Why is the budget correspondence lower hemicontinuous?

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 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?

• 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
Nov 15, 2023 at 8:24
• And the budget correspondence will not be lhc on all of $\mathbb{R}^l_+$. Goods with a price of zero will cause trouble. Nov 15, 2023 at 23:21

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 $$b\in F(a)$$.