First-price auction symmetric equilibria with a general distribution

In this paper I obtain the mixed strategy symmetric equilibria of the independent private values, first-price auction for any distribution of valuations. The equilibrium strategy is a pure strategy at the continuity points of the distribution and bids a mixed strategy at the discontinuity points of the distribution.     

Wage and price controls in the equilibrium sequential search model

In this paper, we study the effects of wage and price controls on employment, output, and welfare in a simplified version of the Bénabou (J. Econom. Theory 60 (1993) 140) equilibrium sequential search model with bilateral heterogeneity. We show that a price ceiling increases output but the change in welfare depends on three effects: the reduction in aggregate search costs, the increase in surplus due to increased output, and the transfer of production to the least efficient firm. The model is formally identical to a standard equilibrium search model of the labor market so analogous results hold for the minimum wage.     

Non-bidding equilibrium in an ascending core-selecting auction

This paper investigates the perfect Bayesian equilibrium in an ascending-price core-selecting auction, which is recently used in some countries? spectrum license auctions. We suppose that there are two identical items, two small bidders, and one large bidder. The small bidders demand only one unit of the item, whereas the large bidder wants both units. Package bidding ensures that the large bidder faces no exposure problem and behaves truthfully. However, one of the small bidders stops bidding at the beginning in the equilibrium. Although small bidders generally face the free-rider problem and have incentives to underbid, once a bidder is the only small one remaining, he bids truthfully. Stopping early induces the remaining bidder to behave truthfully. Hence, each small bidder wants to be the first to stop bidding. The free-rider problem is considerably mitigated when there are many small bidders.     

Rationalizability and equilibrium in N-person sequential bargaining

Summary. This paper deals with N-person sequential bargaining games with complete information. For N-person sequential bargaining games, uniqueness of the SPE has been obtained by allowing the players to exit with partial agreements. Adopting a non-equilibrium approach, we show that N-person sequential bargaining games with exit are solvable by a refinement of rationalizability for multi-stage games (trembling-hand rationalizability) whatever the impatience of the players. That is, once we adopt the non-equilibrium approach, the exit opportunity still fulfils its original aim: we achieve a unique solution by introducing the exit opportunity. Moreover, this unique solution is the unique SPE. Received: October 30, 1996; revised version: July 7, 1998     

Asset price bubbles in Arrow-Debreu and sequential equilibrium

Summary. Price bubbles in an Arrow-Debreu equilibrium in an infinite-time economy are a manifestation of lack of countable additivity of valuation of assets. In contrast, the known examples of price bubbles in a sequential equilibrium in infinite time cannot be attributed to the lack of countable additivity of valuation. In this paper we develop a theory of valuation of assets in sequential markets (with no uncertainty) and study the nature of price bubbles in light of this theory. We define a payoff pricing operator that maps a sequence of payoffs to the minimum cost of an asset holding strategy that generates it. We show that the payoff pricing functional is linear and countably additive on the set of positive payoffs if and only if there is no Ponzi scheme, provided that there is no restriction on long positions in the assets. In the known examples of equilibrium price bubbles in sequential markets valuation is linear and countably additive. The presence of a price bubble means that the dividends of an asset can be purchased in sequential markets at a cost lower than the asset's price. We present further examples of equilibrium price bubbles in which valuation is nonlinear, or linear but not countably additive.     

Sorting equilibrium in a multi-jurisdiction model

This paper analyzes a general model of an economy with heterogeneous individuals choosing among two jurisdictions, such as towns or political parties. Each jurisdiction is described by its constitution, where a constitution is defined as a mapping from all possible population partitions into the (possibly multidimensional) policy space. This study is the first to establish sufficient conditions for existence of sorting equilibria in a two-jurisdiction model for a policy space of an arbitrary dimension.     

Adaptive learning in extensive form games and sequential equilibrium

Summary. This paper studies adaptive learning in extensive form games and provides conditions for convergence points of adaptive learning to be sequential equilibria. Precisely, we present a set of conditions on learning sequences such that an assessment is a sequential equilibrium if and only if there is a learning sequence fulfilling the conditions, which leads to the assessment. Received: November 5, 1996; revised version: May 28, 1997     

Time-inconsistent preferences in a general equilibrium model

This paper introduces time-inconsistent preferences in a multicommodity general equilibrium framework with incomplete markets. The standard concept of competitive equilibrium is extended in order to allow for changes in intertemporal preferences. Depending on whether or not agents recognize that their intertemporal preferences change, agents are called sophisticated or naïve. This paper presents competitive equilibrium notions for economies with naïve agents and economies with sophisticated agents and provides assumptions under which both types of equilibria exist. Surprisingly, the set of naïve equilibria in societies populated by time-consistent households is not allocationally equivalent to the set of competitive equilibria. For sophisticated equilibria, the equivalence holds. Time-inconsistency also raises conceptual issues about the appropriate concept of efficiency. Choices have to be made concerning the incorporation of future preferences and the appropriate instruments to create Pareto improvements. For both naïve and sophisticated societies, we present four possible efficiency concepts. Suitable conditions are specified for which both naïve and sophisticated equilibria satisfy appropriate efficiency concepts.     

Balanced equilibrium in a consumption loans model

In a consumption loans model with many generations, Gale's theorem on the existence of balanced equilibrium is generalized, allowing more general preferences. The new theorem shows that there are plausible conditions under which there exists a Pareto optimal (non-optimal) balanced equilibrium whenever there exists (does not exist) a monetary golden rule equilibrium.     

