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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, 2017 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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Not exactly, you need to assume locally non-satiated preference.

The following example is from Mark Voorneveld's lecture notes:

"Consider a pure exchange economy with two consumers and two commodities. The first consumer’s preferences are represented by the utility function u1(x) = x1x2, the second consumer’s preferences by a constant utility function: he is indifferent between all commodity bundles. If endowment is w1 = w1 = (1,1), then (p,x) = ((1,1),(1,1),(1, 1)) (i.e., prices are equal and each consumer sticks to the initial endowment) is a Walrasian equilibrium. The allocation lies in the core. But the allocation is not Pareto optimal: giving the total endowment to the first consumer makes him better off, while not affecting the happiness of the second consumer."

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