In this paper we argue that indeterminacy of equilibrium is a possibility inherent in economies with a double infinity of agents and goods, large-square economies. We develop a framework that is quite different from the overlapping generations one and that is amenable to analysis by means of differential calculus in linear spaces. The commodity space is a separable, infinite-dimensional Hilbert space, and each of a continuum of consumers is described by means of an individual excess demand function defined on an open set of prices. In this setting we prove an analog of the Sonnenschein-Mantel-Debreu theorem. Using this result, we show that the set of economies whose equilibrium sets contain manifolds of arbitrary dimension is non-empty and open in the appropriate topology. In contrast, we also show that, if the space of consumers is sufficiently small, then local uniqueness is a generic property of economies. The concept of sufficiently small has a simple mathematical formulation (the derivatives of individual excess demands form a uniformly integrable family) and an equally simple economic interpretation (the variation across consumers is not too great).
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*We would like to thank Darrell Duflie, Jean-Francois Mertens, and Joseph Ostroy for helpful conversations. This research was supported in part by NSF Grants SES 81-20790, SES 85-09484, DMS 82-19339, and DMS 86-02839. A large part of the research reported here was conducted while all four authors were at the Mathematical Sciences Research Institute, Berkeley.
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