## European Workshop on General Equilibrium Theory

#### May 22-23, 1998

The conference was organized by CREI jointly with the Caixa de Manresa. The program was prepared by a scientific committee, chaired by professor Luis Corchon, (UPF, Barcelona and U. Carlos III, Madrid), and formed by professors Claude d'Aspremont, CORE (U. Catholique de Louvain), Michele Boldrin, (U. Carlos III, Madrid), Xavier Calsamiglia, (UPF, Barcelona), Egbert Dierker, (U. of Vienne), Birgit Grodal, (U. of Copenhaguen), Michel Lebreton, (GREQAM, Marseille), Andreu Mas-Colell, (UPF, Barcelona) and Walter Trockel, (U. of Bielefeld).

The conference presentations were focused on the recent advances of the General Equilibrium Theory. Issues discussed were the role of alternative ways to coordinate economic decisions, the effects of the market structure and the consequences of asymmetric information. Elements of bounded rationality and fairness considerations were introduced as constraints on the behavior of agents.

### Program

__FRIDAY, MAY 22, 1998__

__INCOMPLETE MARKETS__

**Sunspots and Incomplete Markets with Real Assets **

.N. Laguecir (BETA, Strassbourg).

**A Remark on Rational Expectations Equilibria with Incomplete Markets and Real Assets. **

H. Stahn (BETA, Strassbourg)

__LARGE ECONOMIES__

**Manipulation-Proof Equilibrium in Atomless Economies with Commodity Differentiation **

C. Herves (U. Vigo), E. Moreno and M. R. Pascoa (U. Nova de Lisboa)

**Clubs and the Market: Large Finite Economies**.

B. Ellickson (U. California, L.A.), B. Grodal (U. Copenhagen), S. Scotchmer (U. California, Berkeley) and W. Zame (U. California, L.A.).

**A Note on the Core of a Continuum Economy with Infinitely Many Commodities**

C. Herves (U. Vigo), E. Moreno (U. Nova de Lisboa), C. Moreno (U. Carlos III, Madrid) and M. R. Pascoa (U. Nova de Lisboa).

__MATCHING__

**Unemployment, Optimal Waiting and Queues**

C. Carrera, A. Rodrigo and M. Vazquez (U. Complutense, Madrid)

**A Simple Model of Multiple Equilibria Based on Risk**

J. Costain (Pompeu Fabra, Barcelona).

__MONEY __

**Expectations, Coordination and Business Cycle Stabilization Through Monetary Policy **

S. Gauthier (DELTA, Paris).

**On the Positive Fundamental Value of Money with Short-Sale Constraints: A Comment on two Close Examples**

E. Gimenez-Fernandez (U. Vigo).

__AGGREGATION __

**The Monotonicity of Individual and Market Demand. **

J. K.-H. Quah (Oxford)

**Heterogeneous Households´ Intertemporal Characteristics and the Aggregation Problem **

I. Maret (BETA, Strasbourg)

__SATURDAY, MAY 23, 1998__

__MARKET GAMES__

**Walrasian Allocation without Price-Taking Behavior **

R. Serrano (Brown U., Providence).

**Bid-Ask Competition with Asymmetric Information between Market Maker**s

R. Calcagno and S. Lovo (CORE, Louvain)

**Product Differentiation and Market Power **

E. Dierker and H. Dierker (U. Vienna)

__SOCIAL SECURITY__

**Demography and Pensions. A General Equilibrium Analysis of Spain **

M Montero (U. Vigo).

**Voting on Social Security Reforms with Heterogeneous Agents **

J.C. Conesa (U. Barcelona) and D. Krueger (U. Minnesota)

