# Pareto Optimality and Core

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.

• 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. – kitsune Apr 18 '17 at 1:00