Economic growth and economic development 254

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 240