__FINANCE AND MORAL HAZARD__

**A Sequence of Pareto Improving Financial Innovations **

C. Hara (U. Cambridge).

**Moral Hazard and Nominal, Linear Financial Contracts**

A. Citanna (Carnegie Mellon U.)

**Redistribution as a Selection Device **

H. P. Grüner (U. Bonn)

__DYNAMIC GENERAL EQUILIBRIUM __

**On the Incompatibility between Chaos and Patience**

C. Guerrero (U. Alicante)

**Time Reversibility in overlapping Generations Economies under Extrinsic Uncertainty**

J. Davila (U. Autonoma de Barcelona)

### Report

**Sunspots and Incomplete Markets with Real Asset**

*Nadjette Laguecir (BETA, University Louis Pasteur, Strasbourg)*

**Abstract**: This paper deals with the effects of extrinsic uncertainty or sunspots on competitive equilibrium when markets are incomplete. For the two-period, pure-exchange economy with real assets (yielding returns specified in units of goods and sunspot independent), the generic existence of sunspot equilibria is established. In fact, there is a way of finding an asset structure such that, generically in the space of utility functions, the economy with this asset structure has a regular sunspot equilibrium. The proof consists in using an auxiliary economy (with nominal assets) to detect sunspot equilibria in an economy with real assets. This work is an attempt to generalize the particular case of Gottardi and Kajii (1995) to many assets and many states.

**Discussion**: Jan Werner (University of Minnesota) asked about the case in which the number of states is larger than the number of goods. Nadjette Laguecir said that she did not know how her results would change because the method of proof that she developed in the paper does not work in this case.

**A Remark on Rational Expectation Equilibria with Incomplete Markets and Real Assets**

*Herbert Stahn (Beta, Strasbourg)*

**Abstract**: The purpose of this paper is to establish the generic existence of a fully revealing rational expectation equilibrium in an economy characterized by incomplete markets and real assets. The method of the proof is closed to the one used by Radner and makes extensively use of standard results on economies with real assets which were obtained by Duffie-Shaffer.

**Discussion**: Roberto Serrano (Brown University) asked if full revelation occurred at state 0 or 1. Herbert Stahn replied that it occurred at state 1. Jan Werner (University of Minnesota) asked if agents need to know all prices for full revelation to occur. Herbert Stahn’s answer was no. Finally, Egbert Dierker (University of Vienna) pointed out that it is not clear how the auctioneer is informed of future prices. Herbert Stahn agreed that, in the context of rational expectations, the assumption that anticipated prices are quoted by an auctioneer is problematic since the process of information acquisition by the auctioneer is not modeled.

**Manipulation Proof Equilibrium in Atomless Economies with Commodity Differentiation**

*Carlos Hervés (Universidade de Vigo) (speaker)
Emma Moreno (Universidade Nova de Lisboa)
Mario Rui Pascoa (Universidade Nova de Lisboa)*

**Abstract**: In this paper we introduce a perfect competition test which checks the incentives of arbitrarily small coalitions to behave strategically in endowments and preferences. We apply this coalitional incentive compatibility test to atomless economies with a continuum of differentiated commodities. We show that, under thickness conditions, economies with a finite number of types and economies whose set of agents’ preferences is compact, pass this perfect competition test. Limiting results for replica economies are also presented.

**Discussion**: Roberto Serrano (Brown University) complained that the paper was not couched in the standard implementation framework, but rather in the concept of incentive compatibility. He prompted the authors to modify the paper and to be more specific about hidden assumptions on the information of agents. Luis Corchón (Universitat Pompeu Fabra) pointed out that the definition of manipulability used by the authors imply that agents had complete information about the characteristics of the whole economy. Egbert Dierker (University of Vienna) asked what would happen to the results if consumption bundles were replaced by net trades. Finally Birgit Grodal (University of Copenhagen) asked if the dimensionality of the commodity space matters to the results. Carlos Hervés answered that it does.

**Clubs and the Market: Large Finite Economies**

*Ben Ellickson (University of California, Los Angeles)
Birgit Grodal (University of Copenhagen) (speaker)
Susan Scotchmer (University of California, Berkeley)
William Zame (University of California, Los Angeles)*

**Abstract**: We study large finite club economies, in which agents trade multiple private goods widely in the market, can belong to several clubs, and care about the characteristics of the other members of their clubs. Feasibility requires that the club choices of the agents must be consistent in aggregate. Consequently equilibria and core states might not exist. We show existence of an approximate equilibrium in which all clubs are cost minimizing and have balanced budgets and all agents are in their budget sets, but we allow that at most *S* agents are nonoptimizing, where *S* is independent of the number of agents in the economy. We also define a notion of approximate core, show existence of approximate core states, and show that approximate core states can approximately be decentralized when the economy is large.

**Discussion**: Jim Costain (Universitat Pompeu Fabra) asked what would happen if characteristics were considered to be a continuous variable. Birgit Grodal replied that it would not make any difference to the results reported in the paper. Salvador Barberà (Universitat Autònoma de Barcelona) and Egbert Dierker (University of Vienna) wanted to know what is really new in the approach presented in the paper in relationship with the literature on public goods. Birgit Grodal replied that the novelty of the paper consisted in being more careful and specific when modeling personal characteristics. This was the key point that allowed the authors to derive certain characteristics of the equilibrium set.

