Market equilibrium with heterogeneous recursive-utility-maximizing agents

Summary The paper by C. Ma [1] contains several errors. First, statement and proof of Theorem 2.1 on the existence of intertemporal recursive utility function as a unique solution to the Koopmans equation must be amended. Several additional technical conditions concerning the consumption domain, measurability of certainty equivalent and utility process need to be assumed for the validity of the theorem. Second, the assumptions for Theorem 3.1 need to be amended to include the Feller's condition that, for any bounded continuous functionf C(S × ⁿ ₊), (f(S_t+1, )¦s_t =s) is bounded and continuous in (s, ). In addition, for Theorem 3.1, the pricep, the endowmente and the dividend rate as functions of the state variables S are assumed to be continuous.The Feller's condition for Theorem 3.1 is to ensure the value function to be well-defined. This condition needs to be assumed even for the expected additive utility functions (See Lucas [2]). It is noticed that, under this condition, the right hand side of equation (3.5) in [1] defines a bounded continuous function ins and. The proof of Theorem 3.1 remains valid with this remark in place.A correct version of Theorem 2.1 in [1] is stated and proved in this corrigendum. Ozaki and Streufert [3] is the first to cast doubt on the validity of this theorem. They point out correctly that additional conditions to ensure the measurability of the utility process need to be assumed. This condition is identified as conditionCE ₄ below. In addition, I notice that, the consumption space is not suitably defined in [1], especially when a unbounded consumption set is assumed. In contrast to what claimed in [3], I show that the uniformly bounded consumption setX and stationary information structure are not necessary for the validity of Theorem 2.1.I would like to thank Hiroyuki Ozaki and Peter Streufert for pointing out correctly some mistakes made in the original article. Comments and suggestions from an anonymous referee are gratefully appreciated. Financially support from SSHRC of Canada is acknowledged.     

The Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

An extended Heckscher-Ohlin model with transaction costs and technological comparative advantage

Summary. This paper introduces technological differences and transaction costs into the Heckscher-Ohlin (HO) model and examines the HO theorem, factor price equalization theorem, the Stolper-Samuelson theorem and the Rybczynski theorem. It shows that the HO theorem can be refined, and that the factor price equalisation theorem, the Stolper-Samuelson Theorem and the Rybczynski theorem do not always hold. It also shows that transaction costs play an important role in determining the equilibrium trade pattern.Received: 26 February 2001, Revised: 27 May 2003, JEL Classification Numbers: F10, F11. Correspondence to: Wenli ChengWe are grateful for comments from the anonymous referee, Hugo Sonnenschein, Guangzhen Sun and participants of the seminar on this paper at University of Washington.     

Learning with noiseless information and payoff-relevant signals

Summary. In models of active learning or experimentation, agents modify their actions to affect the distribution of a signal that provides information about future payoffs. A standard result in the experimentation literature is that agents experiment, if at all, to increase information. This finding is a direct consequence of Blackwell's theorem: one experiment is more informative than another if and only if all expected utility maximizers prefer to observe the first. Blackwell's theorem presupposes, however, that the observed signal only conveys information and does not directly affect future payoffs. Often, however, signals are directly payoff relevant, a phenomenon that we call signal dependence. For example, if a firm is uncertain about its demand and uses today's sales as a signal of tomorrow's demand, then that signal may also directly affect tomorrow's profit if the good is durable or if consumers form consumption habits. Datta, Mirman and Schlee [9] and Bertocchi and Spagat [4] show that, if the signal is payoff relevant, experimentation may indeed reduce information. Here we show that, despite the inapplicability of Blackwell's Theorem, agents always experiment to increase information if the information structure is noiseless: given the true value of the unknown parameter, the signal realization is deterministic. We then apply our framework to analyze Lazear's [16] model of retail clearance sales, a model with both signal dependence and noiseless information. Received: February 19, 1999; revised version: August 11, 1999     

Discounting Nordhaus

This paper evaluates Nordhaus's neoclassical complaints about the Stern Review from the vantage point of classical growth theory. Nordhaus criticizes the Stern Review because it uses a discount rate that is well below the market rate of return on capital. From the perspective of classical growth theory, Nordhaus's belief in choosing preference parameters for the social planner based on observed market rates of return is equivalent to assigning the preferences of the capitalist agents to the social planner. This equivalence is an implication of the Cambridge Theorem, which interprets the Ramsey equation as the saving function of the capitalist agents.     

