<p>What one needs are two facts:</p>

<ol>
<li><p><strong>Scalarization/Sufficiency for Pareto Optimality</strong>
Suppose the economy has finitely many agents <span class="math-container">$u_i$</span>, <span class="math-container">$i = 1, \cdots, I$</span>, with feasible allocations given by some <span class="math-container">$X \subset \mathbb{R}^I$</span>. If
<span class="math-container">$x = (x_1, \cdots, x_I) \in X$</span> solves the problem
<span class="math-container">$$
\max_{x \in X} \sum_{i=1}^I \lambda_i u_i(x_i),
$$</span>
where <span class="math-container">$\lambda_i &gt; 0$</span> for all <span class="math-container">$i$</span>, then <span class="math-container">$x$</span> is Pareto optimal. This maximization problem is called the social planner's problem with social weights <span class="math-container">$\lambda_i$</span>.
(The converse is not true. In other words, not all Pareto allocations arise as such, not even when <span class="math-container">$u_i$</span>'s are concave.)</p></li>
<li><p>Continuous functions map compact sets to compact sets. (The Bolzano–Weierstrass gives a characterization of compactness on <span class="math-container">$\mathbb{R}^n$</span>---closed and bounded.) </p></li>
</ol>

<p>So if the preferences are continuous and feasible set <span class="math-container">$X$</span> is compact, existence of Pareto optimal allocation follows immediately. Choose any social weights <span class="math-container">$\lambda_i &gt; 0$</span>. A solution of the resulting social planner's problem is a Pareto optimal allocation.</p>

CC BY-SA 4.0
2019-11-16T06:42:28.717Z
2019-11-20T03:54:58.293Z
2019-11-20T03:54:58.293Z
Michael
<p>Wikipedia says 'The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.' 
However, I couldn't find a compelling simple proof for this theorem anywhere. Could someone provide a proof for the theorem in a simple fashion that is understandable for Undergrad students in Economics who have done a course in Analysis?</p>

CC BY-SA 4.0
2019-07-22T09:57:56.673Z
In order to understand any of the proofs you have to grasp either the concept of "nested intervals" ( https://en.wikipedia.org/wiki/Nested_intervals ) or the concept of a subsequence. If you get those the proof is straight foreward. Anyway these arent complicated. They are taught in intro to analysis in all undergrad programs in math.
Grada Gukovic
CC BY-SA 4.0
2019-07-22T18:08:39.140Z
Perhaps insofar as Bolzano-Weierstrass theorem is [used to prove the extreme value theorem](https://en.wikipedia.org/wiki/Extreme_value_theorem#Proof_of_the_extreme_value_theorem), which in turn can be invoked to easily establish the existence of Pareto optimality. See also: https://www.quora.com/How-is-the-Bolzano-Weierstrass-theorem-used-in-general-equilibrium-theory/answer/Amit-Goyal-135
Herr K.
<mathematical-economics>
Bolzano-Weierstrass Theorem and Pareto Efficient Allocation
Harper Lee
# Bolzano-Weierstrass Theorem and Pareto Efficient Allocation

(Asked by: username_0 on 2019-07-22)

Wikipedia says 'The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.'
However, I couldn't find a compelling simple proof for this theorem anywhere. Could someone provide a proof for the theorem in a simple fashion that is understandable for Undergrad students in Economics who have done a course in Analysis?

**Tags:** mathematical-economics

> **Comments:**
> **username_1**: In order to understand any of the proofs you have to grasp either the concept of "nested intervals" ( https://en.wikipedia.org/wiki/Nested_intervals ) or the concept of a subsequence. If you get those the proof is straight foreward. Anyway these arent complicated. They are taught in intro to analysis in all undergrad programs in math.  
> **username_2**: Perhaps insofar as Bolzano-Weierstrass theorem is [used to prove the extreme value theorem](https://en.wikipedia.org/wiki/Extreme_value_theorem#Proof_of_the_extreme_value_theorem), which in turn can be invoked to easily establish the existence of Pareto optimality. See also: https://www.quora.com/How-is-the-Bolzano-Weierstrass-theorem-used-in-general-equilibrium-theory/answer/Amit-Goyal-135  

---

## 1 answers

### Answer by username_3 on 2019-11-16. Score: 1

What one needs are two facts:

1. **Scalarization/Sufficiency for Pareto Optimality**
   Suppose the economy has finitely many agents $u\_i$, $i = 1, \cdots, I$, with feasible allocations given by some $X \subset \mathbb{R}^I$. If
   $x = (x\_1, \cdots, x\_I) \in X$ solves the problem
   $$
   \max\_{x \in X} \sum\_{i=1}^I \lambda\_i u\_i(x\_i),
   $$
   where $\lambda\_i > 0$ for all $i$, then $x$ is Pareto optimal. This maximization problem is called the social planner's problem with social weights $\lambda\_i$.
   (The converse is not true. In other words, not all Pareto allocations arise as such, not even when $u\_i$'s are concave.)
2. Continuous functions map compact sets to compact sets. (The Bolzano–Weierstrass gives a characterization of compactness on $\mathbb{R}^n$---closed and bounded.)

So if the preferences are continuous and feasible set $X$ is compact, existence of Pareto optimal allocation follows immediately. Choose any social weights $\lambda\_i > 0$. A solution of the resulting social planner's problem is a Pareto optimal allocation.


---

