Economic growth and economic development 252

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
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