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Introduction to Modern Economic Growth
separates the interiors of the set of feasible allocations and the “more preferred”
set. When the number of commodities is finite, a standard separating hyperplane
theorem can be used without imposing additional conditions. When the number of
commodities is infinite, we need to use the Hahn-Banach theorem, which requires
us to check additional technical details (in particular, we need to ensure that the
set Y has an interior point). The normal of the separating hyperplane (the vector orthogonal to the separating hyperplane) gives the price vector p∗∗ . Finally,
we choose the distribution of endowments and shares in order to ensure that the
resulting competitive equilibrium lead to (x , y ).

Ô

The conditions for the Second Welfare Theorem are more difficult to satisfy
than the First Welfare Theorem because of the convexity requirements. In many
ways, it is also the more important of the two theorems. While the First Welfare
Theorem is celebrated as a formalization of Adam Smith’s invisible hand, the Second
Welfare Theorem establishes the stronger results that any Pareto optimal allocation
can be decentralized as a competitive equilibrium. An immediate corollary of this
is an existence result; since the Pareto optimal allocation can be decentralized as
a competitive equilibrium, a competitive equilibrium must exist (at least for the
endowments leading to Pareto optimal allocations).
The Second Welfare Theorem motivates many macroeconomists to look for the
set of Pareto optimal allocations instead of explicitly characterizing competitive
equilibria. This is especially useful in dynamic models where sometimes competitive
equilibria can be quite difficult to characterize or even to specify, while social welfare
maximizing allocations are more straightforward.
The real power of the Second Welfare Theorem in dynamic macro models comes
when we combine it with models that admit a representative household. Recall that
Theorem 5.3 shows an equivalence between Pareto optimal allocations and optimal
allocations for the representative household. In certain models, including many–
but not all–growth models studied in this book, the combination of a representative

consumer and the Second Welfare Theorem enables us to characterize the optimal
growth allocation that maximizes the utility of the representative household and
assert that this will correspond to a competitive equilibrium.
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