Coordination and correlation in Markov rational belief equilibria

Summary This paper studies the effect of correlation in the rational beliefs of agents on the volatility of asset prices. We use the technique of generating variables to study stable and non-stationary processes needed to characterize rational beliefs. We then examine how the stochastic interaction among such variables affects the behavior of a wide class of Rational Belief Equilibria (RBE). The paper demonstrates how to construct a consistent price state space and then shows the existence of RBE for any economy for which such price state space is constructed. Next, the results are used to study the volatility of asset prices via numerical simulation of a two agents model. If beliefs of agents are uniformly dispersed and independent, we would expect heterogeneity of beliefs to have a limited impact on the fluctuations of asset prices. On the other hand, our results show that correlation across agents can have a complex and dramatic effect on the volatility of prices and thus can be the dominant factor in the fluctuation of asset prices. The mechanism generating this effect works through the clustering of beliefs in states of different levels of agreement. In states of agreement the conditional forecasts of the agents tend to fluctuatetogether inducing more volatile asset prices. In states of disagreement the conditional forecasts fluctuatein diverse directions tending to cancel each other's effect on market demand and resulting in reduced price volatility.This research was supported, in part, by the Fondazione Eni Enrico Mattei of Milan, Italy, and by the Research Incentive Fund of Stanford University. The authors thank Carsten K. Nielsen and Ho-Mou Wu for valuable discussions on an earlier draft. Carsten K. Nielsen also made an important contribution to the development of Section 3.     

Asset price volatility and trading volume with rational beliefs

Summary. This paper develops a model of speculative trading in a large economy with a continuum of investors. In our model the investors are assumed to have diverse beliefs which are rational in the sense of being compatible with observed data. We demonstrate the existence of price amplification effects and show that the equilibrium prices can be higher or lower than the rational expectation equilibrium price. It is also shown that trading volume is positively related to the directions of price changes. Moreover, we study how asset price volatility and trading volume are influenced by belief structures, short selling constraints and the amount of fund available for investment.Received: 23 January 2003, Revised: 30 April 2003, JEL Classification Numbers: D84, G12.We are grateful to Professors Mordecai Kurz, Kenneth Arrow, Kenneth Judd, Carsten Nielsen, Maurizio Motolese, Mark Garmaise, Jean-Michel Grandmont, Peter Hammond, Karl Shell, Jan Werner and participants of the Society for the Advancement of Economic Theory (SAET) Conference and Stanford Institute of Theoretical Economics (SITE) Conference for many helpful suggestions. Correspondence to: H.-M. Wu     

Floating exchange rates versus a monetary union under rational beliefs: the role of endogenous uncertainty

Using the concept of ex-post optimality, we compare different exchange rate regimes, including floating exchange rates and fixed exchange rates with a Monetary Union in a two country OLG model with stochastic endowments. The emphasis of this comparison is on the welfare consequences of agents having incorrect beliefs. We do not assume that agents can hold any beliefs, but rather that their beliefs are rational that is consistent with the observed empirical behavior of the economy. We study a large set of possible policies, but two of them have our particular interest. The first policy implies devaluations in reaction to a negative shock, while the other implies a fixed exchange rate. These policies have very different consequences. The first will for generic beliefs not result in an ex-post optimal allocation. The other policy is on the other hand always feasible and results in an ex-post optimal allocation. When the two countries form a Monetary Union, the ex-post optimal allocation is also achieved. The meaning of “endogenous uncertainty” as an institutionally induced uncertainty is illustrated. Received: September 1, 2001; revised version: 24 June 2002 RID="*" ID="*" I would like to thank Horace W. Brock, Gianluca Cassese, Paula Orlando, Ho-Mou Wu as well as seminar participants at Copenhagen Business School, ESEM98, Keio University, Kyoto University, Osaka University, SITE (Stanford) and University of Copenhagen for many useful comments on the paper. I am also grateful to Mark J. Garmaise, Takako Fujiwara-Greve, and an anonymous referee for many helpful suggestions for improving the paper. Without the many discussions about Rational Beliefs and related issues I have had with Mordecai Kurz over the years, the research presented here would not have been possible. Financial support from The Carlsberg Foundation, Danish Social Research Council, University of Copenhagen and SITE is gratefully acknowledged.     

