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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?

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  • $\begingroup$ 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. $\endgroup$ – Grada Gukovic Jul 22 '19 at 9:57
  • $\begingroup$ 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/… $\endgroup$ – Herr K. Jul 22 '19 at 18:08
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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.

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