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I know this is a basic question. Can I get a mathematical proof for "Any allocation in the core of an economy is also Pareto optimal." I found this statement on some PPT online.

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  • $\begingroup$ If you define a game in terms of Cooperative Game Theory, you can show NickCHK's answer "mathematically". For a given game you have the characteristic function $v$ defined on all subsets (coalitions) of $C \subseteq N$ and some imputation $x=(x_1, x_2, . . . , x_n)$ (imputation satisfies usual definitions of individual and group rationality). From core property $\sum_{i \in S} x_i \ge v(C)$ for all $C$, thus $\sum_{i \in N} y_i \ge v(N)$. So all imputations in the core are Pareto Optimal. $\endgroup$ – kitsune Apr 18 '17 at 1:00
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An allocation is defined as being a part of the "core" of an economy if there's no coalition of people that blocks the allocation. A coalition will block an allocation if all of its members could be made better off by another allocation (or, more weakly, if at least one of its members could be made better off without others being made worse off).

And so, by definition, for an allocation to be in the core of the economy, there must not be another allocation which any coalition prefers to the given allocation. And since one coalition that could be formed is the "grand coalition" (i.e. everyone in the economy), any allocation in the core must be Pareto optimal, or else the grand coalition would block it.

This probably isn't much more involved or mathematical than the PPT you pulled it from, but that's because it's basically true by definition.

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