Author index for volume 32

A way of establishing the stationary Pareto optimality of asset equilibria in stochastic settings with age diversity among the agents is suggested. Difficulties in obtaining such results arise because agents' behavior is affected by random future state variables while Pareto optimality, defined in terms of expected lifetime utility of representative agents conditioned on their first period state, involves redistributions of consumption among agents with different histories of realized states. Frobenius' theorem is used to establish the existence of weights for which the so-weighted sum of representative agents' (conditional) expected utilities is maximized at the asset equilibrium allocation.     

The existence of fully rational expectations approximate equilibria with noisy price observations

Nonrevealing fully rational expectations approximate equilibria exist in microeconomic pure exchange economies in which uninformed agents have suitably dispersed noisy price observations. Such traders maximize a state-dependent expected utility conditional on the price vector they observe, the distributions of noisy price observations, and the correct equilibrium relationship between states of the world and prices. In equilibrium, aggregate excess demand is small with high probability in every state of the world (and its expectation is also small); this magnitude diminishes as the noisy price observations become more accurate. Equilibria are obtained by applying a fixed point argument to state-dependent excess demand functions which are smooth because of the noisy price observations.     

Inflation and distribution in a Keynes-Wicksell model of the business cycle

The paper integrates a Kaleckian hypothesis of markup pricing and markup adjustments into a dynamic IS-LM model of Keynes-Wicksell type. Local stability and instability are characterized by different speeds of adjustments in the adaptive expectations of inflation. Application of the Hopf bifurcation theorem proves existence of periodic orbits in the three-dimensional state space. The global dynamics is investigated by means of computer simulations. A suitable modification of the rule of adaptive expectations checks the explosive tendencies and gives rise to an apparently unique and stable limit cycle. It is sketched how the evolution of the most important variables over this growth cycle compares to empirical stylized facts.     

A two-stage core with applications to asset market and differential information economies

Summary. We introduce a new core concept, called the two-stage core, which is appropriate for economies with sequential trade. We prove a general existence theorem and present two applications of the two-stage core: (i) In asset markets economies where we extend our existence proof to the case of consumption sets with no lower bound, in order to capture the case of arbitrary short sales of assets. Further, we show that the two-stage core is non empty in the Hart (1975) example where a rational expectations equilibrium fails to exist. (ii) In differential information economies where we provide sufficient conditions for the incentive compatibility of trades. Namely, that no coalition of agents can misreport the true state and provide improvements to all its members, even by redistributing the benefits from misreporting. Received: December 20, 1995; revised version: December 6, 1996     

Stability of cycles and expectations

The actual dynamics of an economy depends on how agents forecast the future at every date as a function of their information on the past, while possibly learning the structure of their environment. We show in the case of a one-dimensional state variable that under mild conditions on expectations functions, a given cycle with perfect foresight that is stable in the actual dynamics is stable in a fictitious backward perfect foresight dynamics. We exhibit a restricted class of expectations functions for which the converse is true.     

Hopf bifurcation and quasi-periodic dynamics in discrete multisector optimal growth models

This paper discusses the asymptotic stability of the steady state and the existence of a Hopf bifurcation in discrete time multisector optimal growth models. We obtain on the one hand a local turnpike theorem which guarantees the saddle point property for all discount rates. On the other hand, we provide a new proposition which gives some conditions ensuring local stability of the steady state if the impatience rate is not too high. A characterization of the boundδ*, above which the steady state is saddle-point stable, is also proposed in terms of indirect utility function's concavity properties. On this basis, some sufficient conditions for the existence of a Hopf bifurcation are stated. We thus prove the existence of quasi-periodic optimal paths in asymmetric models.     

Limited information minimal state variable learning in a medium-scale multi-country model

Rational expectations has been the dominant way to model expectations, but the literature has quickly moved to a more realistic assumption of boundedly rational learning where agents are assumed to use only a limited set of information to form their expectations. A standard assumption is that agents form expectations by using the correctly specified reduced form model of the economy, the minimal state variable solution (MSV), but they do not know the parameters. However, with medium-sized and large models the closed-form MSV solutions are difficult to attain given the large number of variables that could be included. Therefore, agents base expectations on a misspecified MSV solution. In contrast, we assume that agents know the deep parameters of their own optimising frameworks. However, they are not assumed to know the structure nor the parameterisation of the rest of the economy, nor do they know the stochastic processes generating shocks hitting the economy. In addition, agents are assumed to know that the changes (or the growth rates) of fundament variables can be modelled as stationary ARMA(p,q) processes, the exact form of which is not, however, known by agents. This approach avoids the complexities of dealing with a potential vast multitude of alternative misspecified MSVs.Using a new multi-country euro area model with boundedly estimated rationality we show that this approach is compatible with the same limited information assumption that was used in deriving and estimating the behavioural equations of different optimising agents. We find that there are strong differences in the adjustment path to the shocks to the economy when agents form expectations using our learning approach compared to expectations formed under the assumption of strong rationality. Furthermore, we find some variation in expansionary fiscal policy in periods of downturns compared to boom periods.     

On the stability of the constant relative risk aversion (CRRA) utility under high degrees of uncertainty

Growth models under uncertainty and constant relative risk aversion (CRRA) utility are fragile in explaining consumers’ choice, as equilibrium consumption is dependent on distributional assumptions. We show that, under semi-nonparametric distributions, general equilibrium models are stable, as the existence of expected utility is guaranteed.     

Measuring agents’ reaction to private and public information in games with strategic complementarities

In games with strategic complementarities, public information about the state of the world has a larger impact on equilibrium actions than private information of the same precision, because public signals are more informative about the likely behavior of others. We present an experiment in which agents’ optimal actions are a weighted average of the fundamental state and their expectations of other agents’ actions. We measure the responses to public and private signals. We find that, on average, subjects put a larger weight on the public signal. In line with theoretical predictions, as the relative weight of the coordination component in a player’s utility increases, players put more weight on the public signal when making their choices. However, the weight is smaller than in equilibrium, which indicates that subjects underestimate the information contained in public signals about other players’ beliefs.     

Rational expectations and M2 demand

This essay expands on existing studies of M2 money demand. It differs in that it applies a rational expectations approach to an adaptive expectation model. Unlike the adaptive expectations models, the author includes an explanatory variable for expectations of future inflation. The expectation variables used are: the actual inflation rate (t + 1) and the Livingston Survey from the Philadelphia Fed. By using the different measures of expectations the author is able to compare several adaptive expectations models that appear in the literature and the rational expectations models for fit and forecast ