Comparing Sunspot Equilibrium And Lottery Equilibrium Allocations: The Finite Case*

Abstract

Sunspot equilibrium and lottery equilibrium are two stochastic solution concepts for nonstochastic economies. We compare these concepts in a class of completely finite, (possibly) nonconvex exchange economies with perfect markets, which requires extending the lottery model to the finite case. Every equilibrium allocation of our lottery model is also a sunspot equilibrium allocation. The converse is almost always true. There are exceptions, however: For some economies, there exist sunspot equilibrium allocations with no lottery equilibrium counterpart.

General Equilibrium With Nonconvexities, Sunspots, and Money

“…We do not actually need a continuum here, but it is adopted because in the monetary models discussed below, as in much of the literature, when combined with random matching it generates anonymity. It is worth mentioning that we could get away with a finite number of agents (for the GE results, and also for the monetary results as long as we have some other way to motivate anonymity) because we use sunspots as opposed to lotteries; the latter generally need the law of large numbers while the former do not (Shell and Wright 1993;Garratt, Keister and Shell 2004).…”

Walrasian equilibrium in an exchange economy with indivisibilities

“…Therefore, the Euler solution p(t) defined on [0, b] must be unique since the Euler solution is an absolutely continuous function that satisfies (11). Now it follows from the Upper Envelope Theorem that…”

Introduction to Sunspots and Lotteries

“…However, agents are not allowed to make trades conditional on these events (they are known only by the auctioneer) and hence the extended lottery model is different from the sunspots model where extrinsic-state contingent trades are allowed. For more on this see Garratt, Keister, and Shell [15]. 11 In fact, some SE based on a finite sunspot device do not survive refinement of the state space to (finer) finite partitions.…”

“…For the SE from this restricted model, there is in equilibrium a unique price for each feasible event, prices are linear in events, but the price of a given probability is not necessarily unique. In an unpublished paper, Garratt, Keister, and Shell [15] provide examples of SE allocations based on finite randomizing devices that are not LE allocations. Nonetheless, Garratt, Keister, and Shell [15] show that for most finite randomizing devices the set of LE allocations and the set of SE allocations are equivalent.…”

“…The extensive SE literature (both theoretical and applied) has been reviewed in Shell [38], Shell and Smith [39], and most recently in Spear and Wright [41]; we do not provide a systematic review here. Sources of proper sunspot equilibria include: the overlapping-generations "double-infinity" of consumers and dated commodities 15 , restricted market participation (as naturally arises in OG and other economies), missing markets and other budget restrictions, money and other sources of "indeterminacy", imperfect competition, externalities (an important case of missing markets), private information, non-convexities, regulatory restrictions, bounded rationality, and optimal banking mechanisms.…”