**A Note on the Core of a Continuum Economy with Infinitely Many Commodities”**

*Carlos Hervés (Universidade de Vigo)
Emma Moreno (Universidade Nova de Lisboa) (speaker)
Carmelo Moreno (Universidad Carlos III)
Mario Rui Pascoa (Universidade Nova de Lisboa)*

**Abstract**: We consider a continuum economy with infinitely many commodities and show that, for any positive e the core of the economy coincides with the set of allocations which are not blocked by any coalition with measure less than e . Actually, our main result is an extension of Grodal’s (1972) result and, therefore, Schmeidler’s (1972) result to economies whose commodity space is l ¥ for myopic preferences

**Discussion**: Jan Werner (University of Minnesota) asked about the choice of topology. In particular he wanted to know why the authors chose the weakstar topology instead of the more standard Mackey topology. Carlos Hervés replied that it would not make any difference since consumption sets were compact and under this assumption both topologies coincide. Birgit Grodal (University of Copenhagen) asked if the assumption that consumers are uniformly distributed in the interval [0, 1] could be replaced by the more general assumption of an abstract measure space. Emma Moreno said that such a generalization would affect the nature of results. Finally Roberto Serrano (Brown University) asked if it would be possible to consider coalitions with a finite number of members.

**Unemployment, Optimal Waiting and Queues**

*Carmen Carrera (speaker)
Antonio Rodrigo and Mercedes Vázquez (Universidad Complutense, Madrid)*

**Abstract**: In this article we obtain the steady state distributions of unemployment, the number of vacancies, wages and job durations as a result of the optimal waiting by firms, given the characteristics of technical progress and the distribution of labor qualifications among the unemployed. We model the labor market as a queuing system providing a probabilistic dependence between unemployment and vacancies thus generalizing the usual deterministic relationships established in the literature.

**Discussion**: Joan Maria Esteban (Institut d’Anàlisi Econòmica) asked if, in the model presented here, firms choose how much time they are active. Carmen Carrera answered that they do. Luis Corchón (Universitat Pompeu Fabra) encouraged the authors to be more specific about the kind of problem they address.

**A Simple Model of Multiple Equilibria Based on Risk**

*Jim Costain (Universitat Pompeu Fabra)*

**Abstract**: This paper proposes a business cycle propagation mechanism which may arise in economies with imperfect insurance and multiple assets. We study a two-period job matching model with risk averse agents who act both as workers and as entrepreneurs. They choose between two types of investment; one type is riskless, while the other is a risky activity that creates jobs. We compare several insurance environments. Under perfect insurance, the model has a unique equilibrium, unless there are increasing returns to scale. If investment is fully insured then uninsured unemployment risk implies a negative feedback effect which dampens output fluctuations. However, if both investment and employment are uninsured, a positive feedback effect is also present, because increased unemployment risk causes agents to shift investment away from the risky asset, further increasing unemployment. Thus we construct several examples with multiple, interior, Pareto-ranked equilibria under decreasing returns to scale. Even when the equilibrium is unique, the positive feedback magnifies exogenous shocks to output. The mechanism should extent to other types of risk, such as bankruptcy. It may be quantitatively important, because small changes in output may imply large changes in risk, and because portfolio composition changes are likely to be larger than changes in the level of investment.

**Discussion**: Roberto Serrano (Brown University) pointed out that, contrarily to what was assumed in the paper, an increase in risk may lead to agents to be more prone to take risks as it happened in the Savings & Loans crisis in USA. Jim Costain said that he was aware of this difficulty and expressed his hope of disposing of this assumption in a future version of the paper. Roberto Serrano also asked about asymmetric equilibria. Jim Costain said that there was none. Finally Michele Boldrin (Universidad Carlos III) expressed his doubts about the adequacy of the particular dynamic process used in the paper.