When Does Everyone Contribute in the Private Provision of Local Public Goods?

When does everyone genuinely contribute in the private provision of a local public good? We first introduce a monotonic condition to characterize the relationship between the structure of the network that underlie the noncooperative game of private provision of local public goods on the one hand, and the preferences of the agents on the other, showing that the monotonic condition is a sufficient and necessary condition of existence of a distributed Nash equilibrium (DNE) in which each agent exerts a positive amount of effort to provision of the public good (Theorem 1). We then study the number of equilibria, and, by using the monotonic condition, characterize the condition under which the DNE set is a singleton, a continuum, or null (Theorem 2). As it turns out, the structure of the network and the agents’ preferences jointly shape the effort profile in the provision of local public goods.     

The Partnered Core of a Game without Side Payments

A payoff for a game is partnered if it admits no asymmetric dependencies. We introduce the partnered core of a game without side payments and show that the partnered core of a balanced game is nonempty. The result is a strengthening of Scarf's Theorem on the nonemptiness of the core of a balanced game without side payments. In addition, it is shown that if there are at most a countable number of points in the partnered core of a game then at least one core point isminimallypartnered, meaning that no player requires any other player in particular to obtain his part of the core payoff.Journal of Economic LiteratureClassification Number: C71.     

Perfect communication equilibria in repeated games with imperfect monitoring

This paper introduces an equilibrium concept called perfect communication equilibrium for repeated games with imperfect private monitoring. This concept is a refinement of Myerson's [Myerson, R.B., 1982. Optimal coordination mechanisms in generalized principal agent problems, J. Math. Econ. 10, 67–81] communication equilibrium. A communication equilibrium is perfect if it induces a communication equilibrium of the continuation game, after every history of messages of the mediator. We provide a characterization of the set of corresponding equilibrium payoffs and derive a Folk Theorem for discounted repeated games with imperfect private monitoring.     

INTERTEMPORAL COMPLEMENTARITY AND OPTIMALITY: A STUDY OF A TWO‐DIMENSIONAL DYNAMICAL SYSTEM*

We study the underlying structure of the two‐dimensional dynamical system generated by a class of dynamic optimization models that allow for intertemporal complementarity between adjacent periods, but preserve the time‐additively separable framework of Ramsey models. Specifically, we identify conditions under which the results of the traditional Ramsey‐type theory are preserved even when the intertemporal independence assumption is relaxed. Local analysis of this theme has been presented by Samuelson (Western Economic Journal 9 (1971), 21–26). We establish global convergence results and relate them to the local analysis, by using the mathematical theory of two‐dimensional dynamical systems. We also relate the local stability property of the stationary optimal stock to the differentiability of the optimal policy function near the stationary optimal stock, by using the Stable Manifold Theorem.     

An efficiency characterization of plurality social choice on simple preference domains

Summary. We consider a model of social choice dealing with the problem of choosing a subset from a set of objects (e.g. candidate selection, membership, and qualification problems). Agents have trichotomous preferences for which objects are partitioned into three indifference classes, goods, bads, and nulls, or dichotomous preferences for which each object is either a good or a bad. We characterize plurality-like social choice rules on the basis of the three main axioms, known as Pareto efficiency, anonymity, and independence.Received: 29 August 2003, Revised: 3 June 2004, JEL Classification Numbers: D70, D71, D72.Biung-Ghi Ju: I am grateful to William Thomson and Jianbo Zhang for their helpful comments and discussions. I also thank Brandon Dupont, the participants in seminars at Iowa State University, University of Kansas, and the Midwest Theory Meeting at University of Notre Dame. I thank an anonymous referee for detailed comments and suggestions that were very helpful in simplifying the proof of Theorem 1 and in revising the paper.     

