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.     

Robustness of sunspot equilibria

Summary. A sunspot equilibrium (SSE) is based on some extrinsic randomizing device (RD). We analyze the robustness of SSE. (1) We say that an SSE allocation is robust to refinements if it is also an SSE allocation based on any refinement of its RD. (2) We introduce two core concepts for analyzing the robustness of SSE in the face of cooperative-coalition formation. In the first, the blocking allocations are based on the RD that defines the SSE. In the second (stronger) core concept, coalitions select their own RDs. For the convex economy with restricted market participation, SSE allocations are robust under each of the definitions and the cores converge on replication of the economy to the set of SSE allocations. For the economy with an indivisible good, SSE allocations are not always robust. We provide examples of each of the following: (i) an SSE allocation that is not robust to refinement, (ii) an SSE allocation that is in neither core, (iii) an SSE allocation that is in the first core, but not in the second, and (iv) a core that does not converge upon replication to the set of SSE allocations. Received: July 31, 1995; revised version August 30, 1996     

Manipulation of endowments and sunspot equilibria

We study the connection between occurrence of manipulation via reallocating endowments by coalitions and sunspot equilibria. The uncertainty about which coalition will form introduces extrinsic uncertainty into the economy. Under certain conditions, manipulation of endowments by coalitions can occur if and only if sunspots matter. We would like to thank Bill Ethier, Atsushi Kajii, Cuong Le Van, Karl Shell, Koji Shimomura, Nicholas Yannelis, an anonymous referee, as well as seminar participants at the Second Asian General Equilibrium Theory Workshop, 2005, Tokyo; Workshop on Uncertainty and Information, IMS, NUS, 2005; Kobe Institute of Economic Research; Public Economic Theory 2005 Conference, Marseille; and the South and South East Asia Econometric Society Meetings, 2006, Chennai for helpful discussions and comments.     

Short-sighted managers and learnable sunspot equilibria

This paper assumes that firm managers make choices over a finite horizon while households plan over an infinite horizon. Following Shea (2013), I assume that labor exhibits firm-specific learning by doing so that newly employed labor is less productive than experienced labor. In the model, optimization requires that firm managers make conjectures about how their choices affect the labor demand choices of their successors. The model yields two steady states; one where the firm manager behaves as if she cares only about the present period and another where she is forward looking. The former (myopic) steady state usually exhibits higher output than the non myopic steady state. The non-myopic steady state also exhibits two regions of indeterminacy where extraneous, self-fulfilling expectational errors add volatility. One of these regions of indeterminacy is usually stable under adaptive learning while the other never is stable under learning.     

On the plausibility of sunspot equilibria

This paper presents a one-sector business cycle model with variable capacity utilization and externalities that stem from aggregate economic activity. It uses a new formulation of the endogenous capital utilization rate in which utilization costs appear in the form of variable maintenance expenses. Indeterminacy arises at approximate constant returns to scale. This result challenges the viewpoint that indeterminacy is empirically implausible. Sunspot driven model fluctuations duplicate a number of stylized facts of the business cycle. Plausible parameter space contains regions in which stationary sunspot equilibria are stable under learning.     

Indivisibilities,lotteries, and sunspot equilibria

Summary We analyze economies with indivisible commodities. There are two reasons for doing so. First, we extend and provide some new insights into sunspot equilibrium theory. Finite competitive economies with perfect markets and convex consumption sets do not allow sunspot equilibria; these same economies with nonconvex consumption sets do, and they have several properties that can never arise in convex environments. Second, we provide a reinterpretation of the employment lotteries used in contract theory and in macroeconomic models with indivisible labor. We show how socially optimal employment lotteries can be decentralized as competitive equilibria without lotteries once sunspots are introduced.We thank Kenneth Arrow, Aditya Goenka, Ed Green, Jeremy Greenwood, Walter Heller, Steve Matthews, Herve Moulin, Roger Meyerson, Jim Peck, Patrick Kehoe, Ramon Marimon, Ed Prescott, Richard Rogerson, Nancy Stokey and Raghu Sundaram for their comments. We also thank participants in seminars at Northwestern, Yale, USC, Cornell, Barcelona, Madrid, Santander, and the Canadian Economics Association annual meetings in Victoria. We are grateful to the National Science Foundation (through grants SES-8606944 and SES-8821225), the Center for Analytic Economics, the Thorne Fund, and the University of Pennsylvania Research Foundation for research support. The views expressed here are those of the authors, and not necessarily those of the Federal Reserve System or the Federal Reserve Bank of Minneapolis.     