Modelling exchange of probabilistic opinions

Summary. This paper studies how communication or exchange of opinions influences correlation of beliefs. The paper focuses on a situation in which agents communicate with each other infinitely many times without observing data. It is an extension to the ‘Expert Problem’ in Bayesian theory, where the informational flow is asymmetric. Moreover, this paper generalizes the existing literature of communication that employs the common prior assumption (CPA) by allowing for heterogeneous beliefs. Some basic convergence results are shown in contrast with the results obtained under the CPA. Furthermore, several economic implications of the basic results are provided. Received: August 27, 2001; revised version: April 16, 2002 RID="*" ID="*" The results presented in this paper are taken from my Ph.D. thesis at Stanford University. I gratefully acknowledge the inspiration obtained from innumerable discussions with Mordecai Kurz about this subject. Also, I appreciate comments from Kenneth J. Arrow, Peter J. Hammond, Maurizio Motolese, Carsten K. Nielsen, Ho-Mou Wu and the anonymous referee.     

Endogenous uncertainty and market volatility

We advance the theory that the distribution of beliefs in the market is the most important propagation mechanism of economic volatility. Our model is based on the theory of Rational Beliefs (RB) and Rational Belief Equilibrium (RBE) developed by Kurz (1994, 1997). We argue that the diverse market puzzles which are examined, such as the equity premium puzzle, are all driven by the structure of market expectations. In support of our view, we present an RBE model with which we study financial markets. The model is able to simulate the correct order of magnitude of: (i) the long term mean and standard deviation of the price\dividend ratio; (ii) the long term mean and standard deviation of the risky rate of return on equities; (iii) the long term mean and standard deviation of the riskless rate; (iv) the long term mean equity premium. In addition, the model predicts (v) the GARCH property of risky asset returns; (vi) the observed pattern of the predictability of long returns on assets, and (vii) the Forward Discount Bias in foreign exchange markets. The common economic explanation for these phenomena is the existence of heterogenous agents with diverse but correlated beliefs such that some agents are optimistic and some pessimistic about future capital gains. The model has a unique parameterization under which the model makes all the above predictions simultaneously. The parameterization requires the optimists to be in the majority but the rationality of belief conditions of the RBE require the pessimists to have a higher intensity level. In simple terms, the large equity premium and the low equilibrium riskless rate are the result of the fact that at any moment of time there are agents who hold extreme pessimistic beliefs and they have a relatively stronger impact on the market. The paper also studies the effect of correlation of beliefs among investors. It shows that the main effect of such correlation is on the dynamic patterns of asset prices and returns and is hence important for studying such phenomena as stochastic volatility. Received: May 16, 2000; revised version: November 15, 2000     

Endogenous uncertainty and the non-neutrality of money

Summary. We study some implications of the Theory of Rational Beliefs to monetary policy. We show that monetary policy in a Rational Beliefs environment can have an important effect on the characteristics of economic fluctuations. In Rational Beliefs Equilibria money is generically non-neutral unlike Rational Expectations Equilibria in which money is neutral and monetary policy is ineffective. Under Rational Beliefs Equilibria nominal prices and real output change not only in response to changes in the exogenous growth rate of money but also in response to changes in the state of beliefs. In Rational Beliefs Equilibria monetary shocks have real effects even when they are observed but are not fully anticipated. Furthermore, the non-neutrality of money results in a short run Phillips curve. When money “flutters, real output sputters” [8]. We show that Endogenous Uncertainty and the distribution of market beliefs are the major explanatory variables of such fluctuations. Under Rational Expectations monetary policy is ineffective because agents neutralize it by predicting correctly the effect of the policy. Under Rational Beliefs it is shown instead that inflation and recessions can be substantially aggravated by the distribution of market beliefs. Received: January 14, 2002; revised version: April 5, 2002 RID="*" ID="*" I would like to thank Mordecai Kurz for his constant help and support. Most of the ideas developed hereby have been inspired by innumerable and fruitful discussions with him. I have also greatly benefited from helpful comments by Stanley Black, Luigi Campiglio, Carsten Nielsen and Ho-Mou Wu. I also received valuable remarks from participants at the V meeting of “The Society for the Advancement of Economic Theory” held in Ischia, Italy, on July 2-8, 2001, where an initial draft of the present work was presented.     

