<p>I think you need to recheck the definition you provided for lhc. This is because the definition you provided is always true for any correspondence at any point <span class="math-container">$(a,b)$</span> satisfying <span class="math-container">$b\in F(a)$</span>. To see this, consider constant sequences <span class="math-container">$a_n=a$</span>, and <span class="math-container">$b_n=b$</span>. Clearly, <span class="math-container">$a_n\rightarrow a$</span>, and <span class="math-container">$b_n\rightarrow b$</span>. Additionally, <span class="math-container">$b_n\in F(a_n)$</span> for all <span class="math-container">$n\in \mathbb{N}$</span> also holds because <span class="math-container">$a_n=a$</span>, <span class="math-container">$b_n=b$</span> and <span class="math-container">$b\in F(a)$</span>.</p>

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2023-11-14T05:59:39.897Z
2023-11-14T06:04:52.603Z
2023-11-14T06:04:52.603Z
Amit
<p>Given income <span class="math-container">$y$</span> and a vector of commodity prices <span class="math-container">$p\in R_+^L$</span>, the set of feasible consumption bundles is described by the budget correspondence, <span class="math-container">$B(p,y)=\{x\in R_+^L:px\leq y\}$</span>. <span class="math-container">$B(p,y)$</span> is both upper and lower hemicontinuous, as proved in Chapter 8 Problem 2.2 in Mathematical Methods and Models for Economists by de la Fuente.</p>
<p>Here is the proof: To establish that <span class="math-container">$B$</span> is an lhc correspondence, we need to show that given any price-income seqwuence <span class="math-container">$\{(p_n,y_n)\}$</span> converging to <span class="math-container">$(p,y)&gt;&gt;0$</span> and an arbitrary point <span class="math-container">$x\in B(p,y)$</span>, there exists a companion sequence of consumption bundles <span class="math-container">$\{x_n\}$</span> with <span class="math-container">$x_n\in B(p_n,y_n)$</span> for all <span class="math-container">$n$</span> that converges to <span class="math-container">$x$</span>.</p>
<p>Let <span class="math-container">$x_n=x$</span> if <span class="math-container">$x\in B(p_n,y_n)$</span> and <span class="math-container">$x_n=\frac{y_n}{p_nx}x$</span> if otherwise. Notice that <span class="math-container">$x_n$</span> is feasible for <span class="math-container">$(p_n,y_n)$</span> by construction, because <span class="math-container">$x_n$</span> is defined as the largest fraction of the bundle <span class="math-container">$x$</span> that the consumer can afford with income <span class="math-container">$y_n$</span> and prices <span class="math-container">$p_n$</span>. It's also clear that <span class="math-container">$\{x_n\}\to x$</span>. If <span class="math-container">$x$</span> lies in the interior of the budget set, then we have <span class="math-container">$x_n=x$</span> for <span class="math-container">$n$</span> sufficiently large. Otherwise, <span class="math-container">$y=px$</span> and <span class="math-container">$\lim x_n=\lim\frac{y_n}{p_nx}x=\frac{y}{px}x=x$</span>.</p>
<p>You can also see <a href="https://economics.stackexchange.com/a/37430/33150">here</a> for the proof.</p>
<p><strong>But I'm having trouble understanding this proof.</strong> Here is the definition of lhc: A correspondence is lhc at <span class="math-container">$a$</span> if <span class="math-container">$\forall b\in F(a)$</span>, <span class="math-container">$\exists a_n$</span> and <span class="math-container">$b_n$</span> such that <span class="math-container">$a_n\to a$</span> and <span class="math-container">$b_n\to b$</span>.</p>
<p>But in this proof, <span class="math-container">$p_n$</span> is taken as an arbitrary sequence. <span class="math-container">$p_n$</span> should correspond to the <span class="math-container">$a_n$</span> sequence in the definition of lhc. In the definition, <span class="math-container">$a_n$</span> should be &quot;there exists&quot;, not &quot;for all&quot;. Can someone explain why?</p>

