Stationary measures for some Markov chain models in ecology and economics

Summary. Let continuous, exists in for x in . Let be an i.i.d. sequence from F and X₀ be a nonnegative random variable independent of . Let be the Markov chain generated by the iteration of random maps by . Such Markov chains arise in population ecology and growth models in economics. This paper studies the existence of nondegenerate stationary measures for {X_n}. A set of necessary conditions and two sets of sufficient conditions are provided. There are some convergence results also. The present paper is a generalization of the work on random logistics maps by Athreya and Dai (2000).Received: 20 March 2002, Revised: 4 December 2002, JEL Classification Numbers: C22, D9.The author wishes to thank Professor Mukul Majumdar and the referees for several useful suggestions.     

The nature of the steady state in models of optimal growth under uncertainty

Summary. We study a one-sector stochastic optimal growth model with a representative agent. Utility is logarithmic and the production function is of the Cobb-Douglas form with capital exponent . Production is affected by a multiplicative shock taking one of two values with positive probabilities p and 1-p. It is well known that for this economy, optimal paths converge to a unique steady state, which is an invariant distribution. We are concerned with properties of this distribution. By using the theory of Iterated Function Systems, we are able to characterize such a distribution in terms of singularity versus absolute continuity as parameters and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever . More delicate is the case . Singularity with respect to Lebesgue measure also appears for values such that . For and Peres and Solomyak (1998) have shown that the distribution is a.e. absolutely continuous. Characterization of the invariant distribution in the remaining cases is still an open question. The entire analysis is summarized through a bifurcation diagram, drawn in terms of pairs .Received: 9 April 2002, Revised: 29 October 2002, JEL Classification Numbers: C61, O41.Correspondence to: Tapan MitraThis research was partially supported by CNR (Italy) under the "Short-term mobility" program and by M.U.R.S.T. (Italy) National Group on "Nonlinear Dynamics and Stochastic Models in Economics and Finance" . We are indebted to Rabi Bhattacharya for providing us with the reference to Solomyak's (1995) paper. The present version has benefitted from comments by Mukul Majumdar and two anonymous referees.     

Voting in assemblies of shareholders and incomplete markets

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

The extent of the market and optimal specialization

Summary. In this paper I examine how the socially optimal allocation, and specialization in particular, depends on the extent of the market. I interpret the societys ability to keep transaction records as the extent of the market and measure it by a probability with which the society can update agents past transactions into the public record. The society uses this record to detect potential defections from the optimal allocation and to punish the defectors with autarky. I show that when is small, increasing increases optimal specialization. However, when is close to 1, increasing further has no effect on the optimal allocation. I also show that optimal specialization is gradual over time when there is cost to reduce specialization. Even for small , the process converges to the unconstrained optimum that would occur under .JEL Classification Numbers: D60, C78.An earlier version of this paper was presented at the Society for Economic Dynamics meetings in Stockholm (2001). I thank a referee and the editor for useful comments and suggestions. For financial support, I gratefully acknowledge the Social Sciences and Humanities Research Council of Canada, and the Bank of Canada Fellowship. The opinion expressed here is my own and does not reflect the view of the Bank of Canada.     

An axiomatic approach to \boldsymbol{\varepsilon}-contamination

Summary.  Suppose that an economic agent is 100% certain that uncertainty she faces is characterized by a particular probability measure, but that she has a fear that, with 100% chance, her conviction is completely wrong and she is left perfectly ignorant about the true measure in the present as well as in the future. This situation is often called -contamination of confidence. The purpose of this paper is to provide a simple set of behavioral axioms under which the decision-makers preference is represented by the Choquet expected utility with the -contamination of confidence.Received: 25 November 2002, Revised: 15 November 2004, JEL Classification Numbers:   D81. Correspondence to: Hiroyuki OzakiWe are grateful to an anonymous referee. The referees comments greatly improved the exposition of the paper. The work reported here is partially supported by a grant from the Economic and Social Research Insitute, the Cabinet Office, the Government of Japan.     

