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.     

Die Wahlabhängigkeit wirtschaftspolitischer Maßnahmen

Summary In the last years, some publications appeared where so-called political-economic cycles were postulated.Hubka andObermann in one of the last issues of Empirica tried to find empirical evidence for this theory from the economic development in Austria. An examination of the methods applied in this study shows: The result do not proof as a (secure) evidence for or against the existence of any cyclical setting of administered prices and public expenditures. Finally some conditions are discussed, which must be fulfilled to bring about cyclical economic policies; informational requirements are shown, which are important for empirical verification.     

Communication in dynastic repeated games: ‘Whitewashes’ and ‘coverups’

Summary. We ask whether communication can directly substitute for memory in dynastic repeated games in which short lived individuals care about the utility of their offspring who replace them in an infinitely repeated game. Each individual is unable to observe what happens before his entry in the game. Past information is therefore conveyed from one cohort to the next by means of communication.When communication is costless and messages are sent simultaneously, communication mechanisms or protocols exist that sustain the same set of equilibrium payoffs as in the standard repeated game. When communication is costless but sequential, the incentives to whitewash the unobservable past history of play become pervasive. These incentives to whitewash can only be countered if some player serves as a neutral historian who verifies the truthfulness of others reports while remaining indifferent in the process. By contrast, when communication is sequential and (lexicographically) costly, all protocols admit only equilibria that sustain stage Nash equilibrium payoffs.We also analyze a centralized communication protocol in which history leaves a footprint that can only hidden by the current cohort by a unanimous coverup. We show that in this case the set of payoffs that are sustainable in equilibrium coincides with the weakly renegotiation proof payoffs of the standard repeated game.Received: 30 September 2002, Revised: 5 August 2003, JEL Classification Numbers: C72, C73, D82.We wish to thank an Associate Editor and Dino Gerardi as well as seminar participants at Arizona State, Columbia, Duke, Georgetown, Indiana, Montreal, Princeton, Rochester, Vanderbilt, VPI, the 2001 NSF/NBER Decentralization Conference, the Summer 2001 North American Econometric Society Meetings, and the Midwest Theory Conference, 2000, for useful comments and suggestions. All errors are our own.     

