Recursive structure and equilibria in games with private monitoring

Summary. In each stage of a repeated game with private monitoring, the players receive payoffs and privately observe signals which depend on the players' actions and the state of world. I show that, contrary to a widely held belief, such games admit a recursive structure. More precisely, I construct a representation of the original sequential problem as a sequence of static games with incomplete information. This establishes the ground for a characterization of strategies and, hence, of behavior in interactive-decision settings where private information is present. Finally, the representation is used to give a recursive characterization of the equilibrium payoff set, by means of a multi-player generalization of dynamic programming. Received: February 11, 2002; revised version: July 22, 2002 RID="*" ID="*" I am very grateful to In-Koo Cho, Larry Epstein, Denis Gromb, Stephen Morris, Paolo Siconolfi, Lones Smith and Max Stinchcombe for several insights and suggestions. A referee's comments helped improving the exposition. Finally, I wish to thank the participants to the seminars at MEDS, NYU, Columbia University, Caltech, UCLA, University of Rochester, University of Texas-Austin, Northwestern Summer Microeconomics Conference 98, Summer in Tel Aviv 98, and NASM98.     

Speculative trading with rational beliefs and endogenous uncertainty

Summary. This paper introduces the framework of rational beliefs of Kurz (1994), which makes the assumptions of heterogeneous beliefs of Harrison and Kreps (1978) and Morris (1996) more plausible. Agents hold diverse beliefs that are “rational” in the sense of being compatible with ample observed data. In a non-stationary environment the agents only learn about the stationary measure of observed data, but their beliefs can remain non-stationary and diverse. Speculative trading then stems from disagreements among traders. In a Markovian framework of dividends and beliefs, we obtain analytical results to show how the speculative premium depends on the extent of heterogeneity of beliefs. In addition, we demonstrate that there exists a unique Rational Belief Equilibrium (RBE) generically with endogenous uncertainty (as defined by Kurz and Wu, 1996) and that the RBE price is higher than the rational expectation equilibrium price (REE) under some general conditions Received: March 15, 2001; revised version: April 26, 2002 RID="*" ID="*" We are deeply grateful to Mordecai Kurz for his constant encouragement and inspiring guidance over the years. We wish to express our gratitude to an anonymous referee for the very valuable comments provided. We also thank Kenneth Arrow, Peter Hammond, Roko Aliprantis and Nicholas Yannelis for their helpful suggestions and Academia Sinica and the National Science Council of the R.O.C. for their indispensable support. Correspondence to: H.-M. Wu     

On extensive form implementation of contracts in differential information economies

Summary. In the context of differential information economies, with and without free disposal, we consider the concepts of Radner equilibrium, rational expectations equilibrium, private core, weak fine core and weak fine value. We look into the possible implementation of these concepts as perfect Bayesian or sequential equilibria of noncooperative dynamic formulations. We construct relevant game trees which indicate the sequence of decisions and the information sets, and explain the rules for calculating ex ante expected payoffs. The possibility of implementing an allocation is related to whether or not it is incentive compatible. Implementation through an exogenous third party or an endogenous intermediary is also considered. Received: November 19, 2001; revised version: April 17, 2002 RID="*" ID="*" This paper comes out of a visit by Nicholas Yannelis to City University, London, in December 2000. We are grateful to Dr A. Hadjiprocopis for his invaluable help with the implementation of Latex in a Unix environment. We also thank Leon Koutsougeras and a referee for several, helpful comments. Correspondence to: N.C. Yannelis     

Real exchange rate determination: Empirical observations from East-Asian countries

In this paper the models for the real exchange rate determination are re-examined between Japan and five East-Asian countries. Two important findings are reported. First, the real interest rate-bias model is valid for Korea-, Malaysia-, Indonesia-, and Philippines-Japan, and the productivity-bias model is valid for Indonesia-, and Philippines-Japan: that is, the coefficients of relative variables are stable and statistically significant. Second, there is no evidence that the political risk premium model is valid. First version received: September 2000/Final version received: April 2001 RID="*" ID="*"  This paper was presented at the Tohoku University Economics Conference April 1999. I acknowledge Yoshihiko Tsukuda, Hiroya Akiba, Tadashi Kuriyama, Jiro Akita and Hiroyuki Ozaki for their helpful comments. Also, I am very grateful to two referees of this journal for many valuable comments. The research was supported by the Nomura foundation for Social Science in 2000.     