More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model

Summary. This paper studies monotone risk aversion, the aversion to monotone, mean-preserving increase in risk (Quiggin [21]), in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, characterized by two functions, a utility function u in conjunction with a probability-perception function f. Monotone mean-preserving increases in risk are closely related to the notion of comparative dispersion introduced by Bickel and Lehmann [3,4] in Non-parametric Statistics. We present a characterization of the pairs (u,f) of monotone risk averse decision makers, based on an index of greediness G _u of the utility function u and an index of pessimism P _f of the probability perception function f: the decision maker is monotone risk averse if and only if . The index of greediness (non-concavity) of u is the supremum of taken over . The index of pessimism of f is the infimum of taken over 0 < v < 1. Thus, , with G _u = 1 iff u is concave. If then , i.e., f is majorized by the identity function. Since P _f = 1 for Expected Utility maximizers, forces u to be concave in this case; thus, the characterization of risk aversion as is a direct generalization from EU to RDEU. A novel element is that concavity of u is not necessary. In fact, u must be concave only if P _f = 1.Received: 10 April 2001, Revised: 18 November 2003, JEL Classification Numbers: D81. Correspondence to: Michéle CohenAlain Chateauneuf, Michéle Cohen, Isaac Meilijson: We are most grateful to Mark Machina, Peter Wakker and two anonymous referees for very helpful suggestions and comments.     

Stationary Markov equilibria for overlapping generations

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

The fuzzy core and the (\Pi, \beta)-balanced core

Summary. This note provides a new proof of the non-emptiness of the fuzzy core in a pure exchange economy with finitely many agents. The proof is based on the concept of -balanced core for games without side payments due to Bonnisseau and Iehlé (2003).Received: 8 May 2003, Revised: 8 August 2004, JEL Classification Numbers: D51, C71.Helpful comments of Jean-Jacques Herings and Hans Peters are gratefully acknowledged.     

Location-then-price competition with uncertain consumer tastes

Summary. We investigate Hotellings duopoly game of location-then-price choices with quadratic transportation costs and uniformly distributed consumers under the assumption that firms are uncertain about consumer tastes. When the uncertainty has a uniform distribution on the closed interval , with , we characterize the unique equilibrium and the socially optimal locations. Contrary to the individual-level random utility models, we find that uncertainty is a differentiation force. For small (large) sizes of the uncertainty, there is excessive (insufficient) differentiation. More uncertainty about consumer tastes can have positive or negative welfare effects, depending on the size of the uncertainty.Received: 1 February 2003, Revised: 1 April 2004, JEL Classification Numbers: C72, D43, D81, L10, L13, R30, R39. Correspondence to: Kieron J. MeagherWe thank Simon Anderson, Jörg Borrmann, Vince Crawford, Bernd Engelmann, Catherine de Fontenay, Simon Grant, Stephen King, Preston McAfee, John Miller, Scott Page, Rohan Pitchford, Bill Schworm, Joel Sobel, an anonymous referee and seminar participants at CALTECH and at the 1999 Econometric Society Winter Meetings for their comments and criticisms. Zauner was affiliated with the Department of Economics, University of Sydney, during the earlier stages of this project.     

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

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

Inverse probability weighted M-estimators for sample selection, attrition, and stratification

I provide an overview of inverse probability weighted (IPW) M-estimators for cross section and two-period panel data applications. Under an ignorability assumption, I show that population parameters are identified, and provide straightforward -consistent and asymptotically normal estimation methods. I show that estimating a binary response selection model by conditional maximum likelihood leads to a more efficient estimator than using known probabilities, a result that unifies several disparate results in the literature. But IPW estimation is not a panacea: in some important cases of nonresponse, unweighted estimators will be consistent under weaker ignorability assumptions.JEL Classification: C13, C21, C23I would like to thank Bo Honoré, Christophe Muller, Frank Windmeijer, and the participants at the CeMMAP/ESCR Econometric Study Group Microeconometrics Workshop for helpful comments on an earlier draft.     

Infinite inequality systems and cardinal revelations

Summary. Many economics problems are maximization or minimization problems, and can be formalized as problems of solving linear difference systems of the form and r _k -r _l > c _kl , for r-unknowns, with given c-constants. They typically involve strict as well as weak inequalities, with infinitely many inequalities and unknowns. Since strict inequalities are not preserved under passage to the limit, infinite systems with strict inequalities are notoriously hard to solve. We introduce a unifying tool for solving them. Our main result (Theorem 1 for the countable case, Theorem [2] for the not-necessarily-countable case) introduces a uniform solvability criterion (the -Axiom), and our proof yields a method for solving those that are solvable. The axioms economic intuition extends the traditional ordinal notion of revealed preference to a cardinal notion. We give applications in producer theory, consumer theory, implementation theory, and constrained maximization theory.Received: 21 May 2003, Revised: 29 October 2004, JEL Classification Numbers: C69, D21, D11, D82. Correspondence to: Marcel K. RichterWe thank Professor Leonid Hurwicz for helpful discussions, and a referee for helpful comments.     

