span class="math-container">$(1,1)$</span> and <span class="math-container">$(-1,1)$</span>: we have that <span class="math-container">$U(1,1)=U(-1,1)=1$</span>. However, <span class="math-container">$U(\frac12(1,1) + \frac12(-1,1)) = U(0,1) - 0 &lt; 1 = \min\{U(1,1),U(-1,1)\}$</span>. Hence the function, at least defined globally over <span class="math-container">$\mathbb R^2$</span> is <strong>not</strong> quasi-concave.</p>

CC BY-SA 4.0
2019-11-13T00:10:59.560Z
2019-11-13T00:10:59.560Z
201p
<p>Just to build up on @201p answer:</p>

<hr>

<p>You probably wanted to prove that the utility function <span class="math-container">$u: \mathbb{R^2_{++}} \to \mathbb{R}$</span> given by: <span class="math-container">$u(x_1,x_2) = x^2_1 x_2$</span> is quasiconcave. As @201p pointed - allowing the function to be defined globally over <span class="math-container">$\mathbb{R^2}$</span> gives you trouble. </p>

<p>To see the intuition behind it, let's start with a definition. The function <span class="math-container">$u$</span> of many variables defined on a convex set <span class="math-container">$S$</span> is quasiconcave if every upper level set of function <span class="math-container">$u$</span> is convex.  </p>

<p>Draw a level set of aforementioned utility function allowing it to be defined globally over <span class="math-container">$\mathbb{R^2}$</span>: </p>

<p><a href="https://i.sstatic.net/ChHP6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ChHP6.png" alt="enter image description here"></a></p>

<p>Shaded area is the upper level set (at some level of utility). You can clearly see that it is not convex, therefore utility function defined that way is not quasiconcave. </p>

<p>However, if you allow utility function to be <span class="math-container">$u: \mathbb{R^2_{++}} \to \mathbb{R}$</span> given by: <span class="math-container">$u(x_1,x_2) = x^2_1 x_2$</span>. The level set would consist only with the part on the right, clearly being convex. </p>

<hr>

<p>Let <span class="math-container">$u: \mathbb{R^2_{++}} \to \mathbb{R}$</span> given by: <span class="math-container">$u(x_1,x_2) = x^2_1 x_2$</span>.</p>

<p>Another way of thinking about it could be: </p>

<p><strong>Proposition.</strong> The function <span class="math-container">$f$</span> of many variables defined on a convex set <span class="math-container">$S$</span> is quasiconcave if and only if for all <span class="math-container">$x \in S$</span> and <span class="math-container">$x' \in S$</span> such that <span class="math-container">$f(x) ≥ f(x')$</span> we have <span class="math-container">$f((1−λ)x + λx') ≥ f(x')$</span> for all <span class="math-container">$λ ∈ [0, 1]$</span>.</p>

<p>But all this actually means is that a function is quasiconcave if and only if the line segment joining the points lying on two level curves lies nowhere below the level curve corresponding to the lower value of the function. Higher level curve represents higher utility. Let point <span class="math-container">$A$</span> represent higher utility and <span class="math-container">$B$</span> lower utility. If the segment joining <span class="math-container">$A$</span> and <span class="math-container">$B$</span> lies on or above the indifference curve corresponding to the smaller value of the utility function (speaking about point <span class="math-container">$B$</span>), then the function <span class="math-container">$u$</span> of many variables is quasiconcave.</p>

<p><a href="https://i.sstatic.net/dcj1n.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/dcj1n.png" alt="enter image description here"></a></p>

<p>For more interesting examples you should check out: <a href="https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/qcc/t" rel="nofollow noreferrer">https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/qcc/t</a> </p>

<hr>

<p>Finally, Cobb-Douglas function <span class="math-container">$f: \mathbb{R^2_{++}} \to \mathbb{R}$</span> given by: <span class="math-container">$u(x_1,x_2) = x^a_1 x^b_2$</span> for positive <span class="math-container">$a,b&gt;0$</span> is:</p>

<ul>
<li>strictly concave if <span class="math-container">$a + b &lt; 1$</span></li>
<li>concave if <span class="math-container">$a + b = 1$</span></li>
<li>neither concave nor convex if <span class="math-container">$a + b &gt; 1$</span></li>
<li>quasiconcave for all <span class="math-container">$a,b &gt; 0$</span></li>
</ul>

