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Introduction to Modern Economic Growth and therefore, by multiplying both sides by p∗ and exploiting (5.16), we have ! Ã X X X f p∗ · xˆij ≤ p∗ · ωij + yˆj i∈H i∈H ≤ p∗ · Ã X i∈H f ∈F ωij + X f ∈F ! yjf ∗ , which contradicts (5.15), establishing that any competitive equilibrium allocation (x , y ) is Pareto optimal Ô The proof of the First Welfare Theorem is both intuitive and simple The proof is based on two intuitive ideas First, if another allocation Pareto dominates the competitive equilibrium, then it must be non-affordable in the competitive equilibrium Second, profit-maximization implies that any competitive equilibrium already contains the maximal set of affordable allocations It is also simple since it only uses the summation of the values of commodities at a given price vector In particular, it makes no convexity assumption However, the proof also highlights the importance of the feature that the relevant sums exist and are finite Otherwise, the last step would lead to the conclusion that “∞ < ∞” which may or may not be a contra- diction The fact that these sums exist, in turn, followed from two assumptions: finiteness of the number of individuals and non-satiation However, as noted before, working with economies that have only a finite number of households is not always sufficient for our purposes For this reason, the next theorem turns to the version of the First Welfare Theorem with an infinite number of households For simplicity, here we take H to be a countably infinite set, e.g., H = N The next theorem generalizes the First Welfare Theorem to this case It makes use of an additional assumption to take care of infinite sums Theorem 5.6 (First Welfare Theorem II) Suppose that (x∗ , y∗ , p∗ ) is a competitive equilibrium of the economy E ≡ (H, F, u, ω, Y, X, θ) with H count- ably infinite Assume that all households are locally non-satiated at x∗ and that P∞ ∗ ∗ ∗ ∗ j=0 pj < ∞ Then (x , y , p ) is Pareto optimal Proof The proof is the same as that of Theorem 5.5, with a major difference Local non-satiation does not guarantee that the summations are finite (5.15), since 238