Proof of The First Fundamental Theorem
The first fundamental theorem of welfare economics states that any Walrasian equilibrium is Pareto-efficient. This was first demonstrated graphically by economist Abba Lerner and mathematically by economists Harold Hotelling, Oskar Lange, Maurice Allais, Kenneth Arrow and Gérard Debreu. The theorem holds under general conditions.
The only assumption needed (in addition to complete markets and price-taking behavior) is the relatively weak assumption of local nonsatiation of preferences. In particular, no convexity assumptions are needed.
Read more about this topic: Fundamental Theorems Of Welfare Economics
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