Lotteries, Sunspots, and Incentive Constraints

We study a prototypical class of exchange economies with private information and indivisibilities. We establish an equivalence between lottery equilibria and sunspot equilibria and show that the welfare and existence theorems hold. To establish these results, we introduce the concept of the stand-in consumer economy, which is a standard, convex, finite consumer, finite good, pure exchange economy. With decreasing absolute risk aversion and no indivisibilities, we prove that no lotteries are actually used in equilibrium. We provide a simple numerical example with increasing absolute risk aversion in which lotteries are necessarily used in equilibrium. We also show how the equilibrium allocation in this example can be implemented in a sunspot equilibrium. Journal of Economic Literature Classification Numbers: D11, D50, D82.     

Correlated equilibrium and sunspot equilibrium

Summary We show by an example that the sunspot equilibria of a competitive economy are not equivalent to the correlated equilibria if sunspots generate transfers between (extrinsic) states of nature (through a contingent commodities market). Nevertheless, we prove that the sunspot equilibrium allocations of a standard overlapping generations economy coincide with the (strategic form) correlated equilibrium allocations of a natural market game mimicking the economy.     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

A bargaining model in general equilibrium

The goal of the paper is to present a simple model of rational endogenous household formation in a general equilibrium framework in which Pareto optimality at the economy level is not necessarily obtained. The simplest example of household formation is the case in which pairs of individuals engage themselves in a bargaining process on the division of some wealth: if an agreement on the distribution is (not) reached, we can say that the household is (not) formed. The vast majority of existing bargaining models predicts agreements on an efficient outcome. A seminal paper by Crawford (Econometrica 50:607–637, 1982) describes a very simple game with incomplete information in which, even with rational agents, disagreement causes welfare losses. We embed that model in a general equilibrium framework and present some results on equilibria both in the bargaining game and the associated exchange economy. Crawford’s results support Schelling’s intuition on the reasons of disagreement: it may arise if players’ commitments are reversible. Crawford shows that high probabilities of reversibility tend to favor the bargaining impasse, in fact with low probability. We prove that even if those probabilities are arbitrarily close to zero, disagreement is an equilibrium outcome, with high probability. That conclusion seems to be an even stronger support to Schelling’s original viewpoint. In the exchange economy model with that noncooperative bargaining game as a first stage, we present significant examples of economies for which equilibria exist. Because of disagreement, Pareto suboptimal exchange economy equilibria exist for all elements in the utility function and endowment spaces and they may coexist with Pareto optimal equilibria even at the same competitive prices.     

Do sunspots matter under complete ignorance?

In a two-period, sunspot, pure-exchange economy we analyse the case in which agents do not assign subjective probabilistic beliefs to the ‘sunspot activity’. Two generations, each of which is made up of identical agents, populate this economy. Participation in the Arrow securities market is restricted and the generation, which is allowed to trade in assets, can alternatively face uncertainty via two distribution-free decision rules under ‘complete ignorance’ (axiomatized by Milnor [Milnor, J., 1954. Games against nature. In: Thrall, R. Coombs, C., Davis, R. (Eds.), Decision Processes. John Wiley, London, pp. 49–60]): the ‘minimax regret criterion’ [Savage, L.J., 1954. The Foundations of Statistics. Wiley, New York, ch. 9] and the ‘maxmin return criterion’ [Wald, A., 1950. Statistical Decision Functions. Wiley, New York]. When the former is used, then sunspots can matter. In particular, we prove that, if the economy admits two Walrasian equilibria, then a unique sunspot equilibrium always exists. We pin down this equilibrium, determine the prices of the Arrow securities and show that, at these prices, no trade in securities takes place. In the same framework we prove that, with agents using the maxmin return criterion, sunspots do not matter.     

The Structure of Sunspot Equilibria: The Role of Multiplicity

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.     

