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 ∗

!