Sufficient conditions for the existence of sunspot equilibria

This paper reports on conditions on agents' preferences and endowments sufficient to guarantee the existence of sunspot equilibria in a simple overlapping generations model of pure exchange. Sunspot equilibria are those in which uncertainty extrinsic to the economy operates through expectations to yield a fulfilled expectations competitive equilibrium in which the extrinsic randomness has real effects on prices and allocations. The paper also provides necessary and sufficient conditions for these equilibria to have agents trading in a fixed stock of valued fiat money. The condition derived can be interpreted as requiring that intertemporal income effects appropriately dominate substitution effects.     

Learning of cycles and sunspot equilibria by Genetic Algorithms

We analyze the learning behaviour of a Simple Genetic Algorithm in an overlapping generations model with one consumption good and fiat money. It is shown by simulations, that in cases where periodic equilibria exist the equilibrium of period two is learned by a Genetic Algorithm and not the monetary steady state. We further show that proper coding leads to convergence of the GA towards the sunspot equilibrium. If individuals who believe in the impact of sunspots are brought together with individuals who ignore the sunspots, the sunspot believes will in most cases drive the other individuals out of the population.     

Refining linear rational expectations models and equilibria

This paper develops the case for forward convergence as a model refinement scheme for linear rational expectations models and an associated no-bubble condition as a solution selection criterion. We relate these two concepts to determinacy and characterize the complete set of economically relevant rational expectations solutions to the linear rational expectations models under determinacy and indeterminacy. Our results show (1) why a determinate solution is economically cogent in most, but not all, cases, and (2) that those models that are not forward-convergent have no economically relevant solutions.     

Strict rational expectations equilibria with diffuseness

For smooth static pure-exchange economies under uncertainty, it is shown that a diffuseness condition on the distribution of traders' characteristics as the state of the world varies permits one to use only a weak dimensionality restriction in the proof of the genericity of existence of strict rational expectations equilibria. A generic result on the amount of information conveyed by equilibrium price functions and comparisons with some examples of nonexistence are given.     

On multiple equilibria and the rational expectations hypothesis

Summary This paper analyzes through a simple two-period model the fact that, if some agents hold inside money intertemporally, the second-period normalization matters. Thus, there are several equilibria of the second-period economy, indexed by the level of inflation. A concept of equilibrium acknowledging this fact, and requiring that agents put some weight on any of the possible second-period equilibrium price vectors is developed. Such an equilibrium is shown to exist, and is illustrated by an example.This a revised version of chapter three of my Ph.D. dissertation. I would like to thank David Cass, Atsushi Kajii, George Mailath and Shinichi Suda for helpful discussions and comments. Thanks are also due to Jean-Michel Grandmont who pointed out mistakes in a previous version. All remaining mistakes are of course my own. Financial support from a CARESS scholarship at the University of Pennsylvania is gratefully acknowledged.     

Approximate equilibria in microeconomic rational expectations models

For general equilibrium models in which prices transmit information among asymmetrically informed traders, strict rational expectations approximate equilibria are defined. A state-dependent price function is an ε-equilibrium if, when agents use their own information and that conveyed by prices, aggregate excess demand (in each state of the world) does not exceed ε. For any positive ε, existence requires only very mild assumptions—continuity and compact support. Moreover, there are revealing ε-equilibria for all smooth economies satisfying a dimensionality condition. In an open neighborhood of this case, existence of maximally revealing ε-equilibria holds.     

Arrow-Debreu equilibria with asymptotically heterogeneous expectations exist

Summary. In infinite horizon economies only local equivalence of beliefs is needed to ensure the existence of an Arrow–Debreu equilibrium. In fact, agents can even disagree completely in the long run in the sense that asymptotically, their beliefs are singular. Received: November 3, 2000; revised version: February 13, 2002     

Functional rational expectations equilibria in market games

The rational expectations equilibrium (REE) has been criticized as an equilibrium concept in market game environments. Such an equilibrium may not exist generically, or it may introduce unrealistic assumptions about an economic agent’s knowledge or computational ability. We define an REE as a probability measure over uncertain states of nature which exploits all available information in a market game, and which exists for almost all economies. Furthermore, if retrading is allowed, it is possible for agents to compute such a ‘functional rational expectations equilibrium’ using straightforward numerical fixed point algorithms. The approach is demonstrated in a detailed numerical example.     

