Economic growth and economic development 253

Introduction to Modern Economic Growth
we have the sum over an infinite number of households. However, since endowments
P
∗
are finite, the assumption that ∞
j=0 pj < ∞ ensures that the sums in (5.15) are

indeed finite and the rest of the proof goes through exactly as in the proof of Theorem
5.5.

Ô

Theorem 5.6 will be particularly useful when we discuss overlapping generation
models.
We next briefly discuss the Second Welfare Theorem, which is the converse of
the First Welfare Theorem. It answers the question of whether a Pareto optimal
allocation can be decentralized as a competitive equilibrium. Interestingly, for the
Second Welfare Theorem whether or not H is finite is not important, but we need

to impose much more structure, essentially convexity, for consumption and produc-

tion sets and preferences. This is because the Second Welfare Theorem essentially
involves an existence of equilibrium argument, which runs into problems in the presence of non-convexities. A complete proof of the Second Welfare Theorem utilizes
more advanced tools than those we use in the rest of this book, so we only present
a sketch of the proof of this theorem.
Theorem 5.7. (Second Welfare Theorem) Consider a Pareto optimal allocation (x∗∗ , y∗∗ ) yielding utility allocation {ui∗∗ }i∈H to households. Suppose that all
production and consumption sets are convex and all utility functions {ui (·)}i∈H are
quasi-concave. Then there exists an endowment and share allocation (ω∗∗ , θ ∗∗ ) such

that economy E ≡ (H, F, u, ω ∗∗ , Y, X, θ ∗∗ ) has a competitive equilibrium (x∗∗ , y∗∗ ,p∗∗ ).
Proof. (Sketch) The proof idea goes as follows: we first represent a Pareto

optimum as a point of tangency between a feasibility set constructed from the endowments and the production sets of firms. Convexity of production sets implies
that the feasibility set is convex. We then construct the “more preferred” set, i.e.,
the set of consumption bundles that are feasible and yield at least as much utility
as {ui∗∗ }i∈H to all consumers. Since all consumption sets are convex and utility

functions are quasi-concave, this “more preferred” set is also convex. By construc-

tion, these two sets have (x∗∗ , y∗∗ ) as a common point and have disjoint interiors,
which are also convex sets. We then apply a standard separating hyperplane theorem, which states that there exists a hyperplane passing through this point which
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