Indivisibilities,lotteries, and sunspot equilibria

Summary We analyze economies with indivisible commodities. There are two reasons for doing so. First, we extend and provide some new insights into sunspot equilibrium theory. Finite competitive economies with perfect markets and convex consumption sets do not allow sunspot equilibria; these same economies with nonconvex consumption sets do, and they have several properties that can never arise in convex environments. Second, we provide a reinterpretation of the employment lotteries used in contract theory and in macroeconomic models with indivisible labor. We show how socially optimal employment lotteries can be decentralized as competitive equilibria without lotteries once sunspots are introduced.We thank Kenneth Arrow, Aditya Goenka, Ed Green, Jeremy Greenwood, Walter Heller, Steve Matthews, Herve Moulin, Roger Meyerson, Jim Peck, Patrick Kehoe, Ramon Marimon, Ed Prescott, Richard Rogerson, Nancy Stokey and Raghu Sundaram for their comments. We also thank participants in seminars at Northwestern, Yale, USC, Cornell, Barcelona, Madrid, Santander, and the Canadian Economics Association annual meetings in Victoria. We are grateful to the National Science Foundation (through grants SES-8606944 and SES-8821225), the Center for Analytic Economics, the Thorne Fund, and the University of Pennsylvania Research Foundation for research support. The views expressed here are those of the authors, and not necessarily those of the Federal Reserve System or the Federal Reserve Bank of Minneapolis.     

More on money as a medium of exchange

Summary We extend the analysis of Kiyotaki and Wright, who study economies where the commodities that serve as media of exchange (or, commodity money) are determined endogenously. Kiyotaki and Wright consider only steady-state, pure-strategy equilibria; here we allow dynamic and mixed-strategy equilibria. We demonstrate that symmetric, steady-state equilibria in mixed-strategies always exist, while sometimes no such equilibria exist in pure-strategies. We prove that the number of symmetric steady-state equilibria is generically finite. We also show, however, that for some parameter values there exists a continuum of dynamic equilibria. Further, some equilibria display cycles.We thank the National Science Foundation and the University of Pennsylvania Research Foundation for financial support, as well as seminar participants at Stanford University, the London School of Economics, the Econometric Society World Congress in Barcelona, and the Conference on Monetary Theory and Financial Institutions at the Federal Reserve Bank of Minneapolis for their comments or suggestions. Alberto Trejos provided research assistance. The views expressed here are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.     

Technology usage lags

We present evidence on the differences in the intensity with which ten major technologies are used in 185 countries across the world. We do so by calculating how many years ago these technologies were used in the U.S. with the same intensity as they are used in the countries in our sample. We denote these time lags as technology usage lags and compare them with lags in real GDP per capita. We find that (i) technology usage lags are large, often comparable to lags in real GDP per capita, (ii) usage lags are highly correlated with lags in per-capita income, and (iii) usage lags are highly correlated across technologies. The productivity differentials between the state-of-the-art technologies that we consider and the ones they replace, combined with the usage lags that we document, lead us to infer that differences in the intensity of usage of technologies might account for a large part of cross-country TFP differentials. The views expressed in this paper solely reflect those of the authors and not necessarily those of the National Bureau of Economic Research, Federal Reserve Bank of San Francisco, nor those of the Federal Reserve System as a whole. This research was completed when Emilie Rovito was an economist at the Federal Reserve Bank of New York. We appreciate the financial assistance of the NSF (Grants # SES-0517910 and SBE-738101) and the C.V. Starr Center for Applied Economics. We would like to thank Mark Bils, Hitesh Makhija, Andres Rodríguez-Clare, Romain Wacziarg, and Matt Wiswall for comments and suggestions.     

Estimating substitution elasticities in household production models

Summary Dynamic general equilibrium models that include explicit household production sectors provide a useful framework within which to analyze a variety of macroeconomic issues. However, some implications of these models depend critically on parameters, including the elasticity of substitution between market and home consumption goods, about which there is little information in the literature. Using the PSID, we estimate these parameters for single males, single females, and married couples. At least for single females and married couples, the results indicate a high enough substitution elasticity that including home production will make a significant difference in applied general equilibrium theory.We thank the National Science Foundation for financial support, and Ellen McGrattan, David Runkle, George Jakubson, Zvi Eckstein, Timothy J. Kehoe, Ed Prescott, a referee, and seminar participants at the Cleveland Fed and the Penn Macro Lunch Group for comments. The views expressed here are those of the authors and not necessarily those of any Federal Reserve Bank or the Federal Reserve System.     