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.     

On the constancy of bribe-proof solutions

Summary. We consider the problem of choosing one point in a set of alternatives when monetary transfers are possible. In this context, Schummer (2000) shows that a social choice function must be a constant function if manipulation through bribes is ruled out. But he requires two kinds of domain-richness conditions. One is either smooth connectedness or the finiteness of the set of alternatives and the other is monotonical closedness. However, dispensing with the former condition, we alternatively prove the same result under a weaker condition than monotonical closedness. Received: April 11, 2000; revised version: February 25, 2002 RID="*" ID="*" This paper received the Osaka University Institute of Social and Economic Research Moriguchi Prize in January 2001. I am grateful to Prof. Ryoichi Nagahisa, Prof. Tatsuyoshi Saijo, Prof. Ken-ichi Shimomura, Prof. Ken Urai, and especially two anonymous referees for their useful and helpful comments and suggestions. I am a Research Fellow of the Japan Society for the Promotion of Science.     

Type correlated equilibria for games with payoff uncertainty

Summary Aumann's notion of correlated equilibrium is extended to games with payoff uncertainty. A type correlated equilibrium is a correlated equilibrium for Harsanyi's game in player-types. An equivalent definition is a probability distribution over types and actions which is consistent with the prior distribution over types, such that when each player observes its type and action, the observed action is optimal and no further information about other players' types is obtained. Any such equilibrium can be implemented by a type-independent correlation device when players' observations may be type-dependent. The type correlated equilibrium correspondence is shown to be upperhemicontinuous with respect to player information.Support from NSF grant IRI-8609208 is gratefully acknowledged. I am grateful to Maxwell Stinchcombe for comments on an earlier draft of this paper. Any remaining errors are my own.     

Inefficient ex-post equilibria in efficient auctions

Summary. I consider a single-object English auction with two asymmetric bidders and show that it has a continuum of inefficient undominated ex-post equilibria. The result extends for the generalized VCG mechanism, Dasgupta-Maskin auction and, generally, for every auction that has an efficient ex-post equilibrium. Received: November 5, 2001; revised version: June 10, 2002 RID="*" ID="*"I am grateful to Vijay Krishna, Sergei Izmalkov and anonymous referee for many important comments.     

Closed-loop equilibrium in a multi-stage innovation race

Summary. We examine a multistage model of an R&D race where players have multiple projects. We also develop perturbation methods for general dynamic games that can be expressed as analytic operators in a Banach space. We apply these perturbation methods to solve races with a small prize. We compute second-order asymptotically valid solutions for equilibrium and socially optimal decisions to determine qualitative properties of equilibrium. We find that innovators invest relatively too much on risky projects. Strategic reactions are ambiguous in general; in particular, a player may increase expenditures as his opponent moves ahead of him. Received: January 3, 2002; revised version: June 14, 2002 RID="*" ID="*" This is the final version of Judd (1985). The author gratefully acknowledges the comments of anonymous referees, Paul Milgrom, seminar participants at Northwestern University, the University of Chicago, the 1984 Summer Meetings of the Econometric Society, University of California at Berkeley, Stanford University, and Yale University, and the financial support of the National Science Foundation (SES-8409786, SES-8606581)     

Coalition-proof supply function equilibria under capacity constraints

Whereas in the absence of capacity constraints the Cournot outcome is the unique coalition-proof supply function equilibrium outcome, the presence of capacity constraints may enlarge the set of equilibrium outcomes. Interestingly, if capacities are sufficiently asymmetric the new equilibrium prices are below the Cournot price. These results have important implications for merger and privatization policies: specifically, capacity divestiture will not necessarily imply lower market prices.This article is based on the second chapter of my PhD dissertation. I benefited from the comments and suggestions of Diego Moreno and Bill Hogan. I thank three anonymous referees for helpful comments. Seminar audiences at Carlos III and Harvard are gratefully acknowledged. The author is currently a Repsol YPF fellow at the Harvard John F. Kennedy School of Government. I am grateful to the Repsol YPF-Harvard Fellowship Program for financial support.     

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.     

