Approximation by finite horizons of the undiscounted one-sector growth model

Summary. In this paper I give a method for finding long-run-average policies in the undiscounted economic growth problem using approximations by finite horizons. Required hypothesis is the strong interiority of T-horizon solutions. Received: March 25, 1996; revised version: July 29, 1997     

Stochastic evolution with slow learning

Summary. This paper studies the extent to which diffusion approximations provide a reliable guide to equilibrium selection results in finite games. It is shown that they do for a class of finite games with weak learning provided that limits are taken in a certain order. The paper also shows that making mutation rates small does not in general select a unique equilibrium but making selection strong does. Received: January 19, 2000; revised version: September 25, 2000     

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.     

Discounting long run average growth in stochastic dynamic programs

Summary. Finding solutions to the Bellman equation often relies on restrictive boundedness assumptions. In this paper we develop a method of proof that allows to dispense with the assumption that returns are bounded from above. In applications our assumptions only imply that long run average (expected) growth is sufficiently discounted, in sharp contrast with classical assumptions either absolutely bounding growth or bounding each period (instead of long run) maximum (instead of average) growth. We discuss our work in relation to the literature and provide several examples. Received: July 26, 2000; revised version: July 10, 2002 RID="*" ID="*" I am specially grateful to Cuong Le Van and to anonymous referee for detecting an error in a previous version of this paper and for suggestions that sensibly improved the paper. Comments and suggestions are also acknowledged to Michele Boldrin, Raouf Boucekkine, Fabrice Collard, Tim Kehoe, Omar Licandro, and Luis Puch. I am also indebted to participants to the III Summer School on Economic Theory held at the Universidade de Vigo, the Macroeconomics Workshop at the Universitat Autò}noma de Barcelona, and the Econometrics Seminar at Tilburg University. Financial support from the Belgian government, under project PAI P4/01, at the IRES-UCL, from a European Marie Curie fellowship, Grant HPMF-CT-1999-00410, at the CEPREMAP, and from IVIE and Spanish Ministerio de Ciencia y Tecnología and FEDER, under project BEC2001-0535, at the Universidad de Alicante, is gratefully acknowledged.     

Preference for flexibility in infinite horizon problems

Summary. We consider the extension of the classical problem of preference for flexibility to many periods. Preferences are defined over sets of infinite paths of choices. The main result provides a set of axioms on preferences that yield an additive representation over a subjective state space. This space is the set of preferences over choice today and feasible set tomorrow. The main new axiom, stochastic dominance, is a stronger form of the assumption of monotonicity. Received: September 11 2000; revised version: December 18, 2001     

A simple proof of Ekeland and Scheinkman's result on the necessity of a transversality condition

Summary. Ekeland and Scheinkman (1986) prove the necessity of a standard transversality condition under certain technical conditions. Their result is one of the most powerful on the necessity of a transversality condition currently available in the literature, and their proof involves numerous estimations and relies on Ekeland's variational principle and Fatou's lemma. This note relaxes some of their assumptions and provides a simple proof that uses neither Ekeland's principle nor a convergence result like Fatou's lemma. Received: April 24, 1998; revised version: September 8, 1998     

Stochastic strategy adjustment in coordination games

Summary. The paper explores a model of equilibrium selection in coordination games, where agents from an infinite population stochastically adjust their strategies to changes in their local environment. Instead of playing perturbed best-response, it is assumed that agents follow a rule of ‘switching to better strategies with higher probability’. This behavioral rule is related to bounded-rationality models of Rosenthal (1989) and Schlag (1998). Moreover, agents stay with their strategy in case they successfully coordinate with their local neighbors. Our main results show that both strict Nash equilibria of the coordination game correspond to invariant distributions of the process, hence evolution of play is not ergodic but instead depends on initial conditions. However, coordination on the risk-dominant equilibrium occurs with probability one whenever the initial fraction contains infinitely many agents, independent of the spatial distribution of these agents. Received: March 14, 2000; revised version: June 21, 2001     

