<p>The budget set is always defined given a price vector <span class="math-container">$p=(p_i)_{i\leq l}$</span> (it seems like <span class="math-container">$l$</span> is the number of goods in your problem) and an income <span class="math-container">$w$</span>. We usually implicitly assume that prices are strictly positive and the income is finite.
Otherwise, as you correctly point out, we end up with infinite consumption being optimal and we don't even have an economic problem (loosely defining economics as the science of incentives regarding the allocation SCARCE goods).</p>
<p>A set is bounded if there exists an upper and a lower bound. We have the lower bound of <span class="math-container">$x_i\geq 0$</span> for all goods <span class="math-container">$i$</span> by construction of the problem. We can only have an upper bound on any quantity <span class="math-container">$x_i$</span> if <span class="math-container">$p_i&gt;0$</span> and a finite <span class="math-container">$w$</span>. Suppose some finite <span class="math-container">$\overline x= (\overline x_i)_{i \leq l}$</span> were an upper bound and <span class="math-container">$p_i=0$</span> for some <span class="math-container">$i$</span>. Then you can see that this <span class="math-container">$\overline x_i$</span> cannot be the upper bound, because <span class="math-container">$\overline x_i+\epsilon$</span> is also in the budget set for any <span class="math-container">$\epsilon&gt;0$</span>.</p>
<p>EDIT: Sorry, I ignored the question on non-emptyness. Take any finite income and positive prices and note that <span class="math-container">$x$</span> is a continuous choice. Take the maximum price <span class="math-container">$\overline p= \max_i (p_i)_{i\leq l}$</span> and then consider a quantity vector <span class="math-container">$\widehat x= (\widehat x_i)_{i\leq l}$</span> with <span class="math-container">$\widehat x_i= \frac{w}{l \overline p}$</span> then <span class="math-container">$\sum \widehat x_i p_i \leq \sum \widehat x_i \overline p = l \frac{w}{l \overline p} \underline p=w$</span>. So quantity vector <span class="math-container">$\widehat x$</span> is in the budget set such that it cannot be empty.</p>

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2020-12-08T13:28:46.340Z
Wow! thank you for such a detailed comment. This was very helpful!
Doubts
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2020-12-08T14:16:42.367Z
Sure! Consider accepting the answer if it helped. Then it won't formally remain "unanswered."
Bayesian
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2020-12-08T11:11:51.377Z
2020-12-08T11:34:24.073Z
2020-12-08T11:34:24.073Z
Bayesian
<p>I am fairly new to economics, and we were introduced to budget sets,
The professor mentioned that the budget set <span class="math-container">$B(p,w) = \{x \in R^{l}_{+}: px \leq w\}$</span> is non empty and closed  - I could prove the latter (the closed part) by taking sequences, but how do I claim that it is  non empty?</p>
<p>Also, if the price of one commodity goes to zero, what happens to boundedness of <span class="math-container">$B(p,w)$</span>? - I know zero prices means &quot;free goods&quot;, and person can purchase infinite amounts of it. but how to prove it rigorously?</p>

<microeconomics><consumer-theory><demand><real-analysis>
Budget Set- closed and boundedness
Doubts
# Budget Set- closed and boundedness

(Asked by: username_0 on 2020-12-08)

I am fairly new to economics, and we were introduced to budget sets,
The professor mentioned that the budget set $B(p,w) = \{x \in R^{l}\_{+}: px \leq w\}$ is non empty and closed - I could prove the latter (the closed part) by taking sequences, but how do I claim that it is non empty?

Also, if the price of one commodity goes to zero, what happens to boundedness of $B(p,w)$? - I know zero prices means "free goods", and person can purchase infinite amounts of it. but how to prove it rigorously?

**Tags:** microeconomics, consumer-theory, demand, real-analysis

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## 1 answers

### Answer by username_1 on 2020-12-08. Score: 2 | Accepted answer

The budget set is always defined given a price vector $p=(p\_i)\_{i\leq l}$ (it seems like $l$ is the number of goods in your problem) and an income $w$. We usually implicitly assume that prices are strictly positive and the income is finite.
Otherwise, as you correctly point out, we end up with infinite consumption being optimal and we don't even have an economic problem (loosely defining economics as the science of incentives regarding the allocation SCARCE goods).

A set is bounded if there exists an upper and a lower bound. We have the lower bound of $x\_i\geq 0$ for all goods $i$ by construction of the problem. We can only have an upper bound on any quantity $x\_i$ if $p\_i>0$ and a finite $w$. Suppose some finite $\overline x= (\overline x\_i)\_{i \leq l}$ were an upper bound and $p\_i=0$ for some $i$. Then you can see that this $\overline x\_i$ cannot be the upper bound, because $\overline x\_i+\epsilon$ is also in the budget set for any $\epsilon>0$.

EDIT: Sorry, I ignored the question on non-emptyness. Take any finite income and positive prices and note that $x$ is a continuous choice. Take the maximum price $\overline p= \max\_i (p\_i)\_{i\leq l}$ and then consider a quantity vector $\widehat x= (\widehat x\_i)\_{i\leq l}$ with $\widehat x\_i= \frac{w}{l \overline p}$ then $\sum \widehat x\_i p\_i \leq \sum \widehat x\_i \overline p = l \frac{w}{l \overline p} \underline p=w$. So quantity vector $\widehat x$ is in the budget set such that it cannot be empty.

> **Comments:**
> **username_0**: Wow! thank you for such a detailed comment. This was very helpful!  
> **username_1**: Sure! Consider accepting the answer if it helped. Then it won't formally remain "unanswered."  

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