Sunspots,indeterminacy and Pareto inefficiency in economies with incomplete markets

Summary. We consider two periods economies with both intrinsic and extrinsic uncertainty. Asset markets are incomplete in the certainty economy. If assets are nominal, there are enough commodities and the number of agents is greater than two and smaller than the total number of states of nature tomorrow (minus one), then a sunspot-invariant equilibrium is generically Pareto dominated by some sunspot equilibria. When assets are real, and there are enough commodities, if there are sunspot equilibria, there are sunspot equilibria Pareto dominating sunspot-invariant equilibria under the same restriction on the number of agents (and stronger restrictions on the number of commodities).Received: 20 October 2003, Revised: 1 April 2004, JEL Classification Numbers: D52.I wish to thank Paolo Siconolfi for helpful suggestions and comments. I aknowledge the financial support of M.I.U.R. and the kind hospitality of C.C.D.R. in Summer 2003.     

A note on conspicuous leisure,animal spirits and endogenous cycles

This paper investigates general equilibrium effects of conspicuous leisure. It finds that leisure externalities reduce the degree of other market imperfections needed to generate indeterminacy or sunspot equilibria - endogneous cycles become empirically more plausible. Sunspot equilibria are possible with a downward-sloping labor demand schedule. The economic reasoning behind the result is that with conspicuous-externalities, labor is drawn more easily in and out of leisure to help fulfill agents expectations.Received: June 2003, Accepted: January 2004, JEL Classification: E32Mark Weder: I thank Paulo Brito (the Editor), Michael Burda and an anonymous referee for very helpful comments and suggestions. All remaining errors are my own. Support by the Deutsche Forschungsgemeinschaft in the form of a Heisenberg Fellowshipis gratefully acknowledged.     

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.     

Monetary confidence and asset prices

Three issues are studied in this paper: the existence of sunspot equilibria; the excessive volatility of asset prices; and the possibility that assets may be undervalued relative to their market fundamentals. We show that (i) stationary sunspot equilibria exist in a very general environment; (ii) asset prices may fluctuate despite a constant stream of dividends; and (iii) assets may be undervalued. JEL: 021, 023, 131.     

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.     

Balanced-budget rules: Chaos and deterministic sunspots

Schmitt-Grohé and Uribe [11] illustrate that a balanced-budget rule can lead to aggregate instability. In particular, under such a rule it is possible for a steady state to be locally indeterminate, and therefore sunspot equilibria are possible. In this paper, I extend their analysis to investigate the possibility of chaotic equilibria under a balanced-budget rule. A global analysis reveals Euler equation branching which means that the dynamics going forward are generated by a differential inclusion of the form . Each branch alone will not imply interesting dynamics. However, by switching between the branches, I show that the existence of Euler equation branching in an arbitrarily small neighborhood of a steady state implies topological chaos in the sense of Devaney on a compact invariant set with non-empty interior (the chaos is “thick”). Moreover, the chaos is robust to small C¹ perturbations. This branching under a balanced-budget rule occurs independently of the local uniqueness of the equilibrium around the steady state(s).     

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.     

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 .     

Voting in assemblies of shareholders and incomplete markets

Summary. An economy with two dates is considered, one state at the first date and a finite number of states at the last date. Shareholders determine production plans by voting - one share, one vote - and at -majority stable stock market equilibria, alternative production plans are supported by at most percent of the shareholders. It is shown that a -majority stable stock market equilibrium exists if where S is the number of states at the last date and J is the number of firms. Moreover, an example shows that -majority stable stock market equilibria need not exist for smaller s.Received: 23 December 2002, Revised: 14 June 2004, JEL Classification Numbers: D21, D52, D71, G39. Correspondence to: Hervé CrésThe authors are grateful to an anonymous referee for helpful comments and suggestions. Financial support from the Danish Research Councils and hospitality of HEC is gratefully acknowledged by Mich Tvede and support from Fondation HEC is gratefully acknowledged by Hervé Crés.     

