<p>If the utility function <span class="math-container">$u$</span> is continuous and satisfies local nonsatiation, and walrasian demand function <span class="math-container">$x(p, w)$</span> is a function (i.e. always map to only single values), how to prove that <span class="math-container">$x(p,w)$</span> is continuous in <span class="math-container">$p$</span> (price vector) and <span class="math-container">$w$</span> (wealth)?</p>

CC BY-SA 4.0
2019-10-19T23:35:18.133Z
What are $x(p,w)$ and $w(p,w)$? Are they both the same walrasian demand function?
GabMac
CC BY-SA 4.0
2019-10-20T05:14:51.130Z
Sorry, this was a typo.
Aqqqq
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2019-10-20T15:49:01.667Z
Look at the Theorem of the maximum: https://en.wikipedia.org/wiki/Maximum_theorem
user24622
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2024-01-25T08:20:15.537Z
@user24622 The Theorem of the Maximum will give you upper hemicontinuity of the Walrasian demand correspondence, but AFAICT this does not imply continuity (in the usual sense) even if Walrasian demand is single-valued.
chsk
<microeconomics><utility><consumption><walrasian>
How to prove that the walrasian demand function $x(p,w)$ is continuous in $p$ and $w$?
Aqqqq
# How to prove that the walrasian demand function $x(p,w)$ is continuous in $p$ and $w$?

(Asked by: username_0 on 2019-10-19)

If the utility function $u$ is continuous and satisfies local nonsatiation, and walrasian demand function $x(p, w)$ is a function (i.e. always map to only single values), how to prove that $x(p,w)$ is continuous in $p$ (price vector) and $w$ (wealth)?

**Tags:** microeconomics, utility, consumption, walrasian
> **Comments:**
> **username_1**: What are $x(p,w)$ and $w(p,w)$? Are they both the same walrasian demand function?  
> **username_0**: Sorry, this was a typo.  
> **username_2**: Look at the Theorem of the maximum: https://en.wikipedia.org/wiki/Maximum_theorem  
> **username_3**: @user24622 The Theorem of the Maximum will give you upper hemicontinuity of the Walrasian demand correspondence, but AFAICT this does not imply continuity (in the usual sense) even if Walrasian demand is single-valued.  

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## 0 answers

