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     

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.     

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.     

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.     

The genesis of expectations and of sunspot equilibria

The existence—under certain conditions—of sunspot equilibria in overlapping generations models is a well-known theoretical result, but a lot of research has still to be made to understand whether and how such equilibria may occur as a consequence of a dynamic process. In this paper, we explore a model with individuals having simple utility functions and discover situations in which the dynamic processes of expectations and observed prices do not converge in spite of the existence of static equilibria. In other words, unfounded beliefs do still have a permanent influence on the real economy but induce erratic evolutions.     

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.     

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.     

Indivisibilities, Lotteries, and Monetary Exchange

We introduce lotteries (randomized trading) into search-theoretic models of money. In a model with indivisible goods and fiat money, we show goods trade with probability 1 and money trades with probability τ, where τ<1 iff buyers have sufficient bargaining power. With divisible goods, a nonrandom quantity q trades with probability 1 and, again, money trades with probability τ where τ<1 iff buyers have sufficient bargaining power. Moreover, q never exceeds the efficient quantity (not true without lotteries). We consider several extensions designed to get commodities as well as money to trade with probability less than 1, and to illuminate the efficiency role of lotteries. Journal of Economic Literature Classification Numbers: E40, D83.     

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 .     

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.     

Firm dynamics and markup variations: Implications for sunspot equilibria and endogenous economic fluctuations

This paper analyzes how the interaction between firms’ entry-and-exit decisions and variations in competition gives rise to self-fulfilling, expectation-driven fluctuations in aggregate economic activity and in measured total factor productivity (TFP). The analysis is based on a dynamic general equilibrium model in which net business formation is endogenously procyclical and leads to endogenous countercyclical variations in markups. This interaction leads to indeterminacy in which economic fluctuations occur as a result of self-fulfilling shifts in the beliefs of rational forward looking agents. When calibrated with empirically plausible parameter values and driven solely by self-fulfilling shocks to expectations, the model can quantitatively account for the main empirical regularities characterizing postwar U.S. business cycles and for 65% of the fluctuations in measured TFP.     

Symmetry and impartial lotteries

A prize is to be awarded, so each candidate designates one of his peers on a ballot. The ballots determine the lottery that selects the winner, and impartiality requires that no candidate's choice of designee impacts his own chance of winning, removing incentives for strategic ballot submission. The primary results are (1) a characterization of all impartial rules that treat agents symmetrically as voters, and (2) a characterization of all impartial rules that treat agents symmetrically as candidates. Each rule in either class may be represented as a randomization over a finite set of simple rules. These results have immediate interpretation in a second context: the division of surplus among team members. Corollaries include the constant rule impossibility of Holzman and Moulin (2013), a new dictatorship impossibility, and the first axiomatic characterization of uniform random dictatorship.     

Recursive contracts,lotteries and weakly concave Pareto sets

Marcet and Marimon (1994, revised 1998, revised 2011) developed a recursive saddle point method which can be used to solve dynamic contracting problems that include participation, enforcement and incentive constraints. Their method uses a recursive multiplier to capture implicit prior promises to the agent(s) that were made in order to satisfy earlier instances of these constraints. As a result, their method relies on the invertibility of the derivative of the Pareto frontier and cannot be applied to problems for which this frontier is not strictly concave. In this paper we show how one can extend their method to a weakly concave Pareto frontier by expanding the state space to include the realizations of an end of period lottery over the extreme points of a flat region of the Pareto frontier. With this expansion the basic insight of Marcet and Marimon goes through – one can make the problem recursive in the Lagrangian multiplier which yields significant computational advantages over the conventional approach of using utility as the state variable. The case of a weakly concave Pareto frontier arises naturally in applications where the principal?s choice set is not convex but where randomization is possible.     

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.     

A Model of Commodity Differentiation with Indivisibilities and Production

This paper presents an existence theorem in a general equilibrium model of a production economy with commodity differentiation and indivisibilities. The model is motivated by the existence of markets with indivisible commodities, such as the markets for automobiles and computers. As is standard in the literature, the space of commodity characteristics is described by a compact metric space and a commodity vector is described by an integer-valued Borel measure on the space of commodity characteristics. An atomless measure space of producers and consumers is assumed to overcome the problem of non-convexity of the production and consumption sets induced by indivisibilities. This paper is based on a chapter from James Marton’s dissertation. We would like to thank Marcus Berliant, Wilhelm Neuefeind, and an anonymous referee for their valuable comments and suggestions.     

The set of sunspot equilibria in economies with incomplete financial markets: variable asset prices

Summary. I consider the set of equilibria of two-period economies with S extrinsic states of nature in the second period and I assets with linearly independent nominal payoffs. Asset prices are variable. If the number of agents is greater than (S-I), the payoff matrix is in general position and S 2I, the set of equilibrium allocations generically (in utility function space) contains a smooth manifold of dimension (S-1). Moreover, the map from states o f nature to equilibrium allocations (restricted to this manifold) is one-to-one at each equilibrium. Received: February 23, 1998; revised version: June 1, 2000