Welfare and excess volatility of exchange rates

We study the properties of a GEI model with nominal assets, outside money (injected into the economy as in Magill and Quinzii (J Math Econ 21:301–342, 1992)), and multiple currencies. We analyze the existence of monetary equilibria and the structure of the equilibrium set under two different assumptions on the determination of the exchange rates. If currencies are perfect substitutes, equilibrium allocations are indeterminate and, generically, sunspot equilibria exist. Generically, given a nonsunspot equilibrium, there are Pareto improving (and Pareto worsening) sunspot equilibria associated with an increase in the volatility of the future exchange rates. We interpret this property as showing that, in general, there is no clear-cut effect on welfare of the excess volatility of exchange rates, even when due to purely extrinsic phenomena.     

WELFARE GAINS AND LOSSES IN SUNSPOT EQUILIBRIA*

We study a standard two period exchange economy with one nominal asset. As is well known, there is a continuum of sunspot equilibria around each efficient equilibrium. A sunspot equilibrium is inefficient but some households may gain in sunspot equilibria relative to the efficient equilibrium. We show that a household's equilibrium utility level is either locally maximized or locally minimized at the efficient equilibrium, and derive a condition which identifies whether or not a household's utility is locally minimized or maximized.     

Ergodic chaos, learning and sunspot equilibrium

Summary. In this paper we construct sunspot equilibria that arise from chaotic deterministic dynamics. These equilibria are stationary and have absolutely continuous stationary measures. We prove that they can be learned by a simple rule based on the histograms of past state variables. This work gives a theoretical justification for complex deterministic models that might compete with stochastic models to explain real data. Also we prove the stochastic stability of the indeterminate equilibrium.     

No-arbitrage Equilibria with Differential Information: An Existence Proof

On the example of a pure exchange financial economy with two periods incomplete nominal-asset markets and differential information of the adverse selection’s type, Cornet-De Boisdeffre (J Math Econ 38:393-410, 2002) introduced refined concepts of no-arbitrage prices and equilibria, which extended to the asymmetric information setting the classical concepts of the symmetric information literature. We now assess existence issues and extend a standard property of symmetric information models. Namely, we prove that a no-arbitrage equilibrium always exists, as long as financial markets preclude arbitrage, under the same standard conditions, whether agents have symmetric or asymmetric information.     

Options and sunspots in a simple monetary economy

Summary. We study sunspot immunity in a dynamic monetary economy in which consumers are allowed to trade put and call option contracts on the general price level. We define the concept of strong sunspot immunity to characterize economies that have no sunspot equilibria regardless of the number of extrinsic states and their probabilities of occurrence. We show that a small number of option contracts can make an economy strongly sunspot immune. In addition, we demonstrate how asset re-trading opportunities, and the associated capital gains and losses, reduce the number of options needed for this result to obtain. Received: August 13, 1996; revised version: January 20, 1997     

Correlated equilibria and sunspots

We examine when “sunspots” (uncertainty that has no influence on endowments, preferences, or technology) can affect equilibrium in a simple two-period, two-commodity, two-class economy. We find that such an effect is possibly only if the signals (random variables) that different agents observe are imperfectly correlated (neither perfectly correlated nor independent) and at least one commodity is a Giffen good. For two special cases we characterize the set of equilibria due to sunspots. We conclude by showing the intimate connection between the sunspot equilibria of our finite horizon model and those of the overlapping generations literature.     

Market crashes,speculation and learning in financial markets

A natural conjecture is that if agents’ beliefs are almost correct then equilibrium prices should be close to rational expectations prices. Sandroni (J Econ Theory 82:1–18, 1998) gives a counterexample in an economy with sunspots and complete markets. We extend Sandroni’s result by showing that the conjecture is generically true for economies with complete markets. We consider a standard General Equilibrium model with large but finite horizon and complete markets. We show that, for almost every such economy, if conditional beliefs eventually become correct along a path of events then equilibrium prices of assets traded along this path converge to rational expectations equilibria in the sup-norm. Moreover, we establish that, generically, there exist along any such path local diffeomorphisms between individual beliefs and equilibrium prices. I would like to thank C. Ewerhart and A. Kirman for their comments, as well as the seminar participants at the University of Minho, the General Equilibrium Workshop 2005 in Zurich, and the 15th Asian General Equilibrium Conference 2007 in Singapore. An anonymous referee also provided very valuable comments.     

