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?

  • 1
    $\begingroup$ 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 $b_n \in F (a_n)$. $\endgroup$
    – Amit
    Nov 15, 2023 at 8:24
  • $\begingroup$ And the budget correspondence will not be lhc on all of $\mathbb{R}^l_+$. Goods with a price of zero will cause trouble. $\endgroup$ Nov 15, 2023 at 23:21

1 Answer 1


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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