Stationary Markov equilibria for overlapping generations

Summary.  At a stationary Markov equilibrium of a Markovian economy of overlapping generations, prices at a date-event are determined by the realization of the shock, the distribution of wealth and, with production, the stock of capital. Stationary Markov equilibria may not exist; this is the case with intra-generational heterogeneity and multiple commodities or long life spans. Generalized Markov equilibria exist if prices are allowed to vary also with the realization of the shock, prices and the allocation of consumption and production at the predecessor date-event. (Stationary) Markov -equilibria always exist; as allocations and prices converge to equilibrium prices and allocations that, however, need not be stationary.Received: 2 March 2004, Revised: 2 April 2004, JEL Classification Numbers:   D50, D52, D60, D80, D90.Correspondence to: Felix KublerWe thank participants in seminars in Athens and Lund, at Penn, at IMPA and at Stanford, the 2002 CEME (NBER) General Equilibrium Conference and the 2002 SED meetings, and especially Martin Hellwig, George Mailath and an anonymous referee for very helpful comments.     

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.     

Economic progress and skill obsolescence with network effects

Summary. We construct an OLG model with network effects to examine skill obsolescence when individuals can choose technological vintages. In the absence of transfer payments, some regions of the parameter space have unique stationary equilibria, others have unique cyclical equilibria, and others have multiple stationary equilibria. All equilibria are Pareto efficient. However, rat race equilibria can exist in which all agents currently alive prefer a slower rate of progress than occurs in equilibrium. When contemporaneous transfers are allowed, equilibria are unique everywhere, but a cycle still exists, and a rat race can still arise in equilibrium. Allowing intertemporal transfers (debt) ensures that all equilibria are stationary. In the relevant parameter range, the introduction of debt can eliminate cycles and increase the long-run growth rate. No rat race equilibria exist when debt is allowed.Received: 3 January 2002, Revised: 3 June 2004, JEL Classification Numbers: 041, J24, O33. Correspondence to: Ian P. KingEarly versions of this paper were presented at the West Coast Macro Workshop, the Mid-West Macro Meetings, the Canadian Macroeconomics Study Group Meetings, and the Society for Economic Dynamics Meetings. We would like to thank V. V. Chari, Merwan Engineer, Don Ferguson, John Hillas, John Knowles, Dan Peled, Dan Usher, Linda Welling and Julian Wright for helpful comments. We are especially grateful to an anonymous referee for some very useful and substantive comments.     

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.     

