<p>$u(x, y) = -\max(x, y) = \min(-x, -y)$ is a concave function. Since it is concave, it is also quasiconcave (or equivalently, it represents weakly convex preferences). Here is the indifference map of $u$ :
<a href="https://i.sstatic.net/FxTSy.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/FxTSy.png" alt="enter image description here"></a></p>

<p>$u$ does not represent the same preference as $v(x, y) =\min(x, y)$. Here is the indifference map of $v$ :
<a href="https://i.sstatic.net/xUh3b.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/xUh3b.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2018-05-18T10:04:21.393Z
@B11b What do you mean by "up to today, I never ask a homework"? You have no questions posted today.
Giskard
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2018-05-18T13:11:56.137Z
Thank you! I didn't know that -max(x,y) would be min (-x,-y) instead of min (x, y). Very confusing. I thought the function would mirror itself, e.g. y=x^2 and y=-x^2 So, just to confirm: the curve does not change at all, given the negative sign? (first graph)
TheLight OI
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2018-05-18T13:19:56.987Z
EDIT: will -min (x,y) become max (-x,-y) then? And its curve won't change?
TheLight OI
CC BY-SA 4.0
2018-05-18T03:57:14.803Z
2018-05-18T03:57:14.803Z
Amit
<p>Instead of directly giving you the answers, I'm going to give you a series of hints to help you figure out the answers on your own.</p>

<blockquote>
  <p>1.What would be the shape of the indifference curve?</p>
</blockquote>

<p>Consider three different consumption bundles: $(3,10)$, $(10,10)$, and $(10,3)$. Verify that these three bundles yield the same utility to the consumer. In other words, they should lie on the same indifference curve. Then draw a curve through the three points. Verify that all points described by $(x,10)$ and $(10,y)$ for $x,y\le10$ should also be on this indifference curve.</p>

<blockquote>
  <p>2.Why are these preferences weakly convex?</p>
</blockquote>

<p>First remind yourself what <em>weak convexity</em> means. Then compare the utilities that the individual gets from the following pairs of bundles:</p>

<ul>
<li>$A_1=(10,10)$ and $A_2=\alpha(10,3)+(1-\alpha)(3,10)$ for $\alpha\in(0,1)$ </li>
<li>$B_1=(10,10)$ and $B_2=\alpha(10,3)+(1-\alpha)(10,10)$ for $\alpha\in(0,1)$ </li>
</ul>

<blockquote>
  <p>3.What does the negative sign imply mathematically?</p>
</blockquote>

<p>You said that the consumer <em>hates</em> both goods. Compare the utilities from the bundles $(10,10)$ and $(5,5)$ and see which is higher (and thus more preferred). </p>

<blockquote>
  <p>4.Will it become an min{x,y} function?</p>
</blockquote>

<p>Try sketch an indifference curve for $u=\min\{x,y\}$ and compare to those in part 1.</p>

CC BY-SA 4.0
2018-05-18T13:28:01.270Z
Thanks! It was very confusing for me because both goods are bads. Hence I related them somehow to perfect substitutes, with a positive slope.
TheLight OI
CC BY-SA 4.0
2018-05-18T02:44:55.913Z
2018-05-18T02:44:55.913Z
Herr K.
<p>Consider someone who consume two goods and hates them both. 
Given the utility function:
U(x,y)= -max{x,y}
1.What would be the shape of the indifference curve?
2.Why are these preferences <strong>weakly</strong> convex?
3.What does the negative sign imply mathematically?
4.Will it become an min{x,y} function?</p>

<microeconomics><preferences><perfect-complements>
Relationship between convexity and a perfect complements type utility function
TheLight OI
# Relationship between convexity and a perfect complements type utility function

(Asked by: username_0 on 2018-05-18)

Consider someone who consume two goods and hates them both.
Given the utility function:
U(x,y)= -max{x,y}
1.What would be the shape of the indifference curve?
2.Why are these preferences **weakly** convex?
3.What does the negative sign imply mathematically?
4.Will it become an min{x,y} function?

**Tags:** microeconomics, preferences, perfect-complements

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## 2 answers

### Answer by username_1 on 2018-05-18. Score: 2

$u(x, y) = -\max(x, y) = \min(-x, -y)$ is a concave function. Since it is concave, it is also quasiconcave (or equivalently, it represents weakly convex preferences). Here is the indifference map of $u$ :
[![enter image description here](https://i.sstatic.net/FxTSy.png)](https://i.sstatic.net/FxTSy.png)

$u$ does not represent the same preference as $v(x, y) =\min(x, y)$. Here is the indifference map of $v$ :
[![enter image description here](https://i.sstatic.net/xUh3b.png)](https://i.sstatic.net/xUh3b.png)

> **Comments:**
> **username_2**: @B11b What do you mean by "up to today, I never ask a homework"? You have no questions posted today.  
> **username_0**: Thank you! I didn't know that -max(x,y) would be min (-x,-y) instead of min (x, y). Very confusing. I thought the function would mirror itself, e.g. y=x^2 and y=-x^2 So, just to confirm: the curve does not change at all, given the negative sign? (first graph)  
> **username_0**: EDIT: will -min (x,y) become max (-x,-y) then? And its curve won't change?  

---

### Answer by username_3 on 2018-05-18. Score: 2

Instead of directly giving you the answers, I'm going to give you a series of hints to help you figure out the answers on your own.

> 1.What would be the shape of the indifference curve?

Consider three different consumption bundles: $(3,10)$, $(10,10)$, and $(10,3)$. Verify that these three bundles yield the same utility to the consumer. In other words, they should lie on the same indifference curve. Then draw a curve through the three points. Verify that all points described by $(x,10)$ and $(10,y)$ for $x,y\le10$ should also be on this indifference curve.

> 2.Why are these preferences weakly convex?

First remind yourself what *weak convexity* means. Then compare the utilities that the individual gets from the following pairs of bundles:

* $A\_1=(10,10)$ and $A\_2=\alpha(10,3)+(1-\alpha)(3,10)$ for $\alpha\in(0,1)$
* $B\_1=(10,10)$ and $B\_2=\alpha(10,3)+(1-\alpha)(10,10)$ for $\alpha\in(0,1)$

> 3.What does the negative sign imply mathematically?

You said that the consumer *hates* both goods. Compare the utilities from the bundles $(10,10)$ and $(5,5)$ and see which is higher (and thus more preferred).

> 4.Will it become an min{x,y} function?

Try sketch an indifference curve for $u=\min\{x,y\}$ and compare to those in part 1.

> **Comments:**
> **username_0**: Thanks! It was very confusing for me because both goods are bads. Hence I related them somehow to perfect substitutes, with a positive slope.  

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