Non-stationary,stable Markov processes on a continuous state space

We extend our previous result on simple stable Markov (SSM) processes to the case where the state space is continuous. As anapplication we show the existence of a competitive general equilibrium of a cobweb model where price volatility is generated both by exogenous shocks and by stochastic, so called generating variables (that may be interpreted as sunspots) that govern the correlation of the rational beliefs of individual agents. I would like to thank Danish Social Science Foundation, The Carlsberg Foundation, Stanford Institute for Theoretical Economics (SITE) and Universita Cattolica for financial help. I would also like to thank Trinidad Casasus, Mordecai Kurz, an anonymous referee as well as participants at SITE, ESAM (2004) and ESEM (2004) for helpful discussions and comments. Peter Harremoes provided me with an illuminating counter example and Hiro Nakata provided many comments that helped improve the exposition of the paper. Part of this work is from a paper previously circulated under the title: “Sunspot rational belief structures: anonymity and endogenous uncertainty”.     

Rational belief structures and rational belief equilibria

Summary The paper introduces some simplifying tools and methods for studying Rational Beliefs and for proving existence of Rational Belief Equilibria. We identify a set of stable non-stationary stochastic processes, named SIDS processes. Furthermore we introduce the concept of a Rational Belief Structure, which formulates the Rational Beliefs of the agents as beliefs about the distribution of exogenous variables and the beliefs of other agents. The use of the developed apparatus is demonstrated by showing existence of a set of Rational Belief Equilibria in an Overlapping Generations Model with money and one commodity.The results presented in this paper are taken from my Ph.D. thesis from Stanford University ([13]). The dissertation is devoted to the study of the theory of Rational Beliefs as developed by Mordecai Kurz. I gratefully acknowledge the inspiration obtained from innumerable discussions with him over the years about this subject. His suggestions were instrumental for writing the thesis and this paper. The paper benefited much from a thorough reading by Trinidad Casasus, Mordecai Kurz and Ho-Mou Wu. Financial support from The Academy for Research (Forskerakademiet), Aarhus, Denmark and The University of Copenhagen, during my time as a Ph.D. student and from Danish Social Science Research Council thereafter and in addition from Fondazione Eni Enrico Mattei of Milan, Italy is gratefully acknowledged.     

Random belief equilibrium in normal form games

In defining random belief equilibrium (RBE) in finite, normal form games we assume a player's beliefs about others' strategy choices are randomly drawn from a belief distribution that is dispersed around a central strategy profile, the focus. At an RBE: (1) Each chooses a best response relative to her beliefs. (2) Each player's expected choice coincides with the focus of the other players' belief distributions. RBE provides a statistical framework for estimation which we apply to data from three experimental games. We also characterize the limit-RBE as players' beliefs converge to certainty. When atoms in the belief distributions vanish in the limit, not all limit-RBE (called robust equilibria) are trembling hand perfect Nash equilibria and not all perfect equilibria are robust.     

Endogenous fluctuations in the demand for education

Abstract. Enrollment rates to higher education reveal a quite large variation over time which cannot be explained by productivity shocks alone. We develop a human capital investment model in an overlapping generations framework that features endogenous fluctuations in the demand for education. Agents are heterogeneous in their beliefs about future wage differentials. An evolutionary competition between the heterogeneous beliefs determines the fraction of the newborn generation having a certain belief. Costly access to information on the returns to education induces agents to use potentially destabilizing backward looking prediction rules. Only if previous generations experience regret about their human capital investment decisions, will agents choose a more sophisticated prediction rule that dampens the cycle. Access to information becomes key for stable flows to higher education. RID="*" ID="*"We would like to thank Cars Hommes, Florian Wagener, seminar participants at the University of Amsterdam, participants of the workshop on ‘Skill Needs and Labor Market Dynamics’ at the Wissenschaftszentrum Berlin für Sozialforschung (WZB) for helpful discussions, and an editor of this Journal and three anonymous referees for their comments. Tuinstra's research is supported by the Netherlands Organisation for Scientific Research (NWO) under a MaG-Pionier grant. Neugart acknowledges financial support from the German Ministry of Education. Parts of the research were done while Tuinstra was visiting the WZB and when Neugart was visiting CeNDEF. Correspondence to: The research for this paper was done while the first author was affiliated with the Wissenschaftszentrum Berlin für Sozialforschung.     