Monetary policy and price level determinacy in a cash-in-advance economy

Summary The paper considers the determinacy of the equilibrium price level in the cash-in-advance monetary economy of Lucas and Stokey (1983, 1987), in the case of deterministic fundamentals. The possibilities both of a multiplicity of perfect foresight equilibria and of sunspot equilibria are considered. Two types of monetary policy regimes are considered and compared, one in which the money supply grows at a given exogenous rate (that may be positive or negative), and one in which the nominal interest rate on one-period government debt is pegged at a given non-negative level. In the case of constant money growth rate regimes, it is shown that one can easily have both indeterminacy of perfect foresight equilibrium and existence of sunspot equilibria; indeed, in the case of negative rates of money growth (as called for by Friedman (1969)), both types of indeterminacy necessarily occur. On the other hand, sufficient conditions for uniqueness of equilibrium (and non-existence of equilibria other than a deterministic steady state) are also given, and a class of cases is identified in which a sufficiently high rate of money growth guarantees this. Thus there may be a conflict between the aims of choosing a rate of money growth that results in a high level of welfare in the steady state equilibrium and choosing a rate that makes this steady state the unique equilibrium.) In the case of the interest rate pegging regimes, sufficient conditions are given for uniqueness of equilibrium (and impossibility of sunspot equilibria), and it is shown that these necessarily hold in the case of any low enough nominal interest rate. Thus the nominal interest rate peg allows simultaneous achievement of price level determinacy and a high level of welfare in the unique (steady state) equilibrium.In this paper I consider the consequences of alternative choices of the monetary policy regime for the determinacy of the rational expectations equilibrium value of money, and in particular for the existence or not of sunspot equilibria, i.e., rational expectations equilibria in which fluctuations in the price level occur in response to random events that represent no change in economic fundamentals, simply due to self-fulfilling revisions of people's expectations. I am interested in particular in making the point that a consideration of the complete set of possible equilibria associated with a given policy regime may alter one's evaluation of the relative desirability of alternative policies, relative to the conclusion that one might reach if one considered only a single possible equilibrium associated with each policy regime (perhaps a unique equilibrium involving a minimum set of state variables). In view of this I give particular attention to policy regimes of types that have sometimes been advocated as ways of reducing the inefficiency associated with a rate of return differential between money and other financial assets, and show that policies that might otherwise be desirable (policies that make possible a more desirable equilibrium than would otherwise be possible) can have the unfortunate consequence of rendering equilibrium indeterminate and making possible equilibrium fluctuations in response to sunspot events.Two classes of policy regimes are considered in particular: on the one hand, alternative constant rates of growth or contraction of the money supply, financed through lump sum taxes or transfers, with zero net government assets at all times; and on the other, alternative constant nominal interest rate pegs, to be maintained through open market operations between money and interest-bearing debt, with an exogenously fixed level of net transfer payments. The first class of policies is considered because of Friedman's (1969) well-known proposal that a constant contraction of the money supply of this sort would be welfare improving. I find that while thestationary equilibrium associated with the Friedman regime achieves the maximum possible level of utility for the representative consumer, and while the level of utility associated with stationary equilibrium may be monotonically decreasing in the rate of money growth, lower rates of money growth (in particular, rates near that called for by Friedman) are associated with indeterminacy of equilibrium and the existence of sunspot equilibria, while these problems need not arise in the case of higher rates of money growth.The second class of policies is considered because they represent an obvious alternative approach to the elimination of the same rate of return differential with which Friedman is concerned. Achievement of permanently low nominal interest rates through a simple interest rate peg is not often advocated; one reason is that it is often asserted that such a policy must result in price level indeterminacy. In fact, I find that if the interest rate pegging regime is properly specified, it results in aunique rational expectations equilibrium, regardless of the level at which interest rates are to be pegged. Thus not only does the interest rate peg not result in price level indeterminacy but it allows nominal interest rates to be maintained permanently at a level lower than that which can be obtained through a policy regime of the first sort without creating price level indeterminacy. It would hence appear, at least in the case of the kind of economy modeled here, that interest rate pegging is a more reliable way of trying to reduce the inefficiency associated with consumers being forced to economize on liquidity.This paper represents a revision of Woodford (1988). I would like to thank Leonardo Auernheimer, Buz Brock, Willem Buiter, Peter Howitt, Teh-Ming Huo, David Laidler, David Levine, Bennett McCallum, and an anonymous referee for helpful comments, and the National Science Foundation for research support.     

Note on the non-existence of sunspot equilibrium

In this note, we emphasize the role of consumers’ risk aversion in the non-existence of sunspot equilibria in incomplete market economies. We prove that there are no sunspot equilibria if the fundamentals of the underlying economy admit a unique equilibrium for any distribution of endowments. This substantiates Mas-Colell’s (Economic analysis of markets and games: essays in honor of Frank Hahn. MIT, Cambridge, 1992) conjecture. We also prove that, in a two-consumer economy, no sunspot equilibrium exists under the more relaxed condition that the underlying economy admits a unique equilibrium for the initial endowment. This is a generalization of Corollaries 1 and 2 of Hens and Pilgrim (Econ Theory 24:583–602, 2004).      