**Expectations, Coordination and Business Cycle Stabilization Through Monetary Policy**

*Stéphane Gauthier (Delta, Paris)*

**Abstract**: The paper provides the conditions under which agents coordinate their expectations on a rational expectations equilibrium when information is asymmetrically distributed, in an overlapping generations economy with production. It exhibits a class of monetary policies improving the likelihood of deterministic cycles and Markovian sunspot equilibria. These policies necessarily lead to expand nominal fluctuations with respect to the *laissez-faire* dynamics. Therefore, coordination and stabilization are two inconsistent goals. This implies that nominal magnitudes play a fundamental role, although monetary policies are neutral in real terms, from the coordination view point.

**Discussion**: Herakles Polemarchakis (CORE, Université Catholique de Louvain) asked if the number of goods is a variable or not. Stéphane Gauthier answered that the number of goods was fixed in his model.

**On the Positive Fundamental Value of Money with Short-Sale Constraints: A Comment on Two Close Examples**

*Eduardo Giménez Fernández (Universidade de Vigo)*

**Abstract**: This paper reviews two close examples given by Kocherlakota (1992) and Santos and Woodford (1997). In an intertemporal infinite horizon general equilibrium model with short-sale constraints both articles consider that the positive price of money is a pure bubble pricing. Following recent literature on monetary theory, we argue that in these examples fundamental value of money may be positive, as money represents a claim to an infinite stream of future services represented by its role in the trading process. We also find that Santos and Woodford’s example is a particular case of our own.

**Discussion**: Joan Maria Esteban (Institut d’Anàlisi Econòmica) pointed out that the model presented here was isomorphic to an overlapping generation model.

**The Monotonicity of Individual and Market Demand**

*John Quah (Oxford University)*

**Abstract**: This paper studies the monotonicity of individual and market demand with the aid of the indirect utility function. This is an interesting issue because, amongst other reasons, it has implications for the uniqueness and stability of equilibria in general equilibrium models. We identify conditions on an agent’s indirect utility function which will guarantee that he has a monotonic demand function. A result in demand theory due to Hildenbrand says that market demand is monotonic when the income distribution can be represented by a downward sloping density function (even though individual demand might violate monotonicity). Using the indirect utility function, we introduce a measure of violations of individual monotonicity that allows us to identify a larger class of density functions that will generate a monotonic market demand.

**Discussion**: Luis Corchón (Universitat Pompeu Fabra) pointed out that the assumption that poor agents do not have Giffen goods is unconvincing: According to him most textbook examples of Giffen goods are given in the context of relatively poor agents. Roberto Serrano (Brown University) commented that you need this assumption or something very similar if you want to obtain monotonicity of market demand. However Luis Corchón and Birgit Grodal (University of Copenhagen) believed that the assumption presented in this paper was sufficient, but far from being necessary.

**Heterogeneous Households’ Intertemporal Characteristics and the Aggregation Problem**

*Isabel Maret (BETA, Université Louis Pasteur, Strasbourg)*

**Abstract**: Aggregation in exchange markets is studied by imposing restrictions on the distribution of the households’ intertemporal characteristics. Approximate bounds for the derivatives of market demand that depend upon specific measure of sensitivity of the distribution of households’ intertemporal choices to changes in the real interest rate are provided. This has strong consequences for the prevalence, in the aggregate, of gross substitutability between current and future consumption.

**Discussion**: Joan Maria Esteban (Institut d’Anàlisi Econòmica) pointed out that the results presented in this paper were easily generalizable.

**Walrasian Allocation Without PriceTaking Behavior**

*Roberto Serrano and Oscar Volij (Brown University)*

**Abstract**: Consider an exchange economy with a *finite* number of agents, who are arbitrageurs, in that they try to upset allocations imagining plausible beneficial trades. Their thought process is interactive, in that agents are conscious that the others are also going through the same steps. With this introspective process, each agent constructs a *supermarket*, i.e., a set of bundles that he considers achievable, in the sense that a sequence of plausible trades with other agents yields those bundles. We shed additional light on a result of Dagan’s (1996) by showing that Walrasian allocations can be characterized also as those where each agent chooses optimally from his supermarket. In addition, we extend the analysis to economies without short sales, where the characterization of Walrasian allocations is also obtained. Our analysis provides a different behavioral assumption for Walrasian allocations and connects with the core convergence theorem.

**Discussion**: Beth Allen (University of Minnesota) pointed out that the mechanism presented in this paper resemble the “Greed Process” introduced by Hurwicz. Birgit Grodal (University of Copenhagen) asked if the main argument in the paper was connected to the argument of Schmeidler and Vind showing that Walrasian net trades are fair net trades. Finally, Jan Werner (University of Minnesota) pointed out that the message of the paper would be clearer if the argument were couched in terms of arbitrage and contracts.

