<p>First look at the shape of the indifference curves. (I assume that <span class="math-container">$f$</span> is a utility function here.) The better-sets are convex, so this is consistent with quasi-concavity of the function <span class="math-container">$f$</span>.</p>

<p>Second, consider the convex combination <span class="math-container">$(2,2)=\frac{1}{2}(1,1)+\frac{1}{2}(3,3)$</span>. You have <span class="math-container">$f(2,2)=2$</span>, but <span class="math-container">$\frac{1}{2}f(1,1)+\frac{1}{2}f(3,3)=\frac{1}{2}1+\frac{1}{2}4=2.5&gt;2$</span>, which is inconsistent with the function being concave.</p>

CC BY-SA 4.0
2020-02-16T15:05:20.697Z
Now I get it! I guess my problem was understanding what convex combinations are. Thanks a lot!
mn2609
CC BY-SA 4.0
2020-02-15T14:30:52.420Z
2020-02-15T14:30:52.420Z
VARulle
<p>I am studying for an upcoming exam in microeconomics and I have always struggled with proving concavity, convexity in different exercizes. This one I can't solve for the life of me. It is kind of hard to find the exact point I struggle with, because I'm pretty confused on this one overall.</p>

<p>We are given sets of points for which the value of the function is f(x) = 1, 2, 4.</p>

<p><a href="https://i.sstatic.net/SVVw4.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/SVVw4.png" alt="enter image description here"></a></p>

<p>We are then asked if these curves are consistent or inconsistent with the function being either concave or quasi-concave.</p>

<p>We are given tips: </p>

<ol>
<li>Write down the definition of concavity:</li>
</ol>

<p><span class="math-container">$f: D -&gt; \mathbb R$</span> is concave if for all <span class="math-container">$ x, x' \in D, t \in (0,1):$</span></p>

<p><span class="math-container">$f(tx + (1-t)x') \geq tf(x) + (1-t)f(x')$</span></p>

<ol start="2">
<li>Select two points in the domain (1,3) and write down the convex combination for the points.</li>
</ol>

<p>???</p>

<ol start="3">
<li><p>Evaluate the value of the function.</p></li>
<li><p>Evaluate the value of the convex combination for each of the points for this function.</p></li>
<li><p>Use the defintion to show where the function is concave and/or quasi-concave</p></li>
</ol>

<p>What throws me off here is that there is no actual function given, but only values for the function. If I had a function, say <span class="math-container">$f(x,y) = x^\beta y^{1-\beta}$</span>, I could choose a point and check with a certain t against the concavity defintion, right?</p>

<p>I hope this doesn't fall under the no homework-clause. I don't have to turn this in, I'm just lost :)</p>

CC BY-SA 4.0
2020-02-13T12:31:23.307Z
"*Select two points in the domain (1,3) and write down the convex combination for the points.*" What exactly do you mean by "???" Seems like this could be explained.
Giskard
CC BY-SA 4.0
2020-02-13T17:30:58.850Z
Hint: You don't know the exact form of $f$, but you do know that its values at points $(1,1)$, $(2,2)$ and $(3,3)$ and $(2,2)$ is a convex combination of $(1,1)$ and $(3,3)$.
Herr K.
<microeconomics>
Check for concavity, quasi-concavity with given value of function
mn2609
# Check for concavity, quasi-concavity with given value of function

(Asked by: username_0 on 2020-02-13)

I am studying for an upcoming exam in microeconomics and I have always struggled with proving concavity, convexity in different exercizes. This one I can't solve for the life of me. It is kind of hard to find the exact point I struggle with, because I'm pretty confused on this one overall.

We are given sets of points for which the value of the function is f(x) = 1, 2, 4.

[![enter image description here](https://i.sstatic.net/SVVw4.png)](https://i.sstatic.net/SVVw4.png)

We are then asked if these curves are consistent or inconsistent with the function being either concave or quasi-concave.

We are given tips:

1. Write down the definition of concavity:

$f: D -> \mathbb R$ is concave if for all $ x, x' \in D, t \in (0,1):$

$f(tx + (1-t)x') \geq tf(x) + (1-t)f(x')$

2. Select two points in the domain (1,3) and write down the convex combination for the points.

???

3. Evaluate the value of the function.
4. Evaluate the value of the convex combination for each of the points for this function.
5. Use the defintion to show where the function is concave and/or quasi-concave

What throws me off here is that there is no actual function given, but only values for the function. If I had a function, say $f(x,y) = x^\beta y^{1-\beta}$, I could choose a point and check with a certain t against the concavity defintion, right?

I hope this doesn't fall under the no homework-clause. I don't have to turn this in, I'm just lost :)

**Tags:** microeconomics

> **Comments:**
> **username_1**: "*Select two points in the domain (1,3) and write down the convex combination for the points.*" What exactly do you mean by "???" Seems like this could be explained.  
> **username_2**: Hint: You don't know the exact form of $f$, but you do know that its values at points $(1,1)$, $(2,2)$ and $(3,3)$ and $(2,2)$ is a convex combination of $(1,1)$ and $(3,3)$.  

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## 1 answers

### Answer by username_3 on 2020-02-15. Score: 3 | Accepted answer

First look at the shape of the indifference curves. (I assume that $f$ is a utility function here.) The better-sets are convex, so this is consistent with quasi-concavity of the function $f$.

Second, consider the convex combination $(2,2)=\frac{1}{2}(1,1)+\frac{1}{2}(3,3)$. You have $f(2,2)=2$, but $\frac{1}{2}f(1,1)+\frac{1}{2}f(3,3)=\frac{1}{2}1+\frac{1}{2}4=2.5>2$, which is inconsistent with the function being concave.

> **Comments:**
> **username_0**: Now I get it! I guess my problem was understanding what convex combinations are. Thanks a lot!  

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