The structure of the Nash equilibrium sets of standard 2-player games

Summary. In this paper I study a class of two-player games, in which both players action sets are [0,1] and their payoff functions are continuous in joint actions and quasi-concave in own actions. I show that a no-improper-crossing condition is both necessary and sufficient for a finite subset A of to be the set of Nash equilibria of such a game.Received: 21 November 2002, Revised: 9 September 2004, JEL Classification Numbers: C65, C72.I am grateful to an editor of the journal and an anonymous referee for their very helpful comments. I also would like to thank the seminar participants at City University of Hong Kong, Georgia State University, Northwestern University, and Rice University.     

Asymmetric contests with general technologies

Summary. We investigate the pure-strategy Nash equilibria of asymmetric, winner-take-all, imperfectly discriminating contests, focussing on existence, uniqueness and rent dissipation. When the contest success function is determined by a production function with decreasing returns for each contestant, there is a unique pure-strategy equilibrium. If marginal product is also bounded, limiting total expenditure is equal to the value of the prize in large contests even if contestants differ. Partial dissipation occurs only when infinite marginal products are permitted. Our analysis relies heavily on the use of share functions and we discuss their theory and application.Received: 28 May 2003, Revised: 26 April 2004, JEL Classification Numbers: C72, D72. Correspondence to: Richard CornesMuch of the research in this paper was undertaken while the first author was a Visiting Scholar at the Centre for Economic Studies, University of Munich. The support of the Centre is gratefully acknowledged.     

Strategic market games with a finite horizon and incomplete markets

Summary. We study a strategic market game associated to an intertemporal economy with a finite horizon and incomplete markets. We demonstrate that generically, for any finite number of players, every sequentially strictly individually rational and default-free stream of allocations can be approximated by a full subgame-perfect equilibrium. As a consequence, imperfect competition may Pareto-dominate perfect competition when markets are incomplete. Moreover - and this contrasts with the main message conveyed by the market games literature - there exists a large open set of initial endowments for which full subgame-perfect equilibria do not converge to -efficient allocations when the number of players tends to infinity. Finally, strategic speculative bubbles may survive at full subgame-perfect equilibria.Received: 24 January 2002, Revised: 21 February 2003, JEL Classification Numbers: C72, D43, D52. Correspondence to: Gaël GiraudWe thank Tim Van Zandt for his comments.     

Mistaken self-perception and equilibrium

Summary. Motivated by real-world information economics problems and by experimental findings on overconfidence, this paper introduces a general epistemic construction to model strategic interaction with incomplete information, where the players self-perception may be mistaken. This allows us to rigorously describe equilibrium play, by formulating appropriate equilibrium concepts. We show that there always exist objective equilibria, where the players correctly anticipate each others strategies without attempting to make sense of them, and that these outcomes coincide with the equilibria of an associated Bayesian game with subjective priors. In population games, these equilibria can also be always introspectively rationalized by the players, despite their possibly mistaken self-perception.Received: Received: May, 12, 2003; revised version: 3 December 2004, Revised: 3 December 2004, JEL Classification Numbers: J31, D82, D83.I thank Massimiliano Amarante, Eddie Dekel, Massimo Marinacci, Jean Francois Mertens, Giuseppe Moscarini, Paolo Siconolfi, Marciano Siniscalchi, Joel Sobel, and an anonymous referee.     

Large games with transformed summary statistics

Summary. The existence of pure-strategy Nash equilibrium is shown for a non-cooperative game with a continuum of small players and a compact action space. The players payoffs depend on their own actions and the mean of the transformed strategy profiles. This covers the case when the payoffs depend on players own actions and finitely many summary statistics.Received: 24 November 2003, Revised: 29 March 2004, JEL Classification Numbers: C7, D4. Correspondence to: Haomiao YuThe authors are grateful to Yeneng Sun for his help and guidance. They also thank Ali Khan, Kali Rath, and an anonymous referee for useful comments.