**Bid-Ask Competition with Asymmetric Information between Market Makers**

*Ricardo Calcagno (speaker)
S. Lovo (CORE, Université Catholique de Louvain)*

**Abstract**: We consider the effect of asymmetric information on price formation process in a financial market where private information is held by a market maker. A Bayesian game is proposed in which there is price competition between two market makers with two different information partitions. At each stage players set bid and ask prices simultaneously and then trade occurs between market maker who proposes the most profitable price and liquidity traders. We characterize a set of partially revealing equilibra where the informed market marker’s prices do not convey his private information. Informed player’s equilibrium payoffs are proportional to prior beliefs of the market.

**Discussion**: Beth Allen did not like the assumption that uninformed market makers make zero profits. In her opinion this assumption makes it hard to explain why such traders remain in the market. Ricardo Calcagno said that this was an standard assumption in auction theory.

**Product Differentiation and Market Power**

*Egber Dierker and Hildegard Dierker (speaker) (University of Vienna)*

**Abstract**: Assuming symmetry across firms and constant unit costs Perloff and Salop (1985) show: If product differentiation increases, prices rise in a symmetric equilibrium. This raises the question of whether, in general, more product differentiation leads to higher market prices. Giving up the symmetry and the constant unit costs assumptions we present examples in which at least one firm lowers its equilibrium price when product differentiation increases. We formulate a model of product differentiation and state and discuss, within the theory of supermodular games, conditions ensuring that all firms raise their prices in a Nash equilibrium if product differentiation increases.

**Discussion**: Luis Corchón (Universitat Pompeu Fabra) was curious about the fact that dominant diagonal becomes a necessary and sufficient condition of uniqueness of oligopolistic equilibrium, when in the perfectly competitive framework this condition is only sufficient.

**Demography and Pensions: A General Equilibrium Analysis of the Case of Spain**

*María Montero (Universidade de Vigo)*

**Abstract**: This paper uses an applied general equilibrium model to assess the effects on the viability of the Spanish social security system of changes in demographics, focusing on the balanced growth path. First we assume that the population growth rate is zero, then we simulate the effects of several alternatives policies: hold constant the tax rate, hold constant the initial level of benefit per capita, hold constant the total payments, and delay the retirement age.

**Discussion**: Antonio Cabrales (Universitat Pompeu Fabra) pointed out that in the model presented here when pensions decrease, savings decrease and that this is counterintuitive. Yuri Yegorov (Universitat Pompeu Fabra) said that the assumption of stationarity is both crucial and debatable. In his opinion the welfare losses due to the lack of stationarity were far larger than the welfare losses detected in this study that were small.

**Voting on Social Security Reforms with Heterogeneous Agents**

*Juan Carlos Conesa (Universitat de Barcelona) (speaker)
D. Krueger (University of Minnesota)*

**Abstract**: This paper analyzes the quantitative role of idiosyncratic uncertainty in an economy in which rational agents vote on hypothetical social security reforms. We construct an Overlapping Generations economy in which individuals face idiosyncratic risk with respect to their labor productivity and the labor-leisure decision is endogenous. We find that the role of a pay-as-you-go social security system as a partial insurance device significantly reduces political support for a transition to an economy with a fully funded system. Also, smoother transitions have less political support than more rapid transitions. We conclude that the status quo bias in favor of an unfunded social security system is stronger in economics in which agents of similar age differ significantly with respect to labor earnings and wealth due to idiosyncratic uncertainty.

**Discussion**: Antonio Cabrales (Universitat Pompeu Fabra) said that the model assumes that the policies voted by agents can not be changed by the following generation. He asked what would happen if such a commitment was not feasible. Juan Carlos Conesa answered that the results obtained would change and that policies will switch in time.

**A Sequence of Pareto Improving Financial Innovations**

*Chiaki Hara (University of Cambridge)*

**Abstract**: We consider a simple model of a pure exchange economy under uncertainty with two periods, one physical good, and finitely many states of the world. Denoting the number of the states by S, we prove that there exists a sequence of S securities, all in zero net supply, such that if they are introduced into markets one by one in the order of the sequence, then the security markets eventually become complete while the prices of any security is not affected by subsequent introduction of newer securities. Since there is no pecuniary externality along such a sequence, the sequence is Pareto improving.