Incentives in market games with asymmetric information: the core

Summary. This paper examines the ex ante core of a pure exchange economy with asymmetric information in which state-dependent allocations are required to satisfy incentive compatibility. This restriction on players' strategies in the cooperative game can be interpreted as incomplete contracts or partial commitment. An example is provided in which the incentive compatible core with nontransferable utility is empty; the game fails to be balanced because convex combinations of incentive compatible net trades can violate incentive compatibility. However, randomization of such strategies leads to ex post allocations which satisfy incentive compatibility and are feasible on average. Hence, convexity is preserved in such a model and the resulting cooperative games are balanced. In this framework, an incentive compatible core concept is defined for NTU games derived from economies with asymmetric information. The main result is nonemptiness of the incentive compatible core. Received: December 26, 2001; revised version: June 11, 2002 RID="*" ID"*" This work was financed, in part, by contract No 26 of the programme “P?le d'attraction interuniversitaire” of the Belgian government, and, in part, by research grant SBR93-09854 from the U.S. National Science Foundation. Much of my thinking about this topic was developed during a wonderful visit to CORE for the 1991–1992 academic year (on sabbatical from the University of Pennsylvania). This paper was originally circulated in December 1991 as CARESS Working Paper #91-38, Center for Analytic Research in Economics and the Social Sciences, Department of Economics, University of Pennsylvania and in February 1992 as CORE Discussion Paper 9221, Center for Operations Research and Econometrics, Université Catholique de Louvain, Louvain-la-Neuve, Belgium. RID="*" ID="*" At the very start of my research, Jean-Fran?ois Mertens was almost a co-author. Fran?ois Forges provided detailed comments at a later stage, during my visit to THEMA, Université Cergy-Pontoise, in Spring 1997. They are entitled to the customary disclaimer.     

New insights on earnings trends across skill groups and industries in West Germany

This paper provides an empirical analysis of the structure of earnings in West Germany across skill groups and industries. Our analysis is based on data from the German Socioeconomic Panel for the period 1984 to 1994. We estimate quantile regressions, both for the entire sample period and for each year separately, in order to obtain a finer picture of the earnings structure compared to conventional least squares methods. For robust standard error estimation, this study uses a block bootstrap procedure taking account of heteroskedasticity and autocorrelation in the error term. We also suggest a simple procedure to obtain a consistent estimate of inter-industry earnings variability. Our main findings are: first, pooled estimation comprising a uniform time trend is not rejected by the data, and second, the effects of human capital variables and industry dummies on earnings differ considerably across quantiles. First version: May 1998/Final version: April 2002 RID="*" ID="*"  We are grateful to an anonymous referee as well as to Thiess Büttner, Christian Dustmann, Wolfgang Franz, John Haisken-DeNew, Costas Meghir, Werner Smolny, Peter Winker, Volker Zimmermann, and seminar participants in Heidelberg, Kassel, Konstanz, and Paris for helpful comments. However, all errors are our sole responsibility. RID="*" ID="*"  We are grateful to an anonymous referee as well as to Thiess Büttner, Christian Dustmann, Wolfgang Franz, John Haisken-DeNew, Costas Meghir, Werner Smolny, Peter Winker, Volker Zimmermann, and seminar participants in Heidelberg, Kassel, Konstanz, and Paris for helpful comments. However, all errors are our sole responsibility.     

Ranking investment projects

Summary. This paper describes conditions under which one investment project dominates a second project in terms of net present value, irrespective of the choice of the discount rate. The resulting partial ordering of projects has certain similarities to stochastic dominance. However, the structure of the net present value function leads to characterizations that are quite specific to this context. Our theorems use Bernstein's (1915) innovative results on the representation and approximation of polynomials, as well as other general results from the theory of equations, to characterize the partial ordering. We also show how the ranking is altered when the range of discount rates is limited or the rate varies period by period. Received: January 5, 2002; revised version: October 29, 2002 RID="*" ID="*" We thank Robert Driskill, Andrea Maneschi, Roy Radner, and participants of seminars at NYU, Notre Dame, Purdue, and Washington University for helpful comments. The present version of the paper has benefited from comments by a referee and the editor. Foster is grateful for support from the John D. and Catherine T. MacArthur Foundation through its network on Inequality and Poverty in Broader Perspective. Correspondence to: T. Mitra     

