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.     

Futures market: contractual arrangement to restrain moral hazard in teams

Holmstrom (Bell J Econ 13:324?C340, 1982) argues that a principal is required to restrain moral hazard in a team: wasting output in certain states is required to enforce efficient effort, and the principal is a commitment device for the waste. Under competition in commodity and team-formation markets, I extend his model à la Prescott and Townsend (Econometrica 52(1):21?C45, 1984) to show that competitive contracts can exploit the futures market to transfer output across states instead of wasting it. Thus, the futures market takes the place of a principal as a commitment device. Exploiting the duality of linear programming, I characterize the market environment and the contractual agreements for incentive-constrained efficiency.     

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.     

The type-agent core for exchange economies with asymmetric information

The type-agent core is a new solution concept for exchange economies with asymmetric information. It coincides with the set of subgame-perfect equilibrium outcomes of a simple competitive screening game. Uninformed intermediaries help the agents to cooperate in an attempt to make some profit. The paper extends the work of Perez-Castrillo [Cooperative outcomes through non-cooperative games, Games Econ. Behav. 7 (1994) 428-440] to exchange economies with non-transferable utility and asymmetric information. The type-agent core is a subset of Wilson's coarse core [Wilson, Information, efficiency, and the core of an economy, Econometrica 46 (1978) 807-816]. It is never empty, even though it may be a strict subset of Wilson's fine core. In addition, it converges towards the set of constrained market equilibria as the economy is replicated.     

Competitive and core allocations in large economies with differential information

Summary. We study the core and competitive allocations in exchange economies with a continuum of traders and differential information. We show that if the economy is “irreducible”, then a competitive equilibrium, in the sense of Radner (1968, 1982), exists. Moreover, the set of competitive equilibrium allocations coincides with the “private core” (Yannelis, 1991). We also show that the “weak fine core” of an economy coincides with the set of competitive allocations of an associated symmetric information economy in which the traders information is the joint information of all the traders in the original economy. Received March 22, 2000; revised version: May 1, 2000     

Belief heterogeneity and survival in incomplete markets

In complete markets economies (Sandroni in Econometrica 68:1303–1341, 2000), or in economies with Pareto optimal outcomes (Blume and Easley in Econometrica 74:926–966, 2006), the market selection hypothesis holds, as long as traders have identical discount factors. Traders who survive must have beliefs that merge with the truth. We show that in incomplete markets, regardless of traders’ discount factors, the market selects for a range of beliefs, at least some of which do not merge with the truth. We also show that impatient traders with incorrect beliefs can survive and that these incorrect beliefs impact prices. These beliefs may be chosen so that they are far from the truth.     

Economies with observable types

We study economies of asymmetric information with observable types. Trade takes place in lotteries. Individuals face a standard budget constraint, while the incentive compatibility constraints are imposed on the production set of the intermediaries. This formalization encompasses moral hazard, as in [Jerez, 2003] and [Jerez, 2005], and private information economies. Equilibrium allocations are constrained efficient, but, contrary to what stated for example in Jerez (2005), the set of equilibrium allocations may be empty and the Second Welfare Theorem may fail. This happens for two reasons. First, constrained efficient allocations may violate the necessary and sufficient conditions of price supportability for the individuals. Second, even when constrained efficient allocation are price supportable, they may fail to be a profit maximizing choice of the firm at the individual supporting prices. To restore existence of an equilibrium the firm has to be restricted to supply allocations with support in the set of incentive compatible contracts.     

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.     

Large economies with differential information and without free disposal

We consider exchange economies with a continuum of agents and differential information about finitely many states of nature. It was proved in Einy et al. (Econ Theory 18, 321–332, 2001) that if we allow for free disposal in the market clearing (feasibility) constraints then an irreducible economy has a competitive (or Walrasian expectations) equilibrium, and moreover, the set of competitive equilibrium allocations coincides with the private core. However when feasibility is defined with free disposal, competitive equilibrium allocations may not be incentive compatible and contracts may not be enforceable (see e.g. Glycopantis et al. in Econ Theory 21, 495–526, 2002). This is the main motivation for considering equilibrium solutions with exact feasibility. We first prove that the results in Einy et al. (Econ Theory 18, 321–332, 2001) are still valid without free-disposal. Then, motivated by the issue of contracts’ execution, we adapt the incentive compatibility property introduced in Krasa and Yannelis (Econometrica 62, 881–900, 1994) and we prove that every Pareto optimal exact feasible allocation is incentive compatible, implying that contracts of competitive or core allocations are enforceable. We would like to thank two anonymous reviewers and the Associate Editor for their valuable suggestions and remarks. This work was partially done while V.F. Martins-da-Rocha was visiting the Dipartimento di Matematica e Informatica of the Università degli Studi di Perugia. We thank the audience of the First General Equilibrium Workshop at Rio. Section 6 dealing with contract enforcement and coalitional incentive compatibility has benefited from discussions with J. Correia-da-Silva, W. Daher, F. Forges, C. Hervès-Beloso, E. Moreno-García, K. Podczeck, Y. Vailakis and N.C. Yannelis.     

