<p>I've just encountered the following exercise from GEOFFREY A. JEHLE micro's book</p>
<p><a href="https://i.sstatic.net/4sDqb.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4sDqb.png" alt="enter image description here" /></a></p>
<p>Strict monotonicity comes from the fact that any increase in <span class="math-container">$x_1$</span> or <span class="math-container">$x_2$</span> increases utility, and strict convexity comes from the fact that a convex combination of any pair of bundles is strictly preferred to the pair. However, in this example, we can go in the situation where we get Leontief preferences (<span class="math-container">$\rho \to - \infty$</span>). In that case prefences are weakly monotone and weakly convex. Why preferences can be easily verified to be strictly monotonic and strictly convex? Also, could you provide an example showing how to prove that the CES utility function has strict convex preferences?</p>

CC BY-SA 4.0
2023-02-08T14:49:35.200Z
"*could you provide an example showing how to prove*" An example does not prove anything.
Giskard
CC BY-SA 4.0
2023-02-08T14:50:19.107Z
You wrote "*strict convexity comes from the fact that a convex combination of any pair of bundles is strictly preferred to the pair.*; have you considered using this definition to see if the function has the property?
Giskard
<microeconomics><preferences>
Strict monotonicity and strict convexity - prefences
Dimitru
# Strict monotonicity and strict convexity - prefences

(Asked by: username_0 on 2023-02-08)

I've just encountered the following exercise from GEOFFREY A. JEHLE micro's book

[![enter image description here](https://i.sstatic.net/4sDqb.png)](https://i.sstatic.net/4sDqb.png)

Strict monotonicity comes from the fact that any increase in $x\_1$ or $x\_2$ increases utility, and strict convexity comes from the fact that a convex combination of any pair of bundles is strictly preferred to the pair. However, in this example, we can go in the situation where we get Leontief preferences ($\rho \to - \infty$). In that case prefences are weakly monotone and weakly convex. Why preferences can be easily verified to be strictly monotonic and strictly convex? Also, could you provide an example showing how to prove that the CES utility function has strict convex preferences?

**Tags:** microeconomics, preferences
> **Comments:**
> **username_1**: "*could you provide an example showing how to prove*" An example does not prove anything.  
> **username_1**: You wrote "*strict convexity comes from the fact that a convex combination of any pair of bundles is strictly preferred to the pair.*; have you considered using this definition to see if the function has the property?  

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## 0 answers