On forward induction in money-burning games

Summary. We first consider money-burning games studied by Ben-Porath and Dekel [6]. We show that iterative weak dominance and extensive form rationalizability yield the same unique outcome in this class of games. This result suggests that weak dominance captures the forward induction logic implied by extenisve form rationalizability. Next, we consider an example of entry model by Arvan [1] to demonstrate the power of forward induction. In this example, despite the presence of multiple equilibria, forward induction chooses a unique outcome. Received: January 25, 2000; revised version: January 5, 2001     

Range convexity and ambiguity averse preferences

We show that range convexity of beliefs, a `technical' condition that appears naturally in axiomatizations of preferences in a Savage-like framework, imposes some unexpected restrictions when modelling ambiguity averse preferences. That is, when it is added to a mild condition, range convexity makes the preferences collapse to subjective expected utility as soon as they satisfy structural conditions that are typically used to characterize ambiguity aversion. Received: February 25, 2000; revised version: April 17, 2000     

Dynamic labor contracts with temporary layoffs and permanent separations

Summary. We study the implications of optimal dynamic contracts in private information environments for fluctuations in effort and employment across time and productivity states. To this end, we incorporate temporary layoffs and permanent separations as well as on-the-job effort variations into a dynamic model of moral hazard. We consider two different “commitment” environments. In a “full commitment” environment, although the firm can temporarily lay a worker off, neither party can dissolve the contractual relationship once it has been initiated. On the other hand, in a “limited commitment” environment, both parties can dissolve the relationship at the beginning of any period in order to pursue an outside option. We use our model to study the implications of optimal contracts for incentives, employment histories, layoffs and separations across full information, full commitment and limited commitment settings. We compute solutions to the relevant principal-agent problems, endogenously determining the set of states in which separations occur and the domain of the firm's value function, as well as the value function itself. Received: February 28, 2000; revised version: January 21, 2001     

A simple proof of the Sato proposition on non-homothetic CES functions

Summary. Based on some elementary results on duality, the paper proposes a much simpler way of deriving the class of non-homothetic CES production functions which was derived as a solution to a partial differential equation that defines the elasticity of substitution. Received: February 11, 1998; revised version: April 28, 1998     

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)     

Risk aversion in the Talmud

Summary. Evidence is adduced that the sages of the ancient Babylonian Talmud, as well as some of the medieval commentators thereon, were well aware of sophisticated concepts of modern theories of risk-bearing. Received: April 10, 2002; revised version: May 7, 2002 RID="*" ID="*"Presented at the Institute for Mathematical Studies in the Social Sciences-Economics, Stanford University, August 4, 1981. Subsequent to that presentation, the author's attention was drawn to an article by Zvi Ilani, “Models in the Economics of Uncertainty: The Cost of Concluding a Conditional Contract, according to the Talmud and the Halachic Literature,” Iyunim Bekalkala (Investigations in Economics), The Israel Association for Economics, Jerusalem, Nissan 5740 (April 1980), 246–261 (in Hebrew). Inter alia, Ilani treats the Talmudic passage that forms the subject of this paper, and provides a fairly comprehensive review of the medieval commentaries thereon; undoubtedly, he was the first to recognize in print the relevance of this passage to modern economic theories of uncertainty. It is not clear, though, whether or not his understanding of the passage agrees with ours. The current paper appeared in January 2002 in the Research Bulletin Series of the Research Center on Jewish Law and Economics, Department of Economics, Bar Ilan University.     

Aggregate fluctuations, financial constraints and risk sharing

Summary. Private information and costly state verification often result in credit rationing in models with smooth investment, affecting both loan size and total investment. The optimal contract is derived in a dynamic stochastic growth model with capital for two types of models: one with symmetric information and the other with asymmetric information and costly state verification. When all information is observed costlessly, the equilibrium optimal contract provides complete insurance to risk-averse savers against aggregate fluctuations. When information is asymmetric and there is costly state verification, the equilibrium optimal contract provides only partial insurance against aggregate shocks. The extent of insurance is measured by the marginal rate of transformation of consumption between borrowers and lenders which is closely linked to the user cost of capital. The deadweight monitoring costs create a wedge between a borrower's cost of capital and a lender's stochastic discount factor, with two results: (i) fluctuations in the user cost of capital provides a mechanism by which aggregate shocks can be␣propagated; (ii) the distribution of capital's share of output among borrowers, lenders, and monitoring costs varies even if capital's share is constant. Capital market frictions not only amplify aggregate fluctuations but also generate cross-sectional fluctuations that may not be observable in aggregate data. Received: November 17, 1997; revised version: April 20, 1998     