The option of joint purchase in vertically differentiated markets

Summary. Within the framework proposed by Mussa and Rosen (1978) for modelling quality differentiation, consumers are assumed to make mutually exclusive purchases. A unique pure strategy equilibrium exists in this case. In this note, we allow consumers to buy simultaneously different variants of the differentiated good. We call this the “joint purchase option”. The paper proposes a detailed analysis of price competition when this option is opened: first, we show that either uniqueness, or multiplicity, or absence of price equilibrium arise, depending on the utility derived from joint purchase relative to exclusive purchase. Second, we characterize these equilibria, whenever they exist. Received: July 25, 2001; revised version: October 21, 2002 RID="*" ID="*" The second author gratefully acknowledges the financial support from Interuniversity Attraction Pole Program- Belgian State- Federal Office for Scientific, Technical and Cultural Affairs under contract PAI 5/26. Correspondence to: X.Y. Wauthy     

Mutually acceptable courses of action

We offer a game-theoretic framework that enables the analysis of situations whereby rational individuals with different beliefs and views of the world agree to a shared course of action. We introduce a new solution concept: a mutually acceptable course of action, which can be viewed as an (incomplete) contract or a social norm that free rational individuals would be willing to follow for their own diverse reasons. We show that by varying the degree of completeness of the underlying course of action, our concept can be related to commonly used solutions, such as perfect equilibrium, perfect Bayesian equilibrium, (rationalizable) self-confirming equilibrium, and rationalizable outcomes. We are grateful to the editor and an anonymous referee for very useful and helpful comments and suggestions. Earlier versions of this paper were circulated under the title “Towering over Babel: Worlds Apart but Acting Together”. We thank participants at the World Congress of Game Theory Society (2004), the International Conference on Game Theory (2004), the European Meeting (2004), and the Econometric Society World Congress (2005). Financial support from SSHRC and NSERC of Canada, and the National Science Council of Taiwan are gratefully acknowledged.     

How does the diversity of investors’ beliefs affect stock price informativeness?

A growing number of studies have investigated the role of stock prices in aggregating private information and guiding resource reallocation. However, this article may be the first attempt to study how the diversity of beliefs affects stock price informativeness. The framework of the noisy rational expectations model shows that stock informativeness is determined by both the precision and use of private information in trading. If private beliefs about the value are highly diverse, the aggregate average opinion revealed in a stock’s price will be more accurate and, thus, more informative. As the price becomes more informative, however, individual investors will rely less on their private information. When this occurs, less private information will be absorbed in price, which, in turn, reduces price informativeness. Our model shows that the relationship between belief diversity and price informativeness is U-shaped in equilibrium.     

Pooling surveys in the estimation of income and price elasticities: An application to Tunisian households

This paper presents a methodological extension of Deaton's (1990) model for estimating price elasticities, by pooling Tunisian data from several surveys to improve the inter-cluster variability of unit values which is one of the key elements used in the derivation of these elasticities. Since the surveys cover a relatively long period, possible structural changes in consumption behaviour occurring over time are accounted for by postulating that certain response coefficients of the basic model vary from one survey to the other. The own price and cross price elasticities calculated using appropriate estimates of the extended model are satisfactory both from the economic point of view of their sign and the statistical point of view of their significance and superior to those obtained using a single survey. First version received: April 2000/Final version received: June 2001 RID="*" ID="*"  The authors would like to thank Angus Deaton and anonymous referees for useful comments and suggestions.     

Effect of credible quality investment with Bertrand and Cournot competition

We show how credible revelation and ability to commit to quality choice effect equilibrium qualities and welfare when product market is either Bertrand or Cournot competition. We show that results depend on the type of competition but not generally on the cost of quality function. We show that with Bertrand competition, the equilibrium qualities are lower with credible commitment. Competition is moderated and producer surplus is higher and consumer surplus lower. With Cournot competition, higher quality will be better but lower quality will be worse with credible commitment. Consumer surplus is always greater with credible commitment and if cost does not increase too quickly with quality, producer surplus will also increase. Thus credible commitment is a collusive device with Bertrand competition but it can improve social welfare with Cournot competition. Received: February 8, 2000; revised version: February 14, 2002 RID="*" ID="*" The idea of this paper originated in the weekly workshops of Mordecai Kurz at Stanford. I am forever in debted to Mordecai and fellow students – Luis Cabral, Peter DeMarzo, John Hillas, Michihiro Kandori, Steve Langois, Patrick McAllister, Steve Sharpe, Peter Streufert, Steve Turnbull and Gyu-Ho Wang – for their criticism and encouragement. I also benefited from comments from Yi-Heng Chen, Jin-Li Hu, Kala Krishna, Jinji Naoto, Thomas J. Prusa, and Shyh-Fang Ueng at various later stages of this work. Last but not least, I am grateful for the detailed comments of the referee.     