<p>For proofs you could check out <a href="http://scottmccracken.weebly.com/uploads/9/0/6/6/9066859/convexity-print_version.pdf" rel="nofollow noreferrer">this!</a> </p>

CC BY-SA 4.0
2019-11-13T16:41:40.740Z
2019-11-13T16:41:40.740Z
bajun65537
<p>How do you prove from definition (no Hessians) that <span class="math-container">$U(x_1,x_2)=x_1^2 x_2$</span> is quasi-concave?</p>

CC BY-SA 4.0
2019-11-12T18:40:59.470Z
What is the definition of Quasi-Concavity?
Brennan
<utility>
Prove quasi-concavity of utility function
econdhn
# Prove quasi-concavity of utility function

(Asked by: username_0 on 2019-11-12)

How do you prove from definition (no Hessians) that $U(x\_1,x\_2)=x\_1^2 x\_2$ is quasi-concave?

**Tags:** utility
> **Comments:**
> **username_1**: What is the definition of Quasi-Concavity?  

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## 2 answers

### Answer by username_2 on 2019-11-13. Score: 1

Take $(1,1)$ and $(-1,1)$: we have that $U(1,1)=U(-1,1)=1$. However, $U(\frac12(1,1) + \frac12(-1,1)) = U(0,1) - 0 < 1 = \min\{U(1,1),U(-1,1)\}$. Hence the function, at least defined globally over $\mathbb R^2$ is **not** quasi-concave.


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### Answer by username_3 on 2019-11-13. Score: 0

Just to build up on @201p answer:

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You probably wanted to prove that the utility function $u: \mathbb{R^2\_{++}} \to \mathbb{R}$ given by: $u(x\_1,x\_2) = x^2\_1 x\_2$ is quasiconcave. As @201p pointed - allowing the function to be defined globally over $\mathbb{R^2}$ gives you trouble.

To see the intuition behind it, let's start with a definition. The function $u$ of many variables defined on a convex set $S$ is quasiconcave if every upper level set of function $u$ is convex.

Draw a level set of aforementioned utility function allowing it to be defined globally over $\mathbb{R^2}$:

[![enter image description here](https://i.sstatic.net/ChHP6.png)](https://i.sstatic.net/ChHP6.png)

Shaded area is the upper level set (at some level of utility). You can clearly see that it is not convex, therefore utility function defined that way is not quasiconcave.

However, if you allow utility function to be $u: \mathbb{R^2\_{++}} \to \mathbb{R}$ given by: $u(x\_1,x\_2) = x^2\_1 x\_2$. The level set would consist only with the part on the right, clearly being convex.

---

Let $u: \mathbb{R^2\_{++}} \to \mathbb{R}$ given by: $u(x\_1,x\_2) = x^2\_1 x\_2$.

Another way of thinking about it could be:

**Proposition.** The function $f$ of many variables defined on a convex set $S$ is quasiconcave if and only if for all $x \in S$ and $x' \in S$ such that $f(x) ≥ f(x')$ we have $f((1−λ)x + λx') ≥ f(x')$ for all $λ ∈ [0, 1]$.

But all this actually means is that a function is quasiconcave if and only if the line segment joining the points lying on two level curves lies nowhere below the level curve corresponding to the lower value of the function. Higher level curve represents higher utility. Let point $A$ represent higher utility and $B$ lower utility. If the segment joining $A$ and $B$ lies on or above the indifference curve corresponding to the smaller value of the utility function (speaking about point $B$), then the function $u$ of many variables is quasiconcave.

[![enter image description here](https://i.sstatic.net/dcj1n.png)](https://i.sstatic.net/dcj1n.png)

For more interesting examples you should check out: <https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/qcc/t>

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Finally, Cobb-Douglas function $f: \mathbb{R^2\_{++}} \to \mathbb{R}$ given by: $u(x\_1,x\_2) = x^a\_1 x^b\_2$ for positive $a,b>0$ is:

* strictly concave if $a + b < 1$
* concave if $a + b = 1$
* neither concave nor convex if $a + b > 1$
* quasiconcave for all $a,b > 0$

For proofs you could check out [this!](http://scottmccracken.weebly.com/uploads/9/0/6/6/9066859/convexity-print_version.pdf)


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