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     

General equilibrium without utility functions: how far to go?

How far can we go in weakening the assumptions of the general equilibrium model? Existence of equilibrium, structural stability and finiteness of equilibria of regular economies, genericity of regular economies and an index formula for the equilibria of regular economies have been known not to require transitivity and completeness of consumers’ preferences. We show in this paper that if consumers’ non-ordered preferences satisfy a mild version of convexity already considered in the literature, then the following properties are also satisfied: (1) the smooth manifold structure and the diffeomorphism of the equilibrium manifold with a Euclidean space; (2) the diffeomorphism of the set of no-trade equilibria with a Euclidean space; (3) the openness and genericity of the set of regular equilibria as a subset of the equilibrium manifold; (4) for small trade vectors, the uniqueness, regularity and stability of equilibrium for two version of tatonnement; (5) the pathconnectedness of the sets of stable equilibria.     

More on economies with a finite set of Walrasian equilibria

Proposition. The graph of the Walrasian equilibrium correspondence is a piecewise continuously differentiable manifold. Furthermore, there exists an open dense set of economies, Ω, such that for all w in Ω (a) the number of equilibria of the economy w is finite; and (b) there exists a neighborhood V(w) on which the set of equilibria is represented by a finite family of piecewise continously differentiable functions.     

A general structure theorem for the Nash equilibrium correspondence

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003–1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537–545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46–60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.     

Economies with price-dependent preferences

With the exception of existence, not much is known of economies where individual preferences may depend on prices. We show that the main properties of competitive equilibria that fall under the general heading of qualitative comparative statics remain true when preferences are price-dependent, provided that the total resources are allowed to vary in the parameter space. The equilibrium set is a smooth manifold diffeomorphic to a Euclidean space; the natural projection is a smooth proper map, with topological degree one; equilibria always exist, and are locally unique for an open and dense set of economies.     

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.     

Decentralizing lottery allocations in markets with indivisible commodities

Summary In economies with indivisible commodities, consumers tend to prefer lotteries in commodities. A potential mechanism for satisying these preferences is unrestricted purchasing and selling of lotteries in decentralized markets, as suggested in Prescott and Townsend [Int. Econ. Rev.25, 1–20]. However, this paper shows in several examples that such lottery equilibria do not always exist for economies with finitely many consumers. Other conditions are needed. In the examples, equilibrium and the associated welfare gains are realized if consumptions are bounded or if lotteries are based upon a common sunspot device as defined by Shell [mimeo, 1977] and Cass and Shell [J. Pol. Econ.91, 193–227]. The paper shows that any lottery equilibrium is either a Walrasian equilibrium or a sunspot equilibrium, but there are Walrasian and sunspot equilibria that are not lottery equilibria.This paper is based on Chapter 3 of my doctoral dissertation, written while I was a student at Cornell University. I thank Larry Blume, Yue Yun Chen, David Easley, Aditya Goenka, John Marshall, Bruce Smith, John Wooders and an anonymous referee. I am particularly grateful to Karl Shell and Cheng-Zhong Qin. I thank the Academic Senate at UCSB for financial support.     

Auctions of Heterogeneous Objects

Simultaneous sealed bid auctions of heterogeneous objects are analyzed. Each bidder's reservation value for an object depends upon the other objects he obtains. Bidders' reservation values are common knowledge. In simultaneous first-price auctions, the set of Walrasian equilibrium allocations contains the set of pure strategy Nash equilibrium allocations which in turn contains the set of strict Walrasian equilibrium allocations. Hence, pure strategy Nash equilibria (when they exist) are efficient. Mixed strategy Nash equilibria may be inefficient. In simultaneous second-price auctions, any efficient allocation can be implemented as a pure strategy Nash equilibrium outcome if a Walrasian equilibrium exists.Journal of Economic LiteratureClassification Numbers: D44, D51.     