The bargaining set in strategic market games

This paper presents a hybrid equilibrium notion that blends together the ‘cooperative’ and the ‘noncooperative’ theories of competition. In particular, the Mas-Colell bargaining set has been modified in order to accommodate the features of strategic market games. In other words, allocations, objections and counter objections of the standard bargaining set theory are described for an economy, where trades among groups of individuals are conducted via the Shapley–Shubik mechanism. In the main part of the paper, it is proved that in atomless economies the allocations resulting from this equilibrium notion are competitive.     

Injecting rational bubbles

This paper proves two theorems about economies with a finite number of infinitely lived agents who trade a complete set of one-period Arrow securities and several infinitely lived securities at each date, subject to short-sales constraints. The first theorem in the paper considers an equilibrium to an economy of this kind. It proves that there exists another economy with perturbed short-sales constraints in which there is an allocation-equivalent equilibrium in which asset prices have a bubble. The second theorem extends to the result to the case in short-sales constraints are endogenously determined in the sense of Alvarez and Jermann [Efficiency, equilibrium, and asset pricing with risk of default, Econometrica 68 (2000) 775-797].     

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.     

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.     

ECONOMIC NETWORKING AND STABILITY

Abstract. In this paper we discuss finite economies with the presence of transaction costs and with decreasing, constant or increasing returns. In general such an economy may have no general equilibrium existence and may even have an empty core. We analyse the trading networks of such an economy, introducing the concepts of locally stable network structure, un‐dominated locally stable network structures and most stable network structures. We point out that the set of most stable network structures could be treated as a solution concept for the empty core economies both in theoretical analysis and in application.     

Testing the balanced growth hypothesis: evidence from China

We investigate whether China’s experience during 1952–2004 supports the balanced growth entailment of the neoclassical growth model. Estimation of long-run relations among output, consumption and investment for the full period reject the balanced growth hypothesis for both the national and regional economies. When the economic reforms of the late 1970s are modelled as a structural break by the methods of Johansen et al. (Economet J 3(2):216–249, 2000) and Perron (Econometrica 57(6):1361–1401, 1989), we find some evidence of balanced growth in the pre-break period but in the post-break period the ‘great ratios’ are trend-stationary, precluding fully balanced growth, though permitting a common (stochastic) productivity trend.      

Modelling the sectoral allocation of labour in open economy models

Indivisible labour is not the only type of non‐convexity affecting labour supply decisions. Another type of non‐convexity arises in economies with sectors whenever individuals can work in only one sector at a time. I introduce this restriction into an open economy model with a tradeable and a non‐tradeable sector, and I use lotteries to convexify the consumption possibilities set. This approach implies that the aggregate elasticity of labour supply becomes infinite. I compare the performance of the model with an analogous model in which the labour supply elasticity is finite. I find that there is a disconnect between the response of wages to monetary shocks and the open economy variables. The labour supply elasticity plays a more important role in the transmission of technology and government expenditure shocks to the real exchange rate and the terms of trade.     

Bayesian–Walrasian equilibria: beyond the rational expectations equilibrium

In general rational expectations equilibrium (REE), as introduced in Radner (Econometrica 47:655–678, 1978) in an Arrow–Debreu–McKenzie setting with uncertainty, does not exist. Moreover, it fails to be fully Pareto optimal and incentive compatible and is also not implementable as a perfect Bayesian equilibrium of an extensive form game (Glycopantis et al. in Econ Theory 26:765–791, 2005). The lack of all the above properties is mainly due to the fact that the agents are supposed to predict the equilibrium market clearing price (as agent’s expected maximized utility is conditioned on the information that equilibrium prices reveal), which leads inevitably to the presumption that agents know all the primitives in the economy, i.e., random initial endowments, random utility functions and private information sets. To get around this problematic equilibrium notion, we introduce a new concept called Bayesian–Walrasian equilibrium (BWE) which has Bayesian features. In particular, agents try to predict the market-clearing prices using Bayesian updating and evaluate their consumption in terms of Bayesian price estimates, which are different for each individual. In this framework agents maximize expected utility conditioned on their own private information about the state of nature, subject to a Bayesian estimated budget constraint. Market clearing is not an intrinsic part of the definition of BWE. However, both in the case of perfect foresight and in the case of symmetric information BWE leads to a statewise market clearing; it then becomes an ex post Walrasian equilibrium allocation. This new BWE exists under standard assumptions, in contrast to the REE. In particular, we show that our new BWE exists in the well-known example in Kreps (J Econ Theory 14:32–43, 1977), where REE fails to exist. This work was done in the Spring of 2005, when EJB was a visiting professor at the University of Illinois.     

Understanding some recent existence results for discontinuous games

We introduce a new condition, weak better-reply security, and show that every compact, locally convex, metric, quasiconcave and weakly better-reply secure game has a Nash equilibrium. This result is established using simple generalizations of classical ideas. Furthermore, we show that, when players’ action spaces are metric and locally convex, it implies the existence results of Reny (Econometrica 67:1029–1056, 1999) and Carmona (J Econ Theory 144:1333–1340, 2009) and that it is equivalent to a recent result of Barelli and Soza (On the Existence of Nash Equilibria in Discontinuous and Qualitative Games, University of Rochester, Rochester, 2009). Our general existence result also implies a new existence result for weakly upper reciprocally semicontinuous and weakly payoff secure games that satisfy a strong quasiconcavity property.     

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).