Candidate stability and nonbinary social choice

Summary. The requirement that a voting procedure be immune to the strategic withdrawal of a candidate for election can be formalized in different ways. Dutta, Jackson, and Le Breton (Econometrica, 2001) have recently shown that two formalizations of this candidate stability property are incompatible with some other desirable properties of voting procedures. This article shows that Grether and Plott's nonbinary generalization of Arrow's Theorem can be used to provide simple proofs of two of their impossibility theorems. Received: August 15, 2001; revised version: March 11, 2002 RID="*" ID="*" Parts of this article were previously circulated in somewhat different form in a working paper with the same title by the second author. We are grateful to Michel Le Breton and an anonymous referee for their comments. Correspondence to:J.A. Weymark     

Nash and Walras equilibrium via Brouwer

Summary. The existence of Nash and Walras equilibrium is proved via Brouwer's Fixed Point Theorem, without recourse to Kakutani's Fixed Point Theorem for correspondences. The domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasi-convex preferences. The key idea is to replace optimization with “satisficing improvement,” i.e., to replace the Maximum Principle with the “Satisficing Principle.” Received: July 9, 2001; revised version: February 25, 2002 RID="*" ID="*" I wish to thank Ken Arrow, Don Brown, and Andreu Mas-Colell for helpful comments. I first thought about using Brouwer's theorem without Kakutani's extension when I heard Herb Scarf's lectures on mathematical economics as an undergraduate in 1974, and then again when I read Tim Kehoe's 1980 Ph.D dissertation under Herb Scarf, but I did not resolve my confusion until I had to discuss Kehoe's presentation at the celebration for Herb Scarf's 65th birthday in September, 1995. RID="*" ID="*"Correspondence to: C. D. Aliprantis     

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.     

Markov Equilibria of Stochastic Games with Complementarities

The existence of Markov equilibria for stochastic games with a continuum of states is a complex issue for which no general result holds as yet. In this article, the problem is solved for a class of stochastic games that satisfy assumptions of complementarity and monotonicity. The proof of existence relies on results from lattice programming. In the Markov equilibria singled out by the Theorem of Existence, the policies and continuation values are increasing and Lipschitz continuous functions of the state variable.Journal of Economic LiteratureClassification Numbers: C62, C73.     

Another Graphical Proof of Arrow's Impossibility Theorem

Arrow's (1951) Impossibility Theorem is the idea that, given several well-known assumptions, the social orderings of particular alternatives that are meant to reflect individuals' preferences must match the preferences of an arbitrary individual (the dictator). A social-choice rule other than dictatorship is impossible. Following from Fountain (2000), the author presents another graphical proof of the theorem that is intended to be more accessible to students and teachers of economics. The principal strength of this approach is that the patterns of agreements and conflicts over all possible combinations of two individuals' rankings of alternatives are transparent; appreciating these patterns is the key to intuitively understanding Arrow's theorem. A self-test for readers (or a classroom exercise for students) is included.     

Dictatorial domains

Summary. In this paper, we introduce the notion of a linked domain and prove that a non-manipulable social choice function defined on such a domain must be dictatorial. This result not only generalizes the Gibbard-Satterthwaite Theorem but also demonstrates that the equivalence between dictatorship and non-manipulability is far more robust than suggested by that theorem. We provide an application of this result in a particular model of voting. We also provide a necessary condition for a domain to be dictatorial and use it to characterize dictatorial domains in the cases where the number of alternatives is three. Received: July 12, 2000; revised version: March 21, 2002 RID="*" ID="*" The authors would like to thank two anonymous referees for their detailed comments. Correspondence to: A. Sen     

The critical discount factor as a measure for cartel stability?

This paper considers the stability of tacit collusion in price setting duopolies with repeated interaction. The minimum discount factor above which tacit collusion can be sustained in a subgame perfect equilibrium is called the critical discount factor δ*. In addition, δ* is often used as an intuitive measure for the stability of a tacit cartel, assuming that a collusive equilibrium is more difficult to sustain when δ* increases. However, according to standard theory the distance δ − δ* between the actual and the critical discount factor does not matter for stability as long as δ > δ*. This paper contributes experimental evidence that supports the intuitive idea that a larger critical discount factor makes collusion a less likely outcome.      

Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.     

Condorcet Jury Theorem or Rational Ignorance

We analyze a symmetric model of an election in which voters are uncertain about which of two alternatives is desirable for them. Each voter must incur some cost to acquire information about the alternatives. We show that by focusing on unbiased voting strategies, general symmetric signal structures can be degenerated to a two‐signal model. In addition, we show that for any sequence of unbiased voting equilibria, if the second‐order derivative of the information cost function at no information is zero, then the probability of electing the desirable alternative converges to one, that is, the Condorcet Jury Theorem is valid. Otherwise, this probability converges to some value less than one; that is, the “rational ignorance” hypothesis is valid.