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Introduction to Modern Economic Growth we have the sum over an infinite number of households However, since endowments P ∗ are finite, the assumption that ∞ j=0 pj < ∞ ensures that the sums in (5.15) are indeed finite and the rest of the proof goes through exactly as in the proof of Theorem 5.5 Ô Theorem 5.6 will be particularly useful when we discuss overlapping generation models We next briefly discuss the Second Welfare Theorem, which is the converse of the First Welfare Theorem It answers the question of whether a Pareto optimal allocation can be decentralized as a competitive equilibrium Interestingly, for the Second Welfare Theorem whether or not H is finite is not important, but we need to impose much more structure, essentially convexity, for consumption and produc- tion sets and preferences This is because the Second Welfare Theorem essentially involves an existence of equilibrium argument, which runs into problems in the presence of non-convexities A complete proof of the Second Welfare Theorem utilizes more advanced tools than those we use in the rest of this book, so we only present a sketch of the proof of this theorem Theorem 5.7 (Second Welfare Theorem) Consider a Pareto optimal allocation (x∗∗ , y∗∗ ) yielding utility allocation {ui∗∗ }i∈H to households Suppose that all production and consumption sets are convex and all utility functions {ui (·)}i∈H are quasi-concave Then there exists an endowment and share allocation (ω∗∗ , θ ∗∗ ) such that economy E ≡ (H, F, u, ω ∗∗ , Y, X, θ ∗∗ ) has a competitive equilibrium (x∗∗ , y∗∗ ,p∗∗ ) Proof (Sketch) The proof idea goes as follows: we first represent a Pareto optimum as a point of tangency between a feasibility set constructed from the endowments and the production sets of firms Convexity of production sets implies that the feasibility set is convex We then construct the “more preferred” set, i.e., the set of consumption bundles that are feasible and yield at least as much utility as {ui∗∗ }i∈H to all consumers Since all consumption sets are convex and utility functions are quasi-concave, this “more preferred” set is also convex By construc- tion, these two sets have (x∗∗ , y∗∗ ) as a common point and have disjoint interiors, which are also convex sets We then apply a standard separating hyperplane theorem, which states that there exists a hyperplane passing through this point which 239