Economic growth and economic development 250

Introduction to Modern Economic Growth Definition 5.2 A feasible allocation (x, y) for economy E ≡ (H, F, u, ω, Y, X, θ) is Pareto optimal if there exists no other feasible allocation (ˆ x, y ˆ) such that xˆi ∈ X i , yˆf ∈ Y f for all f ∈ F, X i∈H xˆij ≤ X ωij + i∈H X f ∈F yˆjf for all j ∈ N, and ¡ ¢ ¡ ¢ ui xˆi ≥ ui xi for all i ∈ H with at least one strict inequality Our next result is the celebrated First Welfare Theorem for competitive economies Before presenting this result, we need the following definition Definition 5.3 Household i ∈ H is locally non-satiated at xi if ui (xi ) is strictly increasing in at least one of its arguments at xi and ui (xi ) < ∞ The latter requirement in this definition is already implied by the fact that ui : X i → R, but it is included for additional emphasis, since it is important for the proof and also because if in fact we had ui (xi ) = ∞, we could not meaningfully talk about ui (xi ) being strictly increasing Theorem 5.5 (First Welfare Theorem I) Suppose that (x∗ , y∗ , p∗ ) is a competitive equilibrium of economy E ≡ (H, F, u, ω, Y, X, θ) with H finite Assume that all households are locally non-satiated at x∗ Then (x∗ , y∗ ) is Pareto optimal Proof To obtain a contradiction, suppose that there exists a feasible (ˆ x, y ˆ) xi ) ≥ ui (xi ) for all i ∈ H and ui (ˆ xi ) > ui (xi ) for all i ∈ H0 , where H0 such that ui (ˆ is a non-empty subset of H Since (x∗ , y∗ , p∗ ) is a competitive equilibrium, it must be the case that for all i ∈ H, (5.13) p∗ ·ˆ xi ≥ p∗ · xi∗ Ã = p∗ · ωi + 236 X f ∈F θif y f ∗ !