A generalized theorem of the maximum

Summary This paper generalizes the Theorem of the Maximum (Berge [2]) to allow for discontinuous changes in the domain and the objective function. It also provides a geometrical version of the (generalized) theorem.This research was supported in part by National Science Foundation Grant SES-86-19012. Financial assistance was also provided, for Lawrence Ausubel, by the Lynde and Harry Bradley Foundation, and for Raymond Deneckere, by the Kellogg School of Management's Beatrice/Esmark Research Chair. We thank Alejandro Manelli and Israel Zang for many helpful conversations.     

On characterizing equilibria of economies with externalities and taxes as solutions to optimization problems

Summary We characterize equilibria of general equilibrium models with externalities and taxes as solutions to optimization problems. This characterization is similar to Negishi's characterization of equilibria of economies without externalities or taxes as solutions to social planning problems. It is often useful for computing equilibria or deriving their properties. Frequently, however, finding the optimization problem that a particular equilibrium solves is difficult. This is especially true in economies with multiple equilibria. In a dynamic economy with externalities or taxes there may be a robust continuum of equilibria even if there is a representative consumer. This indeterminacy of equilibria is closely related to that in overlapping generations economies.An earlier version of this paper, entitled Externalities and Taxes in General Equilibrium, was presented at the North American meetings of the Econometric Society, June 1988, at the University of Minnesota. We are grateful to David Backus, Kenneth Judd, Patrick Kehoe, and Rodolfo Manuelli for helpful conversations. National Science Foundation grants SES 86-18325 and SES 87-08616 provided financial support.The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.     

A simple search model of money with heterogeneous agents and partial acceptability

Summary. Simple search models have equilibria where some agents accept money and others do not. We argue such equilibria should not be taken seriously. This is unfortunate if one wants a model with partial acceptability. We introduce heterogeneous agents and show partial acceptability arises naturally and robustly. There can be multiple equilibria with different degrees of acceptability. Given the type of heterogeneity we allow, the model is simple: equilibria reduce to fixed points in [0,1]. We show that with other forms of heterogeneity equilibria are fixed points in set space, and there is no method to reduce this to a problem in R¹.Received: 4 September 2002, Revised: 23 September 2002JEL Classification Numbers: C78, E40.A. Shevchenko, R. Wright: We thank seminar participants at the Federal Reserve Bank of Cleveland, Indiana University, Purdue University, University of Toronto, the 2002 Midwest Macroeconomics Conference at Vanderbilt University, and the 2001 Conference on Economic Dynamics at the University of Essex. The National Science Foundation and the Federal Reserve Bank of Cleveland provided financial support. Braz Ministerio de Camargo and Gabriel Camera provided some helpful suggestions. Correspondence to: R. Wright     

Equilibrium business cycles with idle resources and variable capacity utilization

Summary A real business cycle economy is studied in which some capital is idle each period and the fraction of capital left idle varies in response to technology shocks. Previous equilibrium business cycle models have the characteristic that the entire stock of capital is used for production in each period. Our objective is to determine whether incorporating idle resources, something regularly observed in actual economies, significantly affects the cyclical properties of the model and hence changes our views about the importance of technology shocks for aggregate fluctuations. In our analysis we do not assume an aggregate production function, but instead model production as taking place at individual plants that are subject to idiosyncratic technology shocks. Each period the plant manager must choose whether to operate the plant or to let the plant remain idle. We find that the cyclical properties of this model are surprisingly similar to those of a standard real business cycle economy. One difference is that the model displays variation in factor shares while the standard models does not.The authors acknowledge support from the National Science Foundation.     

Bounded rationality,neural network and folk theorem in repeated games with discounting

Summary The perfect folk theorem (Fudenberg and Maskin [1986]) need not rely on excessively complex strategies. We recover the perfect folk theorem for two person repeated games with discounting through neural networks (Hopfield [1982]) that have finitely many associative units. For any individually rational payoff vector, we need neural networks with at most 7 associative units, each of which can handle only elementary calculations such as maximum, minimum or threshold operation. The uniform upper bound of the complexity of equilibrium strategies differentiates this paer from Ben-Porath and Peleg [1987] in which we need to admit ever more complex strategies in order to expand the set of equilibrium outcomes.I would like to thank Hao Li and John Curran for excellent research assistance. Financial support from National Science Foundation (SES-9223483), Sloan Foundation and the Division of Social Sciences at the University of Chicago is gratefully acknowledged.     