Implementing cooperative solution concepts: a generalized bidding approach

This paper provides a framework for implementing and comparing several solution concepts for transferable utility cooperative games. We construct bidding mechanisms where players bid for the role of the proposer. The mechanisms differ in the power awarded to the proposer. The Shapley, consensus and equal surplus values are implemented in subgame perfect equilibrium outcomes as power shifts away from the proposer to the rest of the players. Moreover, an alternative informational structure where these solution concepts can be implemented without imposing any conditions of the transferable utility game is discussed as well. The authors thank Yukihiko Funaki, Andreu Mas-Colell, David Pérez-Castrillo and Jana Vyrastekova for helpful discussions. We also appreciate the comments from the seminar and conference participants at Keele University, Tilburg University, University of Haifa, University of Warwick, CORE at Louvain-la-Neuve, Catholic University Leuven, Pablo de Olavide University in Seville, University of Vigo, the 2006 Annual Conference of the Israeli Mathematical Union in Neve Ilan, Israel, and the 61st European Meeting of the Econometric Society in Vienna, Austria in 2006. In particular, we are grateful to the associate editor and an anonymous referee. Their valuable comments and constructive suggestions contributed to a significant improvement of the paper. Wettstein acknowledges the financial support of the Pinchas Sapir Center for Development in Tel Aviv University.     

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.     

A class of fair distribution rules à la Rawls and Sen

Summary. This paper discusses and develops “non-welfaristic” arguments on distributive justice à la J. Rawls and A. K. Sen, and formalizes, in cooperative production economies, “non-welfaristic” distribution rules as game form types of resource allocation schemes. First, it conceptualizes Needs Principle which the distribution rule should satisfy if this takes individuals' needs into account. Second, one class of distribution rules which satisfy Needs Principle, a class of J-based Capability Maximum Rules, is proposed. Third, axiomatic characterizations of the class of J-based Capability Maximum Rules are provided. Received: July 30, 1999; revised version: March 11, 2002 RID="*" ID="*" We are grateful to an anonymous referee of this journal, Professors Marc Fleurbaey, Nicolas Gravel, Ryo-ichi Nagahisa, Prasanta Pattanaik, Kotaro Suzumura, Koich Tadenuma, and Yongsheng Xu for their fruitful comments. An earlier version of this paper was published with the title name, “A Game Form Approach to Theories of Distributive Justice: Formalizing Needs Principle” as the Discussion Paper No. 407 of the Institute of Social and Economic Research, Osaka University, and in the proceedings of the International Conference on Logic, Game, and Social Choice held at Oisterwijk in May 1999. That version was also presented at the 3rd Decentralization Conference in Japan held at Hitotsubashi University in September 1997, at the annual meeting of the Japan Association of Economics and Econometrics held at Waseda University in September 1997, and the 4th International Conference of Social Choice and Welfare held at University of British Colombia in July 1998. This research was partially supported by the Japanese Ministry of Education and the Ministry of Health and Welfare. Correspondence to: N. Yoshihara     

Market crashes,speculation and learning in financial markets

A natural conjecture is that if agents’ beliefs are almost correct then equilibrium prices should be close to rational expectations prices. Sandroni (J Econ Theory 82:1–18, 1998) gives a counterexample in an economy with sunspots and complete markets. We extend Sandroni’s result by showing that the conjecture is generically true for economies with complete markets. We consider a standard General Equilibrium model with large but finite horizon and complete markets. We show that, for almost every such economy, if conditional beliefs eventually become correct along a path of events then equilibrium prices of assets traded along this path converge to rational expectations equilibria in the sup-norm. Moreover, we establish that, generically, there exist along any such path local diffeomorphisms between individual beliefs and equilibrium prices. I would like to thank C. Ewerhart and A. Kirman for their comments, as well as the seminar participants at the University of Minho, the General Equilibrium Workshop 2005 in Zurich, and the 15th Asian General Equilibrium Conference 2007 in Singapore. An anonymous referee also provided very valuable comments.     

Indeterminacy of equilibrium in stochastic OLG models

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

Generic inefficiency of equilibria in the general equilibrium model with incomplete asset markets and infinite time

Summary. We consider a Lucas asset-pricing model with heterogeneous agents, exogenous labor income, and a finite number of exogenous shocks. Although agents are infinitely lived, endowments and dividends are time-invariant functions of the exogenous shock alone and are thus restricted to lie in a finite-dimensional space; genericity analysis can be conducted on sets of zero Lebesgue measure. When financial markets are incomplete, that is, there are fewer financial securities than shocks, we show that generically in individual endowments all competitive equilibria are Pareto inefficient. Received: November 22, 1999; revised version: March 4, 2002 RID="*" ID="*" We are grateful to an anonymous referee for very insightful comments on earlier drafts.