Asymptotic methods for asset market equilibrium analysis

Summary. General equilibrium analysis is difficult when asset markets are incomplete. We make the simplifying assumption that uncertainty is small and use bifurcation methods to compute Taylor series approximations for asset demand and asset market equilibrium. A computer must be used to derive these approximations since they involve large amounts of algebraic manipulation. We use this method to analyze the allocative and welfare effects of introducing a new security. We find that adding any nontrivial derivative security will raise the price of the risky security relative to the bond when risks are small. Received: April 1, 2000; revised version: January 10, 2001     

Computable general equilibrium with financial markets

Summary. There are a wide variety of theoretical general equilibrium models with incomplete security markets. In this paper we give a general recipe for using homotopy algorithm to compute equilibria in these models. In many models, taxes, transaction-costs or other market frictions introduce the additional difficulty that equilibrium prices or choices (but not equilibrium allocations) may be undetermined. In order to demonstrate how these difficulties can be dealt with, we develop a globally convergent algorithm to compute equilibria in a model with cash-in-advance constraints, several goods and incomplete financial markets. Furthermore we describe how to implement the algorithm using a publicly available suite of subroutines for homotopy-pathfollowing. Received: October 1, 1999; revised version: December 16, 2000     

The equilibrium set of two-player games with complementarities is a sublattice

Summary. I prove that the equilibrium set in a two-player game with complementarities, and totally ordered strategy spaces, is a sublattice of the joint strategy space. Received: May 31, 2001; revised version: October 4, 2002     

Saving behavior in stationary equilibrium with random discounting

Summary. We study the implications of random discount rates of future generations for saving behavior and capital holdings in a steady state competitive equilibrium with heterogeneous population. A well-known difficulty in deterministic economies with heterogeneous households is that in steady state only the most patient households hold capital. In this paper we state conditions under which this random discounting is sufficient for households other than the most patient ones to save. We thus provide a simple and natural way of overcoming the aforementioned difficulty. Received: December 28, 1998; revised version: May 19, 1999     

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     

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     

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     

Temporary financial equilibrium

Summary. In a two-period pure exchange economy with financial assets, a temporary financial equilibrium is an equilibrium of the current spot and security markets given forecast functions of future prices and payoffs. The temporary equilibrium model can then be interpreted as an Arrow-Debreu economy where preferences depend on prices. This identification implies, among other consequences, the existence and the generic determinateness of the financial temporary equilibria associated with given forecast functions. Received: December 29, 1999; revised version: December 20, 2001     

Information at equilibrium

Summary. In a game with rational expectations, individuals simultaneously refine their information with the information revealed by the strategies of other individuals. At a Nash equilibrium of a game with rational expectations, the information of individuals is essentially symmetric: the same profile of strategies is also an equilibrium of a game with symmetric information; and strategies are common knowledge. If each player has a veto act, which yields a minimum payoff that no other profile of strategies attains, then the veto profile is the only Nash equilibrium, and it is is an equilibrium with rational expectations and essentially symmetric information; which accounts for the impossibility of speculation. Received: June 20, 2001; revised version: January 9, 2002 RID="*" ID="*" We wish to thank Pierpaolo Battigalli, Fran?oise Forges, Franco Donzelli, Leonidas Koutsougeras, Aldo Rustichini, Rajiv Vohra and Nicholas Yannelis for their comments. Correspondence to: H. Polemarchakis     

Global asymptotic stability of a competitive equilibrium with recursive preferences

Summary. This paper demonstrates global stability of a competitive equilibrium in a multi-sector model of many firms, each of which exhibits constant returns to scale technology, and of infinitely lived consumers, whose preferences are recursive but not necessarily additively separable. In the topology induced by a sup-norm, the dominant diagonal blocks condition (Araujo and Scheinkman (Econometrica 45, 1977)) allows us to apply the implicit function theorem to obtain continuity of the equilibrium path. If a stationary equilibrium is locally asymptotically stable, then the continuity of the equilibrium path and smoothness of a weight function on heterogeneous consumers imply that all equilibrium paths converge to the steady state. The dominant diagonal blocks condition is also shown to be sufficient for the local asymptotic turnpike property. Received: December 13, 1996; revised version: June 2, 1999     

Existence of equilibrium in an OLG model with production and altruistic preferences

Summary. This note deals with the existence and uniqueness of a non-trivial steady-state equilibrium in an overlapping generations (OLG) model with productive capital and altruistic agents. We establish a necessary and sufficient condition for operative bequests which extends Abel (1987) and Weil (1987). Interestingly, we prove that the OLG model with production and altruistic agents always experiences a non-trivial steady-state equilibrium. Received: July 16, 1998; revised version: January 29, 1999     

General equilibrium and resource economics

Summary. For a number of reasons a large class of general equilibrium models from the field of resource economics does not allow for an equilibrium analysis along the lines of the theory of infinite dimensional commodity spaces. The reasons concern the choice of the commodity space and the applicability of properness assumptions with respect to preferences and the technology. This paper illustrates the difficulties and shows for a prototype model how the problems can successfully be tackled by the use of a limit argument on equilibria in the truncated economies. Received: May 2, 1996; revised version: May 13, 1998     

How complex are networks playing repeated games?

