<p>Loosely speaking <strong>closed</strong> means that it includes the boundary. The boundary of the budget set is formed by those goods baskets (<span class="math-container">$x$</span>) that imply spending all available income, i.e., such that <span class="math-container">$p·x=w.$</span> It is clear that they belong in the budget set and, hence, it is closed. (To be strict, we'd also need to check that points such as <span class="math-container">$ \vec 0$</span> also belong in the budget set. That's easy and I leave it for you to prove).</p>

CC BY-SA 4.0
2018-12-21T17:14:44.403Z
2018-12-21T17:14:44.403Z
Patricio
<p>Hint: a set is closed if its complement is open, and if a bundle <span class="math-container">$x$</span> is strictly unaffordable, <span class="math-container">$p \cdot x &gt; w$</span>, every nearby bundle is also unaffordable.</p>

CC BY-SA 4.0
2018-12-24T04:41:45.127Z
2018-12-24T04:41:45.127Z
kyle
<p>Mas-Colell, Whinston and Green's Microeconomic Theory affirms the budget set is closed. I would like to know why.</p>

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

<p>The definition of open set in the book is the following:<a href="https://i.sstatic.net/cm1T1.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/cm1T1.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2018-12-21T17:10:49.943Z
What exactly are you asking? Are you asking what 'closed set' means?
Giskard
<microeconomics>
Proof that the budget set is closed
Incognito
# Proof that the budget set is closed

(Asked by: username_0 on 2018-12-21)

Mas-Colell, Whinston and Green's Microeconomic Theory affirms the budget set is closed. I would like to know why.

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

The definition of open set in the book is the following:[![enter image description here](https://i.sstatic.net/cm1T1.png)](https://i.sstatic.net/cm1T1.png)

**Tags:** microeconomics
> **Comments:**
> **username_1**: What exactly are you asking? Are you asking what 'closed set' means?  

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## 2 answers

### Answer by username_2 on 2018-12-21. Score: 1

Loosely speaking **closed** means that it includes the boundary. The boundary of the budget set is formed by those goods baskets ($x$) that imply spending all available income, i.e., such that $p·x=w.$ It is clear that they belong in the budget set and, hence, it is closed. (To be strict, we'd also need to check that points such as $ \vec 0$ also belong in the budget set. That's easy and I leave it for you to prove).


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### Answer by username_3 on 2018-12-24. Score: 0

Hint: a set is closed if its complement is open, and if a bundle $x$ is strictly unaffordable, $p \cdot x > w$, every nearby bundle is also unaffordable.


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