Economic growth and economic development 250

Introduction to Modern Economic Growth Definition 5.2.. Our next result is the celebrated First Welfare Theorem for competitive economies.. Before presenting this result, we need the fol

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 (ˆx, ˆy) such that ˆxi

∈ Xi,

ˆf ∈ Yf for all f ∈ F,

X

i∈H

ˆ

xij ≤X

i∈H

ωij +X

f ∈F

ˆjf for all j ∈ N, and

ui¡ ˆ

xi¢

≥ 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 : Xi → 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 economyE ≡ (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 (ˆx, ˆy) such that ui(ˆxi)≥ ui(xi) for all i∈ H and ui(ˆxi) > ui(xi) for all i∈ H0, whereH0

is a non-empty subset ofH

Since (x∗, y∗, p∗) is a competitive equilibrium, it must be the case that for all

i∈ H,

p∗·ˆxi ≥ p∗· xi∗

(5.13)

= p∗·

Ã

ωi+X

f ∈F

θifyf ∗

!

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