Summary. This paper examines implications of complexity cost in implementing repeated game strategies through networks with finitely many classifiers. A network consists of individual classifiers that summarize the history of repeated play according to a weighted sum of the empirical frequency of the outcomes of the stage game, and a decision unit that chooses an action in each period based on the summaries of the classifiers. Each player maximizes his long run average payoff, while minimizing the complexity cost of implementing his strategy through a network, measured by its number of classifiers. We examine locally stable equilibria where the selected networks are robust against small perturbations. In any locally stable equilibrium, no player uses a network with more than a single classifier. Moreover, the set of locally stable equilibrium payoff vectors lies on two line segments in the payoff space of the stage game. Received: May 9, 1997; revised version: November 18, 1997     

Equilibrium selection in coordination games with simultaneous play

Summary. We apply the dynamic stochastic framework proposed in recent evolutionary literature to a class of coordination games played simultaneously by the entire population. In these games payoffs, and hence best replies, are determined by a summary statistic of the population strategy profile. We demonstrate that with simultaneous play, the equilibrium selection depends crucially on how best responses to the summary statistic remain piece-wise constant. In fact, all the strict Nash equilibria in the underlying stage game can be made stochastically stable depending on how the best response mapping generates piece-wise constant best responses. Received: February 12, 2001; revised version: October 29, 2001     

Non-cooperative games with a continuum of players whose payoffs depend on summary statistics

Summary. We consider static non-cooperative games with a continuum of small players whose payoffs depend on their own actions and finitely many summary statistics of the aggregate strategy profile. We prove the existence of an equilibrium in pure strategies without any convexity restrictions on payoffs or the common action space. We show that this result applies to a broad class of monopolistic competition models. Received: April 13, 2001; revised version: December 18, 2001 RID="*" ID="*" The result in this paper generalizes a result in my PhD dissertation supervised by M. Ali Khan and Joe Harrington. I thank them for support and encouragement. I also thank Sung Kim, Bruce Nanney, Ashvin Rajan, Kali Rath, and an anonymous referee for comments. The usual disclaimer applies.     

Stability in the patent race contest of Lee and Wilde

Summary. To determine how expenditure and profits vary with the number of firms in the patent race contest of Lee and Wilde, it is traditional to impose an ad-hoc stability condition on the best response function. This paper relates the stability condition to the standard myopic adjustment mechanism and shows that a concave hazard rate function with non- increasing hazard rate elasticity is sufficient for the analysis. We provide examples and reveal additional qualitative properties of the Lee and Wilde model. Received: November 15, 1996; revised version: March 25, 1998     

Evolutionary foundations of aggregate-taking behavior

Summary. I show that aggregate-taking behavior is often evolutionarily stable for finite population in symmetric games in which payoff depends only on own strategy and an aggregate. I provide economic examples exhibiting this phenomenon. Received: August 27, 2001; revised version: January 29, 2002 RID="*" ID="*" The paper has profited from the comments of Maria Montero, Burkhard Hehenkamp, Wolfgang Leininger, and Dave Furth. Financial support from the DFG via Postgraduate Programme at the University of Dortmund and via SFB 504 at the University of Mannheim is acknowledged. RID="*" ID="*" Present address: University of Mannheim, SFB 504, L 13, 15, 68131 Mannheim, Germany (e-mail: possajen@sfb504.uni-mannheim.de)     

Moral hazard and general equilibrium in large economies

Summary. The paper analyzes a two period general equilibrium model with individual risk, aggregate uncertainty and moral hazard. There is a large number of households, each facing two individual states of nature in the second period. These states differ solely in the household's vector of initial endowments, which is strictly larger in the first state (good state) than in the second state (bad state). In the first period each household chooses a non-observable action. Higher levels of action give higher probability of the good state of nature to occur, but lower levels of utility. Households' utilities are assumed to be separable in action and the aggregate uncertainty is independent of the individual risk. Insurance is supplied by a collection of firms who behave strategically and maximize expected profits taking into account that each household's optimal choice of action is a function of the offered contract. The paper provides sufficient conditions for the existence of equilibrium and shows that the appropriate versions of both welfare theorems hold. Received: December 7, 1998; revised version: October 25, 1999     

Segmented risk sharing in a continuous-time setting

Summary. The economy we study is comprised of a continuum of individuals. Each has a stochastic endowment that evolves continuously and independently of all other individuals' endowment processes. Individuals are risk averse and would therefore like to insure their endowment processes. The mutual independence of their endowment processes makes it feasible for them to obtain this insurance by pooling their endowments. We investigate whether such a scheme would survive as an equilibrium in a noncooperative setting. Received: October 16, 2000; revised version: August 8, 2001