Relationship between convexity and a perfect complements type utility function

Question

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?

Answer 1

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.

Answer 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 $u$ :

$u$ does not represent the same preference as $v(x, y) =\min(x, y)$. Here is the indifference map of $v$ :