Monetary policy and price level determinacy in a cash-in-advance economy

Summary The paper considers the determinacy of the equilibrium price level in the cash-in-advance monetary economy of Lucas and Stokey (1983, 1987), in the case of deterministic fundamentals. The possibilities both of a multiplicity of perfect foresight equilibria and of sunspot equilibria are considered. Two types of monetary policy regimes are considered and compared, one in which the money supply grows at a given exogenous rate (that may be positive or negative), and one in which the nominal interest rate on one-period government debt is pegged at a given non-negative level. In the case of constant money growth rate regimes, it is shown that one can easily have both indeterminacy of perfect foresight equilibrium and existence of sunspot equilibria; indeed, in the case of negative rates of money growth (as called for by Friedman (1969)), both types of indeterminacy necessarily occur. On the other hand, sufficient conditions for uniqueness of equilibrium (and non-existence of equilibria other than a deterministic steady state) are also given, and a class of cases is identified in which a sufficiently high rate of money growth guarantees this. Thus there may be a conflict between the aims of choosing a rate of money growth that results in a high level of welfare in the steady state equilibrium and choosing a rate that makes this steady state the unique equilibrium.) In the case of the interest rate pegging regimes, sufficient conditions are given for uniqueness of equilibrium (and impossibility of sunspot equilibria), and it is shown that these necessarily hold in the case of any low enough nominal interest rate. Thus the nominal interest rate peg allows simultaneous achievement of price level determinacy and a high level of welfare in the unique (steady state) equilibrium.In this paper I consider the consequences of alternative choices of the monetary policy regime for the determinacy of the rational expectations equilibrium value of money, and in particular for the existence or not of sunspot equilibria, i.e., rational expectations equilibria in which fluctuations in the price level occur in response to random events that represent no change in economic fundamentals, simply due to self-fulfilling revisions of people's expectations. I am interested in particular in making the point that a consideration of the complete set of possible equilibria associated with a given policy regime may alter one's evaluation of the relative desirability of alternative policies, relative to the conclusion that one might reach if one considered only a single possible equilibrium associated with each policy regime (perhaps a unique equilibrium involving a minimum set of state variables). In view of this I give particular attention to policy regimes of types that have sometimes been advocated as ways of reducing the inefficiency associated with a rate of return differential between money and other financial assets, and show that policies that might otherwise be desirable (policies that make possible a more desirable equilibrium than would otherwise be possible) can have the unfortunate consequence of rendering equilibrium indeterminate and making possible equilibrium fluctuations in response to sunspot events.Two classes of policy regimes are considered in particular: on the one hand, alternative constant rates of growth or contraction of the money supply, financed through lump sum taxes or transfers, with zero net government assets at all times; and on the other, alternative constant nominal interest rate pegs, to be maintained through open market operations between money and interest-bearing debt, with an exogenously fixed level of net transfer payments. The first class of policies is considered because of Friedman's (1969) well-known proposal that a constant contraction of the money supply of this sort would be welfare improving. I find that while thestationary equilibrium associated with the Friedman regime achieves the maximum possible level of utility for the representative consumer, and while the level of utility associated with stationary equilibrium may be monotonically decreasing in the rate of money growth, lower rates of money growth (in particular, rates near that called for by Friedman) are associated with indeterminacy of equilibrium and the existence of sunspot equilibria, while these problems need not arise in the case of higher rates of money growth.The second class of policies is considered because they represent an obvious alternative approach to the elimination of the same rate of return differential with which Friedman is concerned. Achievement of permanently low nominal interest rates through a simple interest rate peg is not often advocated; one reason is that it is often asserted that such a policy must result in price level indeterminacy. In fact, I find that if the interest rate pegging regime is properly specified, it results in aunique rational expectations equilibrium, regardless of the level at which interest rates are to be pegged. Thus not only does the interest rate peg not result in price level indeterminacy but it allows nominal interest rates to be maintained permanently at a level lower than that which can be obtained through a policy regime of the first sort without creating price level indeterminacy. It would hence appear, at least in the case of the kind of economy modeled here, that interest rate pegging is a more reliable way of trying to reduce the inefficiency associated with consumers being forced to economize on liquidity.This paper represents a revision of Woodford (1988). I would like to thank Leonardo Auernheimer, Buz Brock, Willem Buiter, Peter Howitt, Teh-Ming Huo, David Laidler, David Levine, Bennett McCallum, and an anonymous referee for helpful comments, and the National Science Foundation for research support.     

On Cournot-Nash equilibrium distributions for games with differential information and discontinuous payoffs

Summary Mas-Colell's model for equilibrium distributions in large anonymous games is extended by the introduction of an abstract, non-topological notion of players' characteristics. The generality of this model allows us to include as a component of a player's characteristic her/his degree of well-informedness (differential information). This makes it possible to address a generalization of a model recently formulated by Khan-Rustichini. We show that one can remove their rather unnatural compactness condition on the space of admissible decision rules, provided that one allows for decision rules which are randomized. Our utility functions need not be continuous; this particular technical aspect solves an open question of Khan concerning Mas-Colell's original model. Our method of proof is based on the noval observation that Cournot-Nash equilibrium distributions are precisely the solutions of a variational inequality for transition probabilities.     