Collusion through mediated communication in repeated games with imperfect private monitoring

Summary. This paper studies repeated games with imperfect private monitoring when there exists a third-party mediator who coordinates play by giving non-binding instructions to players on which action to take and by collecting their private information. The paper presents a Nash-threat folk theorem for a communication equilibrium based on such mediation when monitoring is jointly -perfect in the sense that every player is almost perfectly monitored collectively by other players.JEL Classification Numbers: C72, D82.I am very grateful to Mark Armstrong, V. Bhaskar, and Michihiro Kandori for helpful comments. Part of this research was conducted while I was visiting the University College London. Their hospitality is gratefully acknowledged.     

Strategic market games with a finite horizon and incomplete markets

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

The Zilcha criterion for dynamic inefficiency

Summary. This paper reexamines the condition (1 + n), which Zilcha (1991) presents as a necessary and sufficient condition for dynamic inefficiency of stationary allocations in overlapping generation models with stochastic production. We show that this condition is necessary but not sufficient for a stationary allocation to be dynamically inefficient by Zilchas definition. We also show that there is a narrow but widely studied class of specifications in which the Zilcha test is both necessary and sufficient for dynamic inefficiency of stationary competitive equilibrium allocations. Outside this class, however, counterexamples can be constructed relatively easily.Received: 30 September 2002, Revised: 13 August 2004, JEL Classification Numbers: D51, D90, E13, E22. Correspondence to: Steven RussellWe thank Jon Burke, Subir Chakrabarti, Itzhak Zilcha and an anonymous referee for helpful conversations and/or comments.     

On the generic strategic stability of Nash equilibria if voting is costly

Summary. We prove that for generic plurality games with positive cost of voting, the number of Nash equilibria is finite. Furthermore all the equilibria are regular, hence stable sets as singletons.JEL Classification Numbers: C72, D72.We would like to thank Luisa Bassotto, Marco Celentani, Mamoru Kaneko, and participants at the 6 International Conference on Current Trends in Economics for helpful comments. We also thank an anonymous referee of this journal for precious advices that have substantially improved this paper. This paper was partially written while Francesco was visiting Tor Vergata University, which he thanks for hospitality and financial support. The usual disclaimer applies.     

Sunspot equilibria and the transfer paradox

Summary. We show that for international economies with two countries, in which agents have additively separable utility functions, the existence of sunspot equilibria is equivalent to the occurrence of the transfer paradox. This equivalence enables us to provide some new insights on the relation of the existence of sunspot equilibria and the multiplicity of spot market equilibria.Received: 1 October 2003, Revised: 1 April 2004, JEL Classification Numbers: C62, D52, F3, F20, 012.Correspondence to: Thorsten HensWe would like to thank Anke Gerber for carefully checking and improving our arguments. Moreover, we are grateful to Piero Gottardi for his encouragement and his comments, to Andreu Mas-Colell and Mike Jerison for very fruitful discussions and to Klaus Schenk-Hoppé for his support in using . All remaining errors are ours.     

Regularity of pure strategy equilibrium points in a class of bargaining games

Summary. We develop an index theory for the Stationary Subgame Perfect (SSP) equilibrium set in a class of n-player sequential bargaining games with probabilistic recognition rules. For games with oligarchic voting rules (a class that includes unanimity rule), we establish conditions on individual utilities that ensure that for almost all discount factors, the number of SSP equilibria is odd and the equilibrium correspondence lower-hemicontinuous. For games with general, monotonic voting rules, we show generic (in discount factors) determinacy of SSP equilibria under the restriction that the agreement space is of dimension one. For non-oligarchic voting rules and agreement spaces of higher finite dimension, we establish generic determinacy for the subset of SSP equilibria in pure strategies. The analysis also extends to the case of fixed delay costs. Lastly, we provide a sufficient condition for uniqueness of SSP equilibrium in oligarchic games.Received: 13 May 2004, Revised: 1 March 2005, JEL Classification Numbers: C62, C72, C78.I thank John Duggan and participants of the 2003 annual meeting of the American Political Science Association, Philadelphia, PA, the Political Economy Seminar at Northwestern University, and the Economic Theory seminar at the University of Rochester for helpful comments.