Inequalities and fluctuations in a dynamic general equilibrium model

   Summary. In the present paper a tractable two-sector neo-classical growth model with heterogeneous agents is considered. The local dynamic properties of the equilibrium path are analyzed in relation with the underlying characteristics of the economy. In particular, the existence of fluctuations is related to the degree of heterogeneity in labor and in capital endowments. When applied to international trade theory, the analysis shows that free trade may distabilize a world economy that is originally stable under the regime of autarky. Received: December 28, 1998; revised version: October 29, 1999     

Entry decisions and bidding behavior in sequential first-price procurement auctions: An experimental study

Though many real life auctions are run independently of each other, from the bidders' point of view they often form sequences of auctions. We investigate how behavior responds to the additional incentives that are present in such auction sequences. Comparing subjects' decisions in single first-price procurement auctions with their decisions in a game consisting of two subsequent first-price procurement auctions, we find that, in line with the theoretical prediction, entry and bidding behavior is crucially affected by the opportunity cost of early bid submission. Though, entry decisions and average bids in the auction sequence systematically deviate from the perfect Bayesian equilibrium prediction. While the nature of the opponent (human being or computer) has no significant effect on these findings, giving subjects additional feedback on winners and prices seems to reduce the deviations from the equilibrium prediction.     

Endogenous innovation and imitation in a model of equilibrium growth

This paper provides a growth model in which innovation as well as imitation occurs. Economic growth is due to product innovations. Innovators driven by the possibility to appropriate monopoly profits do not remain in their monopolistic position forever. Latecoming imitators get into possession of the private knowledge of production through investments in R & D. Imitated products are marketed in oligopolistic markets. Imitation proves to be profitable despite a single factor market and positive imitation costs. A steady-state equilibrium with positive imitation and innovation rates as well as different market structures can be derived. Finally, the effects of industrial policy measures are discussed.     

Market and underground activities in a two-sector dynamic equilibrium model

Summary. In this paper a two sector dynamic general equilibrium model is developed in order to evaluate the implications of the underground economy from a business cycle perspective. There are three main results. First, introducing an underground sector improves the fit of the model to the data, especially along several important labor market dimensions. Second, the model produces substantial internal propagation of temporary shocks. Third, it is shown that underground activities offer risk sharing opportunities by allowing households to smooth income through a proper labor allocation between the two sectors.Received: 17 June 2002, Revised: 25 April 2003, JEL Classification Numbers: E320, E260, J22, H200.We have benefited from the comments and suggestions of John Donaldson. We would also like to thank Paolo Siconolfi, Jean Pierre Danthine, Fausto Gozzi, Edmund Phelps, Gustavo Piga, Domenico Tosato, and the participants in the seminars at various universities, David Giles and Stefano Pisani for providing useful information on the underground data, Francesca Caponi for the comments and the information concerning the legal and fiscal aspects involved in the calibration, and Glenn Williams for the research assistance. Finally, we thank two anonymous referees for helpful comments on this and on earlier versions of the paper. Chiarini acknowledges financial support from the Ateneo Research fund of the University of Rome, La Sapienza, Dinamiche dell'integrazione europea e scelta di politica economica. All errors are ours. Correspondence to: F. Busato     

The value of money in a dynamic equilibrium model

Summary. This paper studies the pricing of money in an infinite-horizon economy with heterogeneous agents, incomplete financial markets and arbitrary borrowing restrictions. Purchases of the consumption good are subject to a cash-in-advance constraint. Under general conditions I show that the price of money is equal to its fundamental value, where this value is defined over all state-price processes that are compatible with the existence of no-arbitrage opportunities. This equality implies that the cash-in-advance constraint is binding infinitely often for all agents in the economy. The analysis highlights certain differences in the determination of the price of money with respect to models with money in the utility function that bear on the optimal implementation of economic policies.Received: 23 October 2003, Revised: 26 August 2004 JEL Classification Numbers: D52, E44, G12.M.S. Santos: This paper is an outgrowth of an earlier collaboration with Michael Woodford. I have also benefitted from various discussions with Eduardo Gimenez, Alejandro Hernandez, and Miguel Iraola. Some very useful comments by an anonymous referee are greatly appreciated.     

Existence and uniqueness of stochastic equilibrium in a simple macro model

This paper demonstrates that even the simplest rational expectations macro model may be plagued by nonexistent or multiple equilibria. The reason for these problems is the postulated existence of risk-averse behavior on the part of labor suppliers and purchasers of durable commodities. This risk-averse behavior introduces the price forecast variance into the behavioral parameters of the model, thus leading to a simultaneous nonlinear system.     

A common value auction model with endogenous entry and information acquisition

Summary Much of the auction literature assumes both a fixed number of bidders and a fixed information setting. This sidesteps the important and often costly decisions a potential bidder must make prior to an auction: Should I enter and, if I do, what level of resources should I expend evaluating the good prior to bidding? We answer these questions for a stylized information model of a common value auction. The expected selling price is shown to be the expected value of the good minus the expected aggregate entry and information costs of the bidders. Thus, the seller indirectly pays for these costs to the bidders. There are auctions where the seller seemingly restricts the bidders' information expenditures. While this restriction does influence the entry decision, we demonstrate that the overall effect can be to improve the selling price. Finally, the probability of entry and the chosen accuracy of the information are never more in the second-price auction than in the first-price auction, and the seller prefers the second-price auction.We are grateful for the comments and suggestions of seminar participants at the University of British Columbia, Dartmouth College, the University of Wisconsin, Yale University, and the International Conference on Game Theory and Economics at SUNY Stony Brook.