Peer group effects in applied general equilibrium

Summary. In this paper, I develop an applied general equilibrium environment with peer group effects. The application I consider is schooling. The framework used here is general equilibrium with clubs. I establish the existence of equilibrium for the economy with a finite number of school types. This result is then extended to the case where the set of school types is a continuum. The two welfare theorems are shown to hold for both economies. To compute the equilibrium, I construct a Negishi mapping from the set of weights on individual type's utility to the set of transfers that support the corresponding Pareto allocations as competitive equilibria with transfers. Because this mapping is a correspondence, a version of Scarf's algorithm is used to find a competitive equilibrium. Received: June 9, 1999; revised version: March 13, 2000     

Growth and equilibrium indeterminacy: the role of capital mobility

Summary. The paper presents a human capital driven endogenous growth model which, in general, permits a multiplicity of equilibrium balanced growth paths. It is shown that allowing for perfect capital mobility across countries increases the range of parameter values for which the model permits equilibrium indeterminacy. As opposed to the closed capital markets case, simple restrictions on preferences are no longer sufficient to eliminate the indeterminacy. Intuitively, under perfect capital mobility agents are able to smooth consumption completely. This induces an economy with open capital markets to behave like a closed economy with linear preferences thereby increasing the possibility of equilibrium indeterminacy. Received: 18 November 1998; revised version: 10 August 1999     

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     

On local indeterminacy and endogenous cycles in Ramsey models with heterogeneous households

It is well-known that endogenous cycles can occur in Ramsey models with heterogeneous households and borrowing constraints. In this note, we address the issue of robustness in the more general case of endogenous labor supply and we explain the occurrence of local indeterminacy under progressive taxation.

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     

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     

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     

Generic inefficiency of equilibria in the general equilibrium model with incomplete asset markets and infinite time

Summary. We consider a Lucas asset-pricing model with heterogeneous agents, exogenous labor income, and a finite number of exogenous shocks. Although agents are infinitely lived, endowments and dividends are time-invariant functions of the exogenous shock alone and are thus restricted to lie in a finite-dimensional space; genericity analysis can be conducted on sets of zero Lebesgue measure. When financial markets are incomplete, that is, there are fewer financial securities than shocks, we show that generically in individual endowments all competitive equilibria are Pareto inefficient. Received: November 22, 1999; revised version: March 4, 2002 RID="*" ID="*" We are grateful to an anonymous referee for very insightful comments on earlier drafts.     

Indeterminacy of equilibrium in stochastic OLG models

Summary. This paper studies the equilibria of a stochastic OLG exchange economies consisting of identical agents living for two periods, and having the opportunity to trade a single infinitely-lived asset in constant supply. The agents have uncertain endowments and the stochastic process determining the endowments is Markovian. For such economies, the literature has focused on studying strongly stationary equilibria in which quantities and prices are functions of the exogenous states of nature which describe the uncertainty: such equilibria are generalizations of deterministic steady states, and this paper investigates if they have the same special status as asymptotic limits of other equilibrium paths. The difficulty in extending the analysis of equilibria beyond the class of strongly stationary equilibria comes from the presence of indeterminacy: we propose a procedure for overcoming this difficulty which can be decomposed into two steps. First backward induction arguments are used to restrict the domain of possible prices; then if some indeterminacy is left, expectation functions are introduced to make the forward equilibrium equations determinate. The properties of the resulting trajectories, in particular their asymptotic properties, can then be studied. For the class of models that we study this procedure provides a justification for focusing on strongly stationary equilibria. For the model with positive dividends (equity or land) the justification is complete, since we show that the strongly stationary equilibrium is the unique equilibrium. For the model with zero dividends (money) there is a continuum of self-fulfilling expectation functions resulting in a continuum of equilibrium paths starting from any admissible initial condition: under conditions given in the paper, these equilibrium paths converge almost surely to one of the strongly stationary equilibria-either autarchy or the stochastic analogue of the Golden Rule. Received: November 19, 2001; revised version: March 22, 2002 RID="*" ID="*" We are grateful for the stimulating environment and research support provided by the Cowles Foundation at Yale University during the Fall 2000 when this paper was first conceived. We are also grateful to the participants of the SITE Workshop at Stanford University and the Incomplete Markets Workshop at SUNY Stony Brook during the summer 2001 for helpful discussions. Correspondence to: M. Magill     

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     

On games with identical equilibrium payoffs

Summary. This paper compares the sets of Nash, coalition- proof Nash and strong Nash equilibrium payoffs of normal form games which are closely related. We propose sufficient conditions for equivalent or closely related games to have identical sets of equilibrium payoffs. Received: April 23, 1999; revised version: November 23, 1999     

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     

Existence of a competitive equilibrium in a one sector growth model with heterogeneous agents and irreversible investment

