Asymmetric Factor Substitutability and Indeterminacy

We consider a discrete-time two-sector CES economy with sector specific external effects in which factor substitutability differs across sectors. For this general model, we provide sufficient conditions on the elasticities of capital/labor substitution for local indeterminacy of equilibria.Acknowledgement We would like to thank two anonymous referees, the editor Dieter Bös and Tapan Mitra for comments on an earlier version of this paper, previously entitled Capital Depreciation, Factors Substitutability and Indeterminacy, which allowed us to improve considerably the results.     

The joint exploitation of a productive asset: a game-theoretic approach

Summary In the present paper we explore the set of equilibria in a game-theoretic model in which players can jointly exploit a productive asset. As in repeated games, we find that under certain circumstances there may be efficient as well as inefficient equilibria. In the model we study, efficient trigger-strategy equilibria may exist from some starting states (stocks of assets) but not others. More precisely, there is a stock level, sayy, such that an efficient trigger-strategy equilibrium exists from starting stocks greater than or equal toy, but not from those strictly less thany. (This statement is meant to include the cases in whichy is zero or infinite.) Under some circumstances, there may exist a new kind of equilibrium, which we call aswitching equilibrium. We show that, in our model, whenever y is positive (and finite), there is an open intervalI with upper endpoint y such that, from any starting stock inI there is an equilibrium of the dynamic game with the following structure: the players follow an inefficient but growing path until the stock reaches the levely, and then follow an (efficient) trigger strategy after that. The use of a continuous-time model enables us to conveniently decouple the delay of information from the time interval between decisions.We thank the C. V. Starr Center at New York University for research support. The views expressed here are those of the authors, and not necessarily those of AT&T Bell Laboratories.     

Joint production in teams

Consider Holmström's moral hazard in teams problem when there are n agents, each agent i has an a_i-dimensional strategy space and output is m-dimensional. We show that a compensation mechanism that satisfies budget balance, limited liability and implements an efficient allocation generically exists if and only if . Moreover, under a weak additional condition, the equilibrium implemented by this mechanism is unique in the class of pure strategy Coalition-Proof equilibria.     

Economic progress and skill obsolescence with network effects

Summary. We construct an OLG model with network effects to examine skill obsolescence when individuals can choose technological vintages. In the absence of transfer payments, some regions of the parameter space have unique stationary equilibria, others have unique cyclical equilibria, and others have multiple stationary equilibria. All equilibria are Pareto efficient. However, rat race equilibria can exist in which all agents currently alive prefer a slower rate of progress than occurs in equilibrium. When contemporaneous transfers are allowed, equilibria are unique everywhere, but a cycle still exists, and a rat race can still arise in equilibrium. Allowing intertemporal transfers (debt) ensures that all equilibria are stationary. In the relevant parameter range, the introduction of debt can eliminate cycles and increase the long-run growth rate. No rat race equilibria exist when debt is allowed.Received: 3 January 2002, Revised: 3 June 2004, JEL Classification Numbers: 041, J24, O33. Correspondence to: Ian P. KingEarly versions of this paper were presented at the West Coast Macro Workshop, the Mid-West Macro Meetings, the Canadian Macroeconomics Study Group Meetings, and the Society for Economic Dynamics Meetings. We would like to thank V. V. Chari, Merwan Engineer, Don Ferguson, John Hillas, John Knowles, Dan Peled, Dan Usher, Linda Welling and Julian Wright for helpful comments. We are especially grateful to an anonymous referee for some very useful and substantive comments.     

New complexity results about Nash equilibria

We provide a single reduction that demonstrates that in normal-form games: (1) it is -complete to determine whether Nash equilibria with certain natural properties exist (these results are similar to those obtained by Gilboa and Zemel [Gilboa, I., Zemel, E., 1989. Nash and correlated equilibria: Some complexity considerations. Games Econ. Behav. 1, 80–93]), (2) more significantly, the problems of maximizing certain properties of a Nash equilibrium are inapproximable (unless ), and (3) it is -hard to count the Nash equilibria. We also show that determining whether a pure-strategy Bayes–Nash equilibrium exists in a Bayesian game is -complete, and that determining whether a pure-strategy Nash equilibrium exists in a Markov (stochastic) game is -hard even if the game is unobserved (and that this remains -hard if the game has finite length). All of our hardness results hold even if there are only two players and the game is symmetric.     

Asymptotical behavior of finite and possible discontinuous economies

We consider a finite economy in which the data are depending on an exogenous parameter and the utilities satisfy a condition, previously introduced by the authors and called sequential pseudocontinuity, weaker than sequential lower semicontinuity and than sequential upper semicontinuity. We show that the economy has a nice asymptotical behavior, that is: for any sequence (x _n ) _n of exogenous parameters converging to x _o , any convergent subsequence of a sequence of approximate competitive equilibria of converges to an exact equilibrium of .      

p-Best response set and the robustness of equilibria to incomplete information

In this paper, we use p-best response sets—a set-valued extension of p-dominance—in order to provide a new sufficient condition for the robustness of equilibria to incomplete information: if there exists a set S which is a p-best response set with , and there exists a unique correlated equilibrium μ^* whose support is in S then μ^* is a robust Nash equilibrium.     

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.     

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.     

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.     