Non-existence and inefficiency of equilibria with American options

Summary We analyze two examples of economies with incomplete financial markets. In the first model we consider a stock and an American put option on the stock. Although there is only one commodity and asset payoffs therefore do not depend on spot prices, we derive robust non-existence of equilibria. In the second model we consider an economy where a stock is the only asset available for trade. We show that it is impossible to complete the market by introducing American put options and that equilibria are inefficient. This example is also robust.We wish to thank Jonathan Berk, Virginia France, Flavio Menezes and a referee of this journal for useful comments. This research was partially funded by the National Science Foundation (SES-8821723).     

A new approach to modeling decision-making under uncertainty

This paper presents two axiomatic models of decision making under uncertainty that avoid the use of a state space. The first is a subjective expected utility model with action-dependent subjective probabilities and effect-dependent preferences (the case of effect-independent preferences is obtained as a special instance). The second is a nonexpected utility model involving well-defined families of action-dependent subjective probabilities on effects and a utility representation that is not necessarily linear in these probabilities (a probabilistic sophistication version of this model, with action-dependent subjective probabilities is obtained as a special case). The hospitality of EUREQua, University of Paris 1, and financial support by the National Science Foundation grant SES-0314249 are gratefully acknowledged.     

Selection dynamics and adaptive behavior without much information

This paper uses the experimental method to investigate behavior in a coordination game when the information available to subjects is limited to their feasible choices and their experienced payoffs. In the experiment subjects converge to an absorbing state at rates that are orders of magnitude faster than reinforcement learning algorithms, but slower than under complete information. This state is very close to a mutual best response outcome. All cohorts converged to the market statistic predicted by the interior equilibrium regardless of the information conditions or the stability conditions. Eric Battalio programmed the graphical user interface. The National Science Foundation and Texas Advanced Research Program provided financial support. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or the Texas Advanced Research Program.     

Market clearing and derivative pricing

Summary. We develop a method of assigning unique prices to derivative securities, including options, in the continuous-time finance model developed in Raimondo (2001). In contrast with the martingale method of valuing options, which cannot distinguish among infinitely many possible option pricing processes for a given underlying securities price process when markets are dynamically incomplete, our option prices are uniquely determined in equilibrium in closed form as a function of the underlying economic data.Received: 14 April 2003, Revised: 7 January 2004, JEL Classification Numbers: G13, D52.This paper is dedicated to Birgit Grodal, whose strength and character we greatly admire. We are very grateful to Darrell Duffie, Steve Evans, Botond Koszegi, Roger Purves, Jacob Sagi, Chris Shannon, Bill Zame and an anonymous refereee for very helpful discussions and comments. The work of both authors was supported by Grant SES-9710424, and Andersons work was supported by Grant SES-0214164, from the National Science Foundation.     

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)     

The equilibrium allocation of investment capital in the presence of adverse selection and costly state verification

Summary We consider credit rationing in an environment with adverse selection and costly state verification. The presence of costly state verification permits debt contracts to emerge under conditions that we specify. When debt contracts are observed, so is credit rationing. This rationing occurs even if it is possible for rationed borrowers to bid up expected returns to lenders and hence is voluntary. We also show how the adverse selection and costly state verification problems interact and investigate how improvements in information gathering technology impact on the extent of credit rationing.The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. We have benefitted from comments on an earlier draft of this paper by Franklin Allen, Charlie Calomiris, V. V. Chari, Ed Green, Craig Holden, Jeff Lacker, George Pennachi, Neil Wallace, Anne Villamil, and an anonymous referee and from discussions with Edward Prescott.     