Real indeterminacy in incomplete financial market economies without aggregate risk

Summary We analyze an exchange economy with incomplete financial markets and assets whose returns are fixed in units of account. Moreover, we assume absence of aggregate risk, i.e., that individual preferences and total resources are constrained to be invariant across different states of the world. In this framework we show that the set of (commodity) price-endowment equilibria is diffeomorphic to a Euclidean space. We then exploit this global parameterization to prove that the set of equilibrium allocations associated with each endowment in a generic set contains a smooth manifold, whose dimension is equal to the number of missing assets.     

The indeterminacy of equilibrium city formation under monopolistic competition and increasing returns

We study the indeterminacy of equilibrium in the Fujita–Krugman [When is the economy monocentric?: von Thünen and Chamberlin unified, Reg. Sci. Urban Econ. 25 (1995) 505–528] model of city formation under monopolistic competition and increasing returns. Both the number and the locations of cities are endogenously determined. Assuming smooth transportation costs, we examine equilibria in city-economies where a finite number of cities form endogenously. For any positive integer K, the set of equilibria with K distinct cities has a smooth manifold of dimension K-1 as its interior for almost all parameter values in a regular parameterization. The disjoint union of these sets over all positive integers K constitutes the entire equilibrium set.     

Computing equilibria in finance economies with incomplete markets and transaction costs

Summary. Transaction costs on financial markets may have important consequences for volumes of trade, asset pricing, and welfare. This paper introduces an algorithm for the computation of equilibria in the general equilibrium model with incomplete asset markets and transaction costs. We show that economies with transaction costs can be analyzed with differentiable homotopy techniques and thus in the same framework as frictionless economies despite the existence of non-differentiabilities of agents asset demand functions and the existence of locally non-unique equilibria. We introduce an equilibrium selection concept into the computation of economic equilibria that picks out a specific equilibrium in the presence of a continuum of equilibria.Received: 2 December 2002, Revised: 15 November 2004, JEL Classification Numbers: C61, C62, C63, C68, D52, D58, G11, G12. Correspondence to: P. Jean-Jacques HeringsThis research started when Jean-Jacques Herings enjoyed the generous hospitality of the Cowles Foundation for Research in Economics at Yale University. His research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences and a grant of the Netherlands Organisation for Scientific Research. We thank audiences at Stanford University, UC San Diego, and Venice for discussions on the subject. We are very grateful to an anonymous referee for very helpful comments on an earlier draft.     

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.     

On the Generic Finiteness of Equilibrium Outcomes in Plurality Games

In this paper we prove that for generic (noncooperative) voting games under plurality rule the set of equilibria that induce a mixed distribution over the outcomes (i.e., with two or more candidates elected with positive probability) is finite and, furthermore, each of these equilibria is regular. From that we deduce the finiteness of the set of equilibrium distributions over outcomes. Furthermore we offer an example (S. Govindan and A. McLennan, 1997, “On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms,” mimeo) that shows the impossibility of extending such results to a general framework, even just to voting games. Journal of Economic Literature Classification Numbers: C72, D72.     

p-Best response set and the robustness of equilibria to incomplete information

In this paper, we use p-best response sets—a set-valued extension of p-dominance—in order to provide a new sufficient condition for the robustness of equilibria to incomplete information: if there exists a set S which is a p-best response set with , and there exists a unique correlated equilibrium μ^* whose support is in S then μ^* is a robust Nash equilibrium.     

The joint exploitation of a productive asset: a game-theoretic approach

Summary In the present paper we explore the set of equilibria in a game-theoretic model in which players can jointly exploit a productive asset. As in repeated games, we find that under certain circumstances there may be efficient as well as inefficient equilibria. In the model we study, efficient trigger-strategy equilibria may exist from some starting states (stocks of assets) but not others. More precisely, there is a stock level, sayy, such that an efficient trigger-strategy equilibrium exists from starting stocks greater than or equal toy, but not from those strictly less thany. (This statement is meant to include the cases in whichy is zero or infinite.) Under some circumstances, there may exist a new kind of equilibrium, which we call aswitching equilibrium. We show that, in our model, whenever y is positive (and finite), there is an open intervalI with upper endpoint y such that, from any starting stock inI there is an equilibrium of the dynamic game with the following structure: the players follow an inefficient but growing path until the stock reaches the levely, and then follow an (efficient) trigger strategy after that. The use of a continuous-time model enables us to conveniently decouple the delay of information from the time interval between decisions.We thank the C. V. Starr Center at New York University for research support. The views expressed here are those of the authors, and not necessarily those of AT&T Bell Laboratories.     