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2023-11-15T08:24:46.227Z
This is the definition of lhc: $F$ is lower hemicontinuous at $a$ if for every sequence $(a_n)$ converging to $a$ and every $b \in F(a)$ there exists a sequence $(b_n)$ converging to $b$ with $b_n \in F (a_n)$.
Amit
CC BY-SA 4.0
2023-11-15T23:21:04.533Z
And the budget correspondence will not be lhc on all of $\mathbb{R}^l_+$. Goods with a price of zero will cause trouble.
Michael Greinecker
<microeconomics>
Why is the budget correspondence lower hemicontinuous?
Ludwig Gershwin
# Why is the budget correspondence lower hemicontinuous?

(Asked by: username_0 on 2023-11-14)

Given income $y$ and a vector of commodity prices $p\in R\_+^L$, the set of feasible consumption bundles is described by the budget correspondence, $B(p,y)=\{x\in R\_+^L:px\leq y\}$. $B(p,y)$ is both upper and lower hemicontinuous, as proved in Chapter 8 Problem 2.2 in Mathematical Methods and Models for Economists by de la Fuente.

Here is the proof: To establish that $B$ is an lhc correspondence, we need to show that given any price-income seqwuence $\{(p\_n,y\_n)\}$ converging to $(p,y)>>0$ and an arbitrary point $x\in B(p,y)$, there exists a companion sequence of consumption bundles $\{x\_n\}$ with $x\_n\in B(p\_n,y\_n)$ for all $n$ that converges to $x$.

Let $x\_n=x$ if $x\in B(p\_n,y\_n)$ and $x\_n=\frac{y\_n}{p\_nx}x$ if otherwise. Notice that $x\_n$ is feasible for $(p\_n,y\_n)$ by construction, because $x\_n$ is defined as the largest fraction of the bundle $x$ that the consumer can afford with income $y\_n$ and prices $p\_n$. It's also clear that $\{x\_n\}\to x$. If $x$ lies in the interior of the budget set, then we have $x\_n=x$ for $n$ sufficiently large. Otherwise, $y=px$ and $\lim x\_n=\lim\frac{y\_n}{p\_nx}x=\frac{y}{px}x=x$.

You can also see [here](https://economics.stackexchange.com/a/37430/33150) for the proof.

**But I'm having trouble understanding this proof.** Here is the definition of lhc: A correspondence is lhc at $a$ if $\forall b\in F(a)$, $\exists a\_n$ and $b\_n$ such that $a\_n\to a$ and $b\_n\to b$.

But in this proof, $p\_n$ is taken as an arbitrary sequence. $p\_n$ should correspond to the $a\_n$ sequence in the definition of lhc. In the definition, $a\_n$ should be "there exists", not "for all". Can someone explain why?

**Tags:** microeconomics

> **Comments:**
> **username_1**: This is the definition of lhc: $F$ is lower hemicontinuous at $a$ if for every sequence $(a_n)$ converging to $a$ and every $b \in F(a)$ there exists a sequence $(b_n)$ converging to $b$ with $b_n \in F (a_n)$.  
> **username_2**: And the budget correspondence will not be lhc on all of $\mathbb{R}^l_+$. Goods with a price of zero will cause trouble.  

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## 1 answers

### Answer by username_1 on 2023-11-14. Score: 4 | Accepted answer

I think you need to recheck the definition you provided for lhc. This is because the definition you provided is always true for any correspondence at any point $(a,b)$ satisfying $b\in F(a)$. To see this, consider constant sequences $a\_n=a$, and $b\_n=b$. Clearly, $a\_n\rightarrow a$, and $b\_n\rightarrow b$. Additionally, $b\_n\in F(a\_n)$ for all $n\in \mathbb{N}$ also holds because $a\_n=a$, $b\_n=b$ and $b\in F(a)$.


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