Overview and summary

The ideas treated in this volume have focused on a single unifying idea as a framework for analyzing economic activity: the general equilibrium of a competitive market. General equilibrium – treating all markets and their interactions simultaneously – is thought here to be the appropriate model to decide whether there are well-defined solutions to the economic decision-making mechanism and whether they have efficiency properties making them desirable. This class of questions goes back well over two hundred years in scientific economics. The way we answer them here is the Arrow-Debreu version of the Walrasian economic model (Arrow and Debreu (1954), Arrow (1951)).

We have come to several principal conclusions:

1. (i) The models have well-defined solutions; equilibria exist. Sufficient conditions for this result are scarcity, continuity, and convexity on both the consumer and producer sides. This is true both in settings where demand and supply are characterized as point-valued functions and optimizing behavior is uniquely well defined (Existence of General Equilibrium Theorems 1.2, 7.1, and 11.1) and where they are characterized as set-valued mappings (Theorem 17.7) recognizing the multiplicity of equally profitable production plans a firm may have or the variety of equally satisfactory affordable consumption plans households may consider.

2. (ii) The Brouwer Fixed-Point Theorem, Kakutani Fixed-Point Theorem, or their equivalent are necessary to establish the existence of equilibrium result (Uzawa Equivalence Theorem 11.2).

3. (iii) General equilibrium allocations are Pareto efficient (First Fundamental Theorem of Welfare Economics, Theorem 12.1). The market ensures against the waste of scarce resources. This is a surprisingly robust result, depending on scarcity and continuity, but not requiring convexity.

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