On the existence of equilibria in economies with infinitely many commodities and without ordered preferences

The present paper deals with the existence of equilibria in economies whose commodity space is L_∞(M, M, μ) and where the agents' preferences need not be complete or transitive. Applying a fixed point theorem of Browder, an equilibrium existence theorem for abstract economies (generalized qualitative games) is proven where each agent's choice set is contained in an arbitrary topological vector space. With the help of this theorem the existence of Walrasian general equilibrium for a suitably specified economic model is obtained. The final result is a generalization of T. F. Bewley's (J. Econ. Theory4 (1972), 514–540) equilibrium existence theorem to the case of non-ordered preferences.     

Indeterminacy of equilibrium in stochastic OLG models

Summary. This paper studies the equilibria of a stochastic OLG exchange economies consisting of identical agents living for two periods, and having the opportunity to trade a single infinitely-lived asset in constant supply. The agents have uncertain endowments and the stochastic process determining the endowments is Markovian. For such economies, the literature has focused on studying strongly stationary equilibria in which quantities and prices are functions of the exogenous states of nature which describe the uncertainty: such equilibria are generalizations of deterministic steady states, and this paper investigates if they have the same special status as asymptotic limits of other equilibrium paths. The difficulty in extending the analysis of equilibria beyond the class of strongly stationary equilibria comes from the presence of indeterminacy: we propose a procedure for overcoming this difficulty which can be decomposed into two steps. First backward induction arguments are used to restrict the domain of possible prices; then if some indeterminacy is left, expectation functions are introduced to make the forward equilibrium equations determinate. The properties of the resulting trajectories, in particular their asymptotic properties, can then be studied. For the class of models that we study this procedure provides a justification for focusing on strongly stationary equilibria. For the model with positive dividends (equity or land) the justification is complete, since we show that the strongly stationary equilibrium is the unique equilibrium. For the model with zero dividends (money) there is a continuum of self-fulfilling expectation functions resulting in a continuum of equilibrium paths starting from any admissible initial condition: under conditions given in the paper, these equilibrium paths converge almost surely to one of the strongly stationary equilibria-either autarchy or the stochastic analogue of the Golden Rule. Received: November 19, 2001; revised version: March 22, 2002 RID="*" ID="*" We are grateful for the stimulating environment and research support provided by the Cowles Foundation at Yale University during the Fall 2000 when this paper was first conceived. We are also grateful to the participants of the SITE Workshop at Stanford University and the Incomplete Markets Workshop at SUNY Stony Brook during the summer 2001 for helpful discussions. Correspondence to: M. Magill     

On the (non-)lattice structure of the equilibrium set in games with strategic substitutes

This paper studies models where the optimal response functions under consideration are not increasing in endogenous variables, and weakly increasing in exogenous parameters. Such models include games with strategic substitutes, and include cases where additionally, some variables may be strategic complements. The main result here is that the equilibrium set in such models is a non-empty, complete lattice, if, and only if, there is a unique equilibrium. Indeed, for a given parameter value, a pair of distinct equilibria are never comparable. Therefore, with multiple equilibria, some of the established techniques for exhibiting increasing equilibria or computing equilibria that use the largest or smallest equilibrium, or that use the lattice structure of the equilibrium set do not apply to such models. Moreover, there are no ranked equilibria in such models. Additionally, the analysis here implies a new proof and a slight generalization of some existing results. It is shown that when a parameter increases, no new equilibrium is smaller than any old equilibrium. (In particular, in n-player games of strategic substitutes with real-valued action spaces, symmetric equilibria increase with the parameter.)