Arbitrary small indivisibilities

Summary. Arbitrary small indivisibilities may play an important role when the strong survival assumption does not hold. A hierarchic price is a finite ordered family of price vectors . It extends the notion of exchange values proposed by Gay [15]. These price notions were introduced in order to establish the existence of a generalized competitive equilibrium without the strong survival assumption. We show that a hierarchic price models phenomena related to small indivisibilities which the standard approach may not capture. More precisely, we prove in the framework of linear exchange economies that a hierarchic price may be seen as a standard price of an economy with arbitrary small indivisibilities. Received: September 25, 2001; revised version: November 25, 2002 RID="*" ID="*" This paper was partly written at Departamento de Matemáticas of Universidad de Vigo.     

Estimation of technical inefficiency effects using panel data and doubly heteroscedastic stochastic production frontiers

In previous studies, measures of technical inefficiency effects derived from stochastic production frontiers have been estimated from residuals which are sensitive to specification errors. This study corrects for this inaccuracy by extending the doubly heteroscedastic stochastic cost frontier suggested by Hadri (1999) to the model for technical inefficiency effects. This model is a stochastic frontier production function for panel data as proposed by Battese and Coelli (1995). The study uses, for illustration of the techniques, data on 101 mainly cereal farms in England. We find that the correction for heteroscedasticity is supported by the data. Both point estimates and confidence intervals for technical efficiencies are provided. The confidence intervals are constructed by extending the “Battese-Coelli” method reported by Horrace and Schmidt (1996) by allowing the technical inefficiency to be time varying and the disturbance terms to be heteroscedastic. The confidence intervals reveal the precision of technical efficiency estimates and show the deficiencies of making inferences based exclusively on point estimates. First version received: March 2000/Final version received: Oct. 2001 RID="*" ID="*"  The authors are grateful to the Economic and Social Research Council for access to their Data Archive which has provided the data for this research. We are indebted to Badi Baltagi and two anonymous referees for their helpful comments and suggestions. The usual caveat applies.     

Do national business cycles have an international origin?

This paper analyses a set of output data for 113 countries and identifies common sources of fluctuations, to estimate a world business cycle. We also analyze the multiplier effects of worldwide or global shocks and their implications for the persistence and amplitude of cyclical fluctuations. We find that a higher relative importance of global shocks leads to lower persistence and greater volatility. Finally, we compare some regional integration agreements and find that the EC emerges as the most integrated block. The analytical framework used is that of Forni, Hallin, Lippi and Reichlin (1999). First version: November 1999/Final version accepted: November 2001 RID="*" ID="*"  Université Libre de Bruxelles, DULBEA – CP 140, 50 av. Franklin Roosevelt, 1050 Bruxelles, Belgium. We are grateful towards two anonymous referees for their really helpful remarks. We also wish to express our deepest gratitude towards Lucrezia Reichlin for her very precious advice. We also would like to thank Jorge Rodigez and Marco Lippi for providing large pieces of code and Khalid Sekkat, Christophe Croux, Frederic Pivetta, and Francois Rycx for helpful comments and discussions.     

The proper panel econometric specification of the gravity equation: A three-way model with bilateral interaction effects

We argue that the proper specification of a panel gravity model should include main (exporter, importer, and time) as well as time invariant exporter-by-importer (bilateral) interaction effects. In a panel of 11 APEC countries, the latter are highly significant and account for the largest part of variation. First version received: February 2001/Final version received: June 2002 RID="*" ID="*"  We are grateful to two anonymous referees and Robert Kunst for their helpful comments.     

Cournot-Nash and Lindahl equilibria in pure public “bad” economies

Summary. Using a general equilibrium framework, this paper analyzes the equilibrium provision of a pure public bad commodity (for example pollution). Considering a finite economy with one desired private good and one pure public “bad” we explicitly introduce the concept of Lindahl equilibrium and the Lindahl prices into a pure public bad economy. Then, the Lindahl provision is analyzed and compared with the Cournot-Nash provision. The main results for economies with heterogeneous agents state that the asymptotic Lindahl allocation of the pure public bad is the null allocation. In contrast, the asymptotic Cournot-Nash provision of the public bad might approach infinity. Other results were obtained in concert with the broad analysis of the large finite economies with pure public bad commodities. Received: July 26, 2001; revised version: March 12, 2002 RID="*" ID="*" We are indebt to Nicholas Yannelis and anonymous referee for their valuable comments and suggestions. Correspondence to: B. Shitovitz