Summary. We prove existence of a competitive equilibrium in a version of a Ramsey (one sector) model in which agents are heterogeneous and gross investment is constrained to be non negative. We do so by converting the infinite-dimensional fixed point problem stated in terms of prices and commodities into a finite-dimensional Negishi problem involving individual weights in a social value function. This method allows us to obtain detailed results concerning the properties of competitive equilibria. Because of the simplicity of the techniques utilized our approach is amenable to be adapted by practitioners in analogous problems often studied in macroeconomics. Received: September 13, 2001; revised version: December 9, 2002 RID="*" ID="*" We are grateful to Tapan Mitra for pointing out errors as well as making very valuable suggestions. Thanks are due to Raouf Boucekkine and Jorge Duran for additional helpful discussions. We also thank an anonymous referee for his/her helpful comments. The second author acknowledges the financial support of the Belgian Ministry of Scientific Research (Grant ARC 99/04-235 “Growth and incentive design”) and of the Belgian Federal Goverment (Grant PAI P5/10, “Equilibrium theory and optimization for public policy and industry regulation”). Correspondence to: C. Le Van     

Sensitivity analysis for applied general equilibrium models in the presence of multiple Walrasian equilibria

Summary. Pagan and Shannon's (1985) widely used approach employs local linearizations of a system of non-linear equations to obtain asymptotic distributions for the endogenous parameters (such as prices) from distributions over the exogenous parameters (such as estimates of taste, technology, or policy variables, for example). However, this approach ignores both the possibility of multiple equilibria as well as the problem (related to that of multiplicity) that critical points might be contained in the confidence interval of an exogenous parameter. We generalize Pagan and Shannon's approach to account for multiple equilibria by assuming that the choice of equilibrium is described by a random selection. We develop an asymptotic theory regarding equilibrium prices, which establishes that their probability density function is multimodal and that it converges to a weighted sum of normal density functions. An important insight is that if a model allows multiple equilibria, say , but multiplicity is ignored, then the computed solution for the i-th equilibrium generally no longer coincides with the expected value of that i-th equilibrium. The error can be large and correspond to several standard deviations of the mean's estimate. Received: December 7, 1999; revised version: December 4, 2000     

Equilibria in infinite dimensional commodity spaces revisited

Summary. The aim of the paper is to provide a new proof of the Mas-Colell–Richard existence of equilibrium result when preferences are non-transitive and incomplete. Our proof generalizes the main ideas of the Negishi approach to the case of unordered preferences. Received: January 10, 1996; revised version: November 23, 1999     

Nash equilibrium without mutual knowledge of rationality

Summary. In a Nash equilibrium, players' rationality is mutual knowledge. However, both intuition and experimental evidence suggest that players do not know for sure the rationality of opponents. This paper proposes a new equilibrium concept, cautious equilibrium, that generalizes Nash equilibrium in terms of preferences in two person strategic games. In a cautious equilibrium, players do not necessarily know the rationality of opponents, but they view rationality as infinitely more likely than irrationality. For suitable models of preference, cautious equilibrium predicts that a player might take a “cautious” strategy that is not a best response in any Nash equilibrium. Received: January 28, 1998; revised version October 2, 1998     

The convergence of least squares learning in stochastic temporary equilibrium models

Summary. This paper provides conditions for the almost sure convergence of the least squares learning rule in a stochastic temporary equilibrium model, where regressions are performed on the past values of the endogenous state variable. In contrast to earlier studies, (Evans and Honkapohja, 1998; Marcent and Sargent, 1989), which were local analyses, the dynamics are studied from a global viewpoint, which allows one to obtain an almost sure convergence result without employing projection facilities. Received: April 7, 2001; revised version: September 5, 2001     

Non-computability of competitive equilibrium

Summary. We provide a “computable counterexample” to the Arrow-Debreu competitive equilibrium existence theorem [2]. In particular, we find an exchange economy in which all components are (Turing) computable, but in which no competitive equilibrium is computable. This result can be interpreted as an impossibility result in both computability-bounded rationality (cf. Binmore [5], Richter and Wong [35]) and computational economics (cf. Scarf [39]). To prove the theorem, we establish a “computable counterexample” to Brouwer's Fixed Point Theorem (similar to Orevkov [32]) and a computable analogue of a characterization of excess demand functions (cf. Mas-Colell [26], Geanakoplos [16], Wong [50]). Received: September 9, 1997; revised version: December 17, 1997     

Stability of the cooperative equilibrium in N-person prisoners' dilemma with sequential moves

Summary. Nishihara [3] showed that N-person prisoners' dilemma has a cooperative Nash equilibrium, if the players decide their actions sequentially in the order determined by Nature under a certain information structure, and if each player's payoffs satisfy a certain inequality. This paper examines the stability of this cooperative equilibrium against two matters: players' slight mistakes and deviations by coalitions. The main results are as follows: (i) if the inequality on each player's payoffs strictly holds, then the cooperative equilibrium is a strictly proper equilibrium; (ii) if N≤3, and if full cooperation is Pareto efficient in N-person prisoners' dilemma, then the cooperative equilibrium is a strong Nash equilibrium; (iii) the cooperative equilibrium is in general a coalition-proof Nash equilibrium. Received: June 23, 1997; revised version: December 2, 1997     

On the structure of choice under different external references

Summary. In this paper we fully characterize an individual's choice behaviour according to three different so–called external references. The first system which we describe axiomatically is standard utility maximization or preference optimization. The second approach characterizes the choice of the second largest element as an optimal choice, the third system is the choice of a medium element, also as a first best choice. For all three approaches, we have established a common axiomatic structure which allows us to point out rather precisely congruences and divergences among the different systems considered. Received: December 12, 1997; revised version: September 15, 1998