The structure of the set of sunspot equilibria in economies with incomplete financial markets

Summary This paper considers the set of equilibria of two-period, sunspot economies withS purely extrinsic states of nature in the second period andI assets with linearly independent nominal payoffs. The span of the payoff matrix contains the vector [1, ... , 1] (i.e., inside money). The set of economies is described in terms of (sunspot-invariant) utility functions. IfS>I> 0, there is an open, dense set of economies such that, given a vector of no arbitrage asset prices, the set of equilibrium allocations contains a smooth manifold of dimensionS—I. Such a manifold contains at least one nonsunspot equilibrium (and at most a finite number of such equilibria).The paper was written while I was a visitor at C.O.R.E., Universitè Catholique de Louvain, with the financial support of a S.P.E.S. fellowship. I would like to thank D. Cass, H. Polemarchakis and P. Siconolfi for their helpful comments.     

Rational expectations equilibria of economies with local interactions

We consider general economies in which rational agents interact locally. The local aspect of the interactions is designed to represent in a simple abstract way social interactions, that is, socioeconomic environments in which markets do not mediate all of agents’ choices, which might be in part determined, for instance, by family, peer group, or ethnic group effects. We study static as well as dynamic infinite horizon economies; we allow for economies with incomplete information, and we consider jointly global and local interactions, to integrate e.g., global externalities and markets with peer and group effects. We provide conditions under which such economies have rational expectations equilibria. We illustrate the effects of local interactions when agents are rational by studying in detail the equilibrium properties of a simple economy with quadratic preferences which captures, in turn, local preferences for conformity, habit persistence, and preferences for status or adherence to aggregate norms of behavior.     

On multiple equilibria and the rational expectations hypothesis?

Summary. This paper analyzes through a simple two-period model the fact that, if some agents hold inside money intertemporally, the second-period “normalization” matters. Thus, there are several equilibria of the second-period economy, indexed by the level of inflation. A concept of equilibrium acknowledging this fact, and requiring that agents put some weight on any of the possible second-period equilibrium price vectors is developed. Such an equilibrium is shown to exist, and is illustrated by an example.     

Real indeterminacy of equilibria in a sunspot economy with inside money

Summary In a two-period sunspot economy with inside money andS possible realizations of the sunspot, we prove that, genetically in the space of utility functions, there areS — 1 degrees of real indeterminacy. This result generalizes the previously known result for sunspot models that, generically in endowments, there is at least one degree of real indeterminacy. The proof involves showing that generically the equilibrium allocation is different across states for some household. This property allows us to perturb the utility function in a simple way and to apply standard transversality arguments to prove our main theorem.We would like to thank David Cass, Atsushi Kajii, Michael Mandler and Paolo Siconolfi for helpful discussions.     

Existence of adaptively stable sunspot equilibria near an indeterminate steady state

We examine the nonlinear model x_t=E_tF(x_t+1). Markov stationary sunspot equilibria (SSEs) exist near an indeterminate steady state, , provided . Despite the importance of indeterminacy in macroeconomics, earlier results have not provided conditions for the existence of adaptively stable SSEs near an indeterminate steady state. We show that there exist Markov SSEs near x? that are E-stable, and therefore locally stable under adaptive learning, if .     

The existence of fully rational expectations approximate equilibria with noisy price observations

Nonrevealing fully rational expectations approximate equilibria exist in microeconomic pure exchange economies in which uninformed agents have suitably dispersed noisy price observations. Such traders maximize a state-dependent expected utility conditional on the price vector they observe, the distributions of noisy price observations, and the correct equilibrium relationship between states of the world and prices. In equilibrium, aggregate excess demand is small with high probability in every state of the world (and its expectation is also small); this magnitude diminishes as the noisy price observations become more accurate. Equilibria are obtained by applying a fixed point argument to state-dependent excess demand functions which are smooth because of the noisy price observations.