Dynamic coordination games

Summary Gains from coordination provide incentives for delay. In this paper, the extent of delay is studied in a dynamic,N-person, coordination game. There is no social gain from delay, so an equilibrium with delay is always inefficient. For fixedN, there is no coordination failure when the period length is short: all equilibrium outcomes converge to the Pareto efficient outcome as the period length converges to zero. On the other hand, holding period length fixed, there exist equilibria in which delay is proportional toN, for arbitrarily large values ofN. In addition, it can be shown that the possibility of delay depends on the timing of strategic complementarities. However, under certain conditions, delay is shown to be a robust phenomenon, in the sense that well-behaved equilibria exhibit infinite delay forN sufficiently large.This paper grew out of discussions with Christophe Chamley. While writing it I benefited from discussions with Ken Binmore, Russell Cooper, Bob Rosenthal and Michael Manove. Joe Farrell, Drew Fudenberg, Martin Hellwig and Sawoong Kang made very useful comments on an earlier version that led to substantial improvements. Helpful comments were also made by seminar participants at the London School of Economics, the SUNY at Stoney Brook, the NBER Summer Institute, Northwestern University, and the University of Chicago. I would like to thank Nick Yannelis and an anonymous referee for their editorial advice. Financial support for this research was provided by the National Science Foundation under Grant No. SES 9196061.     

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 number and type of long-term equilibria

We consider a balanced-growth model with finitely many processes, a uniform rate of profits, and a given composition of final demand. Two types of equilibria are distinguished, black and white: the color of an equilibrium is defined by the relative sign of two determinants associated with the operated methods, and can be interpreted as a local property of a cross-dual dynamic process. Under standard economic assumptions, and flukes apart, the number of white equilibria exceeds that of black equilibria by one. In particular the total number of long-term equilibria is odd.     

Money,price dispersion and welfare

Summary. We introduce heterogeneous preferences into a tractable model of monetary search to generate price dispersion, and then examine the effects of money growth on price dispersion and welfare. With buyers search intensity fixed, we find that money growth increases the range of (real) prices and lowers welfare as agents shift more of their consumption to less desirable goods. When buyers search intensity is endogenous, multiple equilibria are possible. In the equilibrium with the highest welfare level, money growth reduces welfare and increases the range of prices, while having ambiguous effects on search intensity. However, there can be a welfare-inferior equilibrium in which an increase in money growth increases search intensity, increases welfare, and reduces the range of prices.Received: 25 July 2003, Revised: 12 December 2003JEL Classification Numbers: E31, D60.B. Peterson, S. Shi: We thank Gabriele Camera, Aleksander Berentsen and an anonymous referee for useful suggestions. We have also received valuable comments from the participants of the workshop at Michigan State, the Purdue Conference on Monetary Theory (2003) and the Midwest Macro Meeting (Chicago, 2003). Shi gratefully acknowledges financial support from the Bank of Canada Fellowship and the Social Sciences and Humanities Research Council of Canada. The opinion expressed here is the authors own and does not reflect the view of the Bank of Canada.Correspondence to: S. Shi     

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.     

Austrian economics and the analysis of labor markets

Theory and policy relating to labor markets is dominated by the mainstream labor market model, although a less well-known, socioeconomic version can also be identified. The mainstream model is methodologically flawed and forced, thereby, to relegate any (serious) investigation of labor market institutions and/or social structures to the margins of its analysis. The socioeconomic account is not so much methodologically flawed, as methodologically ambivalent. While this ambivalence does not actually prevent the investigation of institutions and/or social structures, it does promote ambiguity whenever we inquire into the precise nature of the interaction between them and labor markets. Insights from Austrian economics, when used in collaboration with critical realist methodology, can play a part in augmenting the socioeconomic account, generating a totally new approach to the analysis of labor markets.

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.     

On disequilibrium savings and public consumption

Conclusions A particular aspect of the present paper is the introduction of specific policy measures for the government, whose behavior on the goods market was described in earlier work as purely exogenous, like in Malinvaud. In our context, the government appears as an active economic agent, acting at absorbing any excess supply or reducing any excess demand on the goods market. Though this behavior may look somewhat arbitrary, it has the advantage to force the state of the economy towards a SME if combined with natural endogenous behavior of the other agents! Furthermore, it does not contradict observed policies through which governments stimulate or restrain economic activity via purchases or fiscal and monetary policies. Perhaps alternative policies, like direct actions on the labor market by supplying (non-productive) jobs or unemployment compensations, could have done as well: this remains an open door for further research. The preceding feature also contrasts with the recent work done by V. Böhm (1978) in a macroeconomic set up. In this paper, he studies the stability of stationary Keynesian unemployment or stationary repressed inflation states but without imposing a particular policy on the government. Comparing his work with ours, it is easily verified that if the government would keep its consumption at the levelg ^*, the SME would be stable if the economy starts out in Keynesian unemployment and unstable if the economy starts out in repressed inflation, confirming Böhm's result.Our analysis is related to an earlier work of Archibald and Lipsey (1958), dealing with the adjustment of the economy to a Stationary Equilibrium after a change in real balances. The Quantity Theory of Money postulates that along a SME, a change in the price and the wage rate from (p,w) to (p,w) leads to an adjustment in the level of stationary money holdings from m_i ^* to m_i ^*, m_i ^*=_i ^* (p,w). In this paper, Archibald and Lipsey suggest that the economy follows a sequence of temporary market equilibria: Starting from a change in real balances,prices adjust at each period through a tâtonnement process so as to match supply and demand. Our paper proposes an alternative path: At each period,quantities adjust through a tâtonnement process at constant prices.This paper is a revised version of CORE Discussion Paper 7701. We wish to thank Paul Champsaur, Jacques Drèze, Werner Hildenbrand and Reinhard John for stimulating discussions. We are grateful to Volker Böhm for valuable comments and criticisms.Research supported by the Fonds National Belge de la Recherche Scientifique.     

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.