Equilibria and Pareto optima of markets with adverse selection

Summary This paper examines the efficiency properties of competitive equilibrium in an economy with adverse selection. The agents (firms and households) in this economy exchange contracts, which specify all the relevant aspects of their interaction. Markets are assumed to be complete, in the sense that all possible contracts can, in principle, be traded. Since prices are specified as part of the contract, they cannot be used as free parameters to equate supply and demand in the market for the contract. Instead, equilibrium is achieved by adjusting the probability of trade. If the contract space is sufficiently rich, it can be shown that rationing will not be observed in equilibrium. A further refinement of equilibrium is proposed, restricting agents' beliefs about contracts that are not traded in equilibrium. Incentive-efficient and constrained incentive-efficient allocations are defined to be solutions to appropriately specified mechanism design problems. Constrained incentive efficiency is an artificial construction, obtained by adding the constraint that all contracts yield the same rate of return to firms. Using this notion, analogues of the fundamental theorems of welfare economics can be proved: all refined equilibria are constrained incentive-efficient and all constrained incentive-efficient allocations satisfying some additional conditions can be decentralized as refined equilibria. A constrained incentive-efficient equilibrium is typically not incentive-efficient, however. The source of the inefficiency is the equilibrium condition that forces all firms to earn the same rate of return on each contract.Notation ={ ₁,..., _k } set of outcomes - : ₊ generic contract or lottery - A = () ; - A_o A{, where denotes the null contract or no trade - S={1,...,¦S¦} set of seller types - L(s) number of type-s sellers - M number of buyers - u: × S seller's utility function, which can be extended toA× S by puttingu(, s) ; - v. × S buyer's utility function, which can be extended toA × S by puttingv(, s) ; - f:A ₀ ×S ₊ allocation of sellers - g:A ₀ ×S ₊ allocation of buyers - A ₊ sellers' trading function - :A ×S ₊ buyers' trading function This paper has had a long gestation period, during which I have been influenced by helpful conversations with many persons, by their work, or both. Among those who deserve special mention are Martin Hellwig, Roger Myerson, Edward Prescott, Robert Townsend and Yves Younés. Earlier versions were presented to the NBER/CEME Conference on Decentralization at the University of Toronto and the NBER Conference on General Equilibrium at Brown University. I would like to thank John Geanakoplos, Walter Heller, Andreu Mas Colell, Michael Peters, Michel Poitevin, Lloyd Shapley, John Wooders, Nicholas Yannelis and an anonymous referee for their helpful comments and especially Robert Rosenthal for his careful reading of two drafts. The financial support of the National Science Foundation under Grant No. 912202 is gratefully acknowledged.     

Endogenously determined price rigidities

Summary There exists an extensive literature about economies with price rigidities, where some constraints on the set of admissible price systems are exogenously given. In this paper a general equilibrium model extended by a political system is described where the price rigidities are endogenously chosen by political candidates. Sufficient conditions for the existence of a mixed strategy and a pure strategy equilibrium are given. Finally an example is discussed, where in equilibrium both political candidates propose price rigidities excluding the Walrasian equilibrium price system.The author would like to thank Dolf Talman, Pieter Ruys, Jan van Geldrop, and an anonymous referee for their valuable comments on previous drafts of this paper. The author is financially supported by the Cooperation Centre Tilburg and Eindhoven Universities, The Netherlands. This research is part of the VF-program Competition and Cooperation.     

Three brief proofs of Arrow’s Impossibility Theorem

Summary. Arrows original proof of his impossibility theorem proceeded in two steps: showing the existence of a decisive voter, and then showing that a decisive voter is a dictator. Barbera replaced the decisive voter with the weaker notion of a pivotal voter, thereby shortening the first step, but complicating the second step. I give three brief proofs, all of which turn on replacing the decisive/pivotal voter with an extremely pivotal voter (a voter who by unilaterally changing his vote can move some alternative from the bottom of the social ranking to the top), thereby simplifying both steps in Arrows proof. My first proof is the most straightforward, and the second uses Condorcet preferences (which are transformed into each other by moving the bottom alternative to the top). The third proof proceeds by reinterpreting Step 1 of the first proof as saying that all social decisions are made the same way (neutrality).Received: 9 July 2001, Revised: 2 September 2004, JEL Classification Numbers: D7, D70, D71.John Geanakoplos: I wish to thank Ken Arrow, Chris Avery, Don Brown, Ben Polak, Herb Scarf, Chris Shannon, Lin Zhou, and especially Eric Maskin for very helpful comments and advice. I was motivated to think of reproving Arrows theorem when I undertook to teach it to George Zettler, a mathematician friend. After I presented this paper at MIT, a graduate student there named Luis Ubeda-Rives told me he had worked out the same neutrality argument as I give in my third proof while he was in Spain nine years ago. He said he was anxious to publish on his own and not jointly, so I encourage the reader to consult his forthcoming working paper. The proofs appearing here appeared in my 1996 CFDP working paper. Proofs 2 and 3 originally used Mays notation, which I have dropped on the advice of Chris Avery.     

