# Bolzano-Weierstrass Theorem and Pareto Efficient Allocation

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?

• In order to understand any of the proofs you have to grasp either the concept of "nested intervals" ( 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 Jul 22 '19 at 9:57
• Perhaps insofar as Bolzano-Weierstrass theorem is used to prove the extreme value theorem, which in turn can be invoked to easily establish the existence of Pareto optimality. See also: quora.com/… – Herr K. Jul 22 '19 at 18:08

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.