Measurement distortion and missing contingencies in optimal contracts

Summary Theory suggests that optimal contracts should include many contingencies to achieve optimal risk sharing. However, in practice, few contracts are as complex as theory suggests. This paper develops a model which is consistent with this observation. The lack of risk sharing results from the interplay of two factors. First, contingencies must be based on information produced by measurement systems, which may be manipulable. Second, when two parties to a contract meet, they often have incomplete information. The type of contract offered may reveal information about the party who proposes it. Different types of agents have different preferences over contingent contracts, because they have different abilities to manipulate the measurement system. These differences in preferences allow the parties to signal their types through the contracts they offer. Noncontingent contracts may be chosen in equilibrium because they are the only contracts which do not give any type an incentive to distort the measurement system and, hence, do not reveal information about the party proposing the contract.We have benefited from conversations with Oliver Hart, Rick Lambert, Michael Riordan and Jean Tirole and the comments of Michel Habib, Nick Yannelis and two anonymous referees. Financial support from the National Science Foundation under grants SES-8920048 and SES-8720589 is gratefully acknowledged.     

Survival auctions

Dynamic clock auctions with drop-out information typically yield outcomes closer to equilibrium predictions than do comparable sealed-bid auctions. However, clock auctions require congregating bidders for a fixed time interval, which has limited field applicability and introduces inefficiencies of its own given the time cost of congregating bidders. In this experiment we explore the effects of removing these inefficiencies through survival auctions—a multi-round sealed-bid auction which is theoretically isomorphic to the dynamic clock auction with drop-out information. Kagel’s research was partially supported by National Science Foundation Grants No. 0136925 and 0136928. Any opinions, findings, and conclusions or recommendations in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We thank the associate editor, an anonymous referee, seminar participants at Purdue University, especially Tim Cason, participants at International Industrial Organization April 2004 Conference, the ESA June 2003 meetings, and our discussant, Tim Salmon, for valuable comments. Kirill Chernomaz provided valuable research assistance. The usual caveat applies.     

Cyclical variation in the short-run Federal Reserve reaction function, 1969–1978

This note reports an empirical study of the Federal Reserve reaction function, as indicated by changes in the Federal funds interest rate, for the period from 1969–1978. In particular we examine whether changes in the funds rate have shown responses to underlying economic conditions which vary on a cyclical basis. We find that the Fed reacted more strongly to downturns than to recovery, with its response during the early recovery phase perhaps even procyclical. We also find that response to changes in inflation was less significant than reaction to output fluctuations.     

Vickrey auctions with reserve pricing

Summary. We generalize the Vickrey auction to allow for reserve pricing in a multi-unit auction with interdependent values. In the Vickrey auction with reserve pricing, the seller determines the quantity to be made available as a function of the bidders' reports of private information, and then efficiently allocates this quantity among the bidders. Truthful bidding is a dominant strategy with private values and an ex post equilibrium with interdependent values. If the auction is followed by resale, then truthful bidding remains an equilibrium in the auction-plus-resale game. In settings with perfect resale, the Vickrey auction with reserve pricing maximizes seller revenues.Received: 31 December 2002, Revised: 5 May 2003, JEL Classification Numbers: D44, C78, D82.Correspondence to: Lawrence M. AusubelThe authors gratefully acknowledge the generous support of National Science Foundation Grants SES-97-31025, SES-01-12906 and IIS-02-05489. We appreciate valuable comments from Ilya Segal. Special thanks go to Mordecai Kurz, who served as Larry's dissertation advisor and who introduced both authors to the economics profession back at IMSSS at Stanford. Congratulations and best wishes are extended to Mordecai and his family on the happy occasion of the publication of Assets, Beliefs, and Equilibria in Economic Dynamics: Essays in Honor of Mordecai Kurz, in which this article also appears.     

A general-equilibrium intertemporal model of an open economy

Summary This paper develops a very general (general-equilibrium) intertemporal model of a country engaged in international trade which can be used to address a wide variety of issues of interest — in particular, econometric application — under the assumption that prices of tradable commodities (consumer goods and capital goods) and the interest rate are exogenous to the country. It allows for an arbitrarily large number of commodities which are distinguished into seven categories and for finite or infinite periods of time. This model can be used to draw various policy conclusions. We investigate how current net imports, the balance of payments on current account, current consumption expenditure, next-period bondholdings, current wealth, and current internal prices will react to exogenous changes in current external prices, the current interest rate, current taxes, current factor endowments, and current-period bondholdings. This paper also considers the integrability of net-import demand functions.Work supported by National Science Foundation grant SES-8607652. We wish to thank the editor and an anonymous referee for useful comments and suggestions.