The Structure of Sunspot Equilibria: The Role of Multiplicity

Abstract

This paper examines the structure of sunspot equilibria in a standard two period exchange economy with real assets. We show that for a generic choice of utility functions and endowments, there exists an open set of real asset structures whose payoffs are independent of sunspots such that the economy with this asset structure has a regular sunspot equilibrium. An important implication of our result is that the multiplicity of non-sunspot equilibria is not necessary for the existence of sunspot equilibria. Our technique is general and can be applied to show the existence of sunspot equilibria in other frameworks. This paper analyses the structure of sunspot equilibria in a standard two period competitive exchange economy, with more than one consumption good in each period and a real asset which pays a bundle of commodities. There is no uncertainty except for sunspots. We show that for an open and dense subset of the set of utility functions and endowments, there exists an open set of real asset structures, with payoffs independent of sunspots, such that the resulting economy with an asset structure in this set has a regular sunspot equilibrium. We also show that this result is tight: in the space of utility functions, endowments and asset structures there is in fact an open set of economies for which sunspot equilibria do not exist. Moreover, our argument provides a systematic method which allows us to find, generically, all the sunspot equilibria of a given economy. Thus it can be used as a tool for studying the global structure of the set of sunspot equilibria, not just the existence of a sunspot equilibrium. The dependence of the existence of sunspot equilibria on the particular specification of the payoff of the asset provides an interesting contrast with the fact that the set of equilibrium allocations of the underlying certainty economy does not depend on the asset structure. Recall that when markets are complete, preferences and endowments completely determine the set of competitive equilibria. In our setting, markets are complete in the underlying certainty economy. So if payoffs change, the price of the asset will change accordingly but there is no effect on the equilibrium consumption of the certainty economy. Thus the fundamentals of the certainty economy are the distribution of endowments and preferences among consumers, and our result implies that the existence of sunspot