A dynamic analysis of an endogenous growth model with leisure

Summary. The main goal in this paper is to analyze an economic model of endogenous growth where human capital accumulation acts as the engine propelling economic activity. The added ingredient in our model is that agents derive utility from consumption and leisure, where leisure is defined as the amount of time devoted to those activities augmented by the level of education. Under regular conditions we show that there is a unique globally stable balanced growth path. We also provide a characterization of the behavior of our economic variables along the transition. Received: May 26, 1998; revised version: September 9, 1999     

Satisficing and optimality in 2×2 common interest games

Summary. The paper studies the evolution of cooperation when satisficing players repeatedly play a symmetric two-by-two game of common interest. We show that if initial aspiration levels are sufficiently close to the efficient payoff and aspiration adjusts at a sufficiently slow speed then the unique long run state will be the efficient outcome. In the special case of coordination games, the more tension there is between payoff dominance and risk dominance, the longer it takes for the system to lock into the payoff dominant outcome. Received: June 23, 1997; revised version: November 19, 1997     

Rate-of-return dominance and efficiency in an experimental economy

Summary. One of the main challenges for monetary economics is to explain the use of assets that are dominated in rate-of-return as media of exchange. We use experimental methods to study how a fiat money might come to be used in transactions when an identically marketable, dividend-bearing asset, a consol, is also available. Our experimental economies, which have an overlapping generations structure, have the property that the only stationary rational expectations equilibria (SREE) require exclusive use of the consol as the medium of exchange. In a baseline treatment, agents use the consol exclusively, as would occur in an SREE. However, in other treatments, we observe episodes of rate-of-return dominance,with consistent use of fiat money as a medium of exchange. The results show that two properties of our economies are associated with the rate of return dominance anomaly. The first is a history of trading with fiat money, prior to the introduction of the consol. The second is the timing of the dividend payment; when the dividend payment follows the execution of trades between generations, hoarding of the consol occurs on the part of the old, who earn dividends by hoarding. In our economies, settling transactions with a dividend-bearing asset does not improve allocations over those resulting from trading with fiat money. Received: July 11, 2002; revised version: July 25, 2002 RID="*" ID="*"We thank Anne Villamil, participants in the 2000 Purdue University Conference on Monetary Economics, the Summer 2000 meetings of the Economic Science Association, and a referee, for very helpful comments. We thank the Krannert School of Management and the Purdue University Center for International Business, Education and Research for financial support and Vivian Lei for research assistance. We also thank Ron Michener for referring us to the historical account of the early introduction of money into the American colonies, as reported by Benjamin Franklin. Correspondence to: G. Camera     

A simple proof of the necessity of the transversality condition

Summary. This note provides a simple proof of the necessity of the transversality condition for the differentiable reduced-form model. The proof uses only an elementary perturbation argument without relying on dynamic programming. The proof makes it clear that, contrary to common belief, the necessity of the transversality condition can be shown in a straightforward way. Received: January 22, 2001; revised version: April 2, 2001     

The set of sunspot equilibria in economies with incomplete financial markets: variable asset prices

Summary. I consider the set of equilibria of two-period economies with S extrinsic states of nature in the second period and I assets with linearly independent nominal payoffs. Asset prices are variable. If the number of agents is greater than (S-I), the payoff matrix is in general position and S 2I, the set of equilibrium allocations generically (in utility function space) contains a smooth manifold of dimension (S-1). Moreover, the map from states o f nature to equilibrium allocations (restricted to this manifold) is one-to-one at each equilibrium. Received: February 23, 1998; revised version: June 1, 2000