**Discussion**: John Quah (Oxford Univeristy) asked if the sequence of Pareto improving financial innovations was unique. Chiaki Hara said that it was essentially unique. Antonio Cabrales (Universitat Pompeu Fabra) and Herakles Polemarchakis (CORE, Université de Louvain) asked if equilibrium was supposed to be unique. The answer was that equilibrium was not necessarily unique in this model but that there was an exogenously given mechanism of selecting equilibria.

**Moral Hazard and Nominal, Linear Financial Contracts**

*Alessandro Citanna (Carnegie Mellon University)*

**Abstract**: We study competitive equilibria with moral hazard in economies with aggregate risk and where trading occurs with an incomplete set of financial assets (this version preliminarly analyzes economies with trading in only one asset). The main conclusion of the paper is that, contrary to the individual risk economies, moral hazard is compatible with trading in competitive linear financial contracts, and gives rise to no manipulation problem. We establish existence of nonmanipulable equilibria provided that there are no relative price effects (e.g. a one-commodity economy), and that financial markets display nonlinearly homogeneous payoffs (e.g., nominal) and are sufficiently incomplete. Finally, we justify the linear contract as the optimal pricing schedule in a specific trading game with an auctioneer.

**Discussion**: Luis Corchón (Universitat Pompeu Fabra) asked the author about the relationship of his findings with the existing literature.

**Redistribution as a Selection Device**

*HansPeter Grüner (University of Bonn)*

**Abstract**: This paper studies the role of the wealth distribution for the market selection of entrepreneurs when agents differ in talent. It argues that the redistribution of initial endowments can increase an economy’s surplus because more talented individuals get credit for their risky investment projects. Moreover, the redistribution of initial endowments may lead to a Pareto-improvement although all agents are nonsatiable. In my model an agent’s entrepreneurial ability is his private information. Moral hazard in production creates rents for entrepreneurs if they are believed to be both talented and willing to provide entrepreneurial effort. I find conditions such that unproductive rich entrepreneurs crowd out productive poor ones on the capital market. Then redistribution of initial endowments leads to the selection of better entrepreneurs, increases the economy’s surplus, and -in some cases- makes all agents better off.

**On the Incompatibility of Chaos and Patience**

*Cesar Guerrero (Universidad de Alicante)*

**Abstract**: We prove in a precise formulation that chaos has to disappear as the discount factor tends to one. This statement is formulated as a general theorem, which is then applied to some usual indicators of chaos. This generalizes and proves a conjecture stated in Nishimura, Sorger and Yano (Economic Theory, 1994), where they provide a family of strictly concave optimal growth problems such that, for any value of the discount factor there exists a member of the family displaying ergodic chaos, and they then prove that, for their example, some indicators of chaos -topological chaos, among others- tend to zero as the discount factor tends to one. Nevertheless, they do not analyze the case of ergodic chaos, which is one of the applications of the main theorem of this paper.

**Discussion**: Herakles Polemarchakis (CORE, Université Catholique de Louvain) asked about the meaning of genericity assumptions in the paper. Cesar Guerrero said that this assumptions were merely technical.

**Time reversibility and extrinsic uncertainty in overlapping generations economies**

*Julio Dávila (Universitat Autònoma de Barcelona)*

**Abstract**: This paper investigates how time and uncertainty interweave. Specifically, it studies the appearance of symmetries and asymmetries -with respect to the direction in which time flows- in the set of equilibria of simple overlapping generations economies which are open-ended in both directions, when there is extrinsic uncertainty and when there isn’t any. It is shown in what follows that, while all the perfect foresight equilibria of any such overlapping generations economy can be reversed in time -which amounts to look at it as an equilibrium of an economy which is the mirror image of the original one- such general reversibility is lost for equilibria under uncertainty. More specifically, there is no robust overlapping generations economy of this class whose set of reversible finite markovian stationary sunspot equilibria is not negligible, although for any such economy this set is nonempty. This result shows the arrow of time in these overlapping generations economies to be meaningful only under uncertainty.

**Discussion**: Herakles Polemarchakis (CORE, Louvain) commented about the nature of results. Antonio Cabrales and Luis Corchón (Universitat Pompeu Fabra) made the point that a tree representing uncertainty can not be inverted and that this underlies the results obtained in this paper.