Contests with a stochastic number of players

We study Tullock's (1980) n-player contest when each player has an independent probability 0 < p 1 of participating. A unique symmetric equilibrium is found for any n and p and its properties are analyzed. In particular, we show that for a fixed n > 2 individual equilibrium spending as a function of p is single-peaked and satisfies a single-crossing property for any two different numbers of potential players. However, total equilibrium spending is monotonically increasing in p and n. We also demonstrate that ex-post over-dissipation is a feature of the pure-strategy equilibrium in our model. It turns out that if the contest designer can strategically decide whether to reveal the actual number of participating players or not, then the actual number of participants is always revealed.     

On the existence of involuntary unemployment equilibria

Conclusion We have proved the existence of equilibria implying involuntary unemployment of labour in non-competitive economies with fixed wage rate. We notice that model (II) and its extensions have not been presented as aninterpretation of Keynes' theory. We may define them as Keynesian models only in the sense that they admit involuntary unemployment equilibrium, which concept is basic to Keynes' theory. In any case, we think that these models are a first step in the development of more realistic microeconomic foundations for short-run macroeconomic theory.A first draft of this paper was presented in December 1972 at the Institute of Political Economy, Faculty of Economics and Banking, Siena University.     

Strength of the Social Dilemma in a Public Goods Experiment: An Exploration of the Error Hypothesis

We present the results of an experiment on voluntary contributions to a public good with a unique dominant strategy equilibrium in the interior of the strategy space. The treatment variable is the equilibrium contribution level. By increasing the equilibrium contribution level, we reduce the strength of the social dilemma. Though we observe that the average level of contribution rises with the equilibrium contribution level, the average rate of over-contribution is not affected in a systematic way. Over-contribution is statistically significant only at the lower level of equilibrium contribution but not at the higher levels. We show that the Anderson et al. (1998, Journal of Public Economics. 70, 297–323) logit equilibrium model which combines altruism and decision errors fits quite well our laboratory data.     

Markovian equilibrium in a class of stochastic games: existence theorems for discounted and undiscounted models

Summary We study a strategic version of the neoclassical growth model under possible production uncertainty. For a general specification of the problem, we establish (i) the existence of stationary Markov equilibria in pure strategies for the discounted game, and (ii) the convergence, under a boundedness condition, of discounted equilibrium strategies to a pure strategy stationary Markovian equilibrium of the undiscounted game as the discount factor tends to unity. The same techniques can be used to prove that such convergence also obtains in all finitestate, finite-action stochastic games satisfying a certain full communicability condition. These results are of special interest since there are well known examples in the literature in which the limit of discounted equilibria fails to be an equilibrium of the undiscounted game.We are grateful to Marcus Berliant, M. Ali Khan, Mukul Majumdar, and an anonymous referee for helpful suggestions, and to Bonnie Huck for technical assistance. The first author acknowledges research support from the Columbia University Council for the Social Sciences.     

Price rigidities and institutional variations in markets with posted prices

Summary An exchange economy with price rigidities and rationing is considered. The rationing systems allowed are very general. Several characterizations of the set of constrained equilibria are given, and new equilibrium existence results are provided. More specifically, well-known properties like the existence of equilibria without rationing of the numeraire commodity, and the existence of supply and demand constrained equilibria without rationing on the market of at least one commodity follow as special cases from the theorems proved. Finally it is shown that the equilibrium correspondence is upper semi-continuous, while it is continuous on a residual set of points. In order to prove these results a new continuity result for the budget correspondence is given.The author would like to thank Dolf Talman, Gerard van der Laan, Jan van Geldrop, and an anonymous referee for their valuable comments on previous drafts of this paper. The author is financially supported by the Cooperation Centre Tilburg and Eindhoven Universities, The Netherlands. This research is part of the VF-program Competition and Cooperation.     

Tax Interaction Effects,Environmental Regulation,and “Rule of Thumb” Adjustments to Social Cost

Preexisting distortions in factor markets complicate the estimation of the social welfare effects of regulatory interventions. The existence of these tax interaction effects (TIEs) suggests that general equilibrium (GE) approaches should be used to evaluate regulatory policies. However, formal GE analysis is not always feasible for the numerous environmental regulations proposed by federal, state, and local agencies. The question addressed in this paper is whether an empirically based rule of thumb upward adjustment factor is appropriate to properly scale social cost estimates in environmental policy. We argue that such rule of thumb adjustments are significantly less reliable than estimates based on a detailed general equilibrium analysis because of the uncertainty about both the magnitude and sign of the social cost distortion. In addition to addressing this question, the paper gives an overview of TIEs and their relevance to environmental policy.     

On the theory of cost sharing

This paper investigates efficiency and equitability issues given a cost sharing method in an economy with a public commodity. We study the concept of a -cost share equilibrium and examine the set of all equilibrium allocations. Finally, we devise a mechanism to implement -cost share equilibria as strong equilibria of an associated non-cooperative game.This paper was written while Shlomo Weber was visiting the University of Bonn. Financial support from theSonderforschungsbereich 303 is gratefully acknowledged. We also would like to thank Dieter Bös for valuable comments.     

Existence of an equilibrium in first price auctions

Summary We study the first price auction game with an arbitrary number of bidders when the bidders' valuations are independent from each other. In technical words, we work within the independent private value model. We show that if the supports of the valuation probability distributions have the same minimum and if this minimum is not a mass point of any of these distributions, then a Nash equilibrium of the first price auction exists. We then modify the first price auction game by adding a closed interval of messages. Every bidder has to send a message with the bid he submits. These messages are used in the resolution of the ties. The winner of the auction is chosen randomly among the highest bidders with the highest value of the message among the highest bidders. In the general case, we prove the existence of a Nash equilibrium for this augmented first price auction.I wish to thank Mamoru Kaneko and a referee for their comments on an earlier draft.     

Budget constraint of a firm and economic theory

Summary Firms in reality are subject to budget constraints which general equilibrium theorists have paid little attention. Using Morishima (1950, 1992) model, this paper deals with firms that are subject to budgets pertaining to sales and investment decisions, and proves the existence of a general equilibrium. We show that an economy with firms subject to budgets does not necessarily satisfy the efficiency proposition, and clarify how the total profit maximum condition in the Arrow-Debreu (1954) type economy ensures an efficiency in a limited dynamic sense.The author is grateful to Professors J. Iritani, H. Nagatani, and K. Urai who gave useful comments on occasions of Saturday Workshop on Economic Theory and Mathematics. He is also grateful to Professor M. Kaneko for his useful comments on an earlier version of the paper Kuga (1993), to which this article is closely related.     

Large games with transformed summary statistics

Summary. The existence of pure-strategy Nash equilibrium is shown for a non-cooperative game with a continuum of small players and a compact action space. The players payoffs depend on their own actions and the mean of the transformed strategy profiles. This covers the case when the payoffs depend on players own actions and finitely many summary statistics.Received: 24 November 2003, Revised: 29 March 2004, JEL Classification Numbers: C7, D4. Correspondence to: Haomiao YuThe authors are grateful to Yeneng Sun for his help and guidance. They also thank Ali Khan, Kali Rath, and an anonymous referee for useful comments.     

Walrasian versus quasi-competitive equilibrium and the core of a production economy

Summary. This paper presents very general conditions guaranteeing that a quasi-competitive equilibrium is a Walrasian equilibrium. We also develop a generalization (and a simplified proof) of Nikaidos and McKenzies extensions of the classic Debreu-Scarf theorem on core convergence, and apply the first result to obtain an equivalence between the set of Edgeworth equilibria and the set of Walrasian equilibria in a production economy.Received: 6 September 2002, Revised: 14 November 2003, JEL Classification Numbers: C71, D50, D51.     

Some unfamiliar dynamics of a familiar macro model a note

The idea that for small disturbances the full employment equilibrium is stable while for large disturbances it is unstable was coined by Leijonhufvud in the notion of a corridor. We discuss the existence of a corridor in the standard Keynesian-monetarist textbook macro-model. It turns out that though the full employment steady state of this model may be locally stable — which is the case when the well-known Cagan condition holds — the model is never globally stable. This is due to the inherent non-linearity in the demand for money function, arising from non-negativity of the nominal rate of interest. Thus, perhaps surprisingly, the Cagan condition is both necessary and sufficient for the existence of a corridor in the Keynesian-monetarist model.This note is adapted from a paper presented at the European Meeting of the Econometric Society, Bologna, August 1988. I would like to thank Søren Bo Nielsen and Peter Birch Sørensen (Copenhagen Business School), Thomas Lux (University of Bamberg), and two anonymous referees for helpful comments and suggestions. Remaining errors and shortcomings are mine.