On Rothschild–Stiglitz as Competitive Pooling

Dubey and Geanakoplos (Q J Econ 117:1529–1570, 2002) have developed a theory of competitive pooling, which incorporates adverse selection and signaling into general equilibrium. By recasting the Rothschild–Stiglitz model of insurance in this framework, they find that a separating equilibrium always exists and is unique. We prove that their uniqueness result is not a consequence of the framework, but rather of their definition of refined equilibria. When other types of perturbations are used, the model allows for many pooling allocations to be supported as such: in particular, this is the case for pooling allocations that Pareto dominate the separating equilibrium.     

A note on external habits and efficiency in the AK model

This paper examines the implications for equilibrium efficiency of alternative formulations of habit formation in the AK model with external habits. Agent’s utility depends on how her current consumption compares to a reference consumption level—the habits stock—determined by average past consumption in the economy. We first prove that perfect substitutability between consumption and habits along the equilibrium path is necessary and sufficient for efficiency. Then, we prove that the competitive equilibrium is efficient if and only if habits enter utility in a subtractive form.     

Trend inflation and macroeconomic stability in a small open economy

This paper estimates a small open economy New Keynesian model for Australia with positive trend inflation while allowing for multiple equilibria. We first show the extent to which positive trend inflation can shrink the determinacy region. We then conduct a Bayesian estimation over two separate periods: from 1983Q1 to 1993Q1, covering the pre-inflation-targeting regime, and from 1993Q2 to 2018Q4, covering the inflation-targeting regime. We find that Australian monetary policy before the adoption of inflation-targeting permitted multiple equilibria and self-fulfilling inflation expectations to arise, resulting in exacerbated macroeconomic volatility. The implementation of inflation-targeting in Australia in the early 1990s successfully eradicated equilibrium multiplicity and sunspot shocks, and thereby drove the economy towards greater stability.     

Time and Uncertainty in Overlapping Generations Economies

This paper shows the general reversibility of every perfect foresight equilibrium of an overlapping generations economy. It then shows and characterizes the existence of reversible sunspot equilibria in these economies as well, which seems to be at odds with our intuition about the irreversibility of a tree of events. Although the paper establishes also that such reversible stochastic equilibria constitute a negligible subset of all the equilibria of their class, their mere existence may be considered somewhat puzzling for this intuition. Journal of Economic Literature Classification Numbers: D50, D80, D90.     

Rationalizable expectations and sunspot equilibria in an overlapping-generations economy

There are two theories for the treatment of market uncertainty: rationalizable expectations and sunspot equilibria. This paper shows how the game-theoretic solution concept of rationalizable expectations can be applied to an overlapping-generations exchange economy. Some general properties of these equilibria are discussed. It is shown that rationalizable-expectations equilibria are the predictions yielded by considering sunspot equilibria in which probability beliefs may differ across individuals. This result allows for a new interpretation of sunspot equilibria and helps to understand their relevance.     

Notes on optimal transportation

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

An elasticity approach to equilibrium and preference concentration in the Hotelling game

We consider the standard Hotelling price-and-location game with two firms, and show how equilibrium is affected by the shape of the consumers’ density. By describing the latter in terms of elasticity (Esteban in Int Econ Rev 27:439–444, 1986), we develop a framework which enables us to prove a sufficient condition for the existence of multiple asymmetric equilibria when the density is symmetric, in terms of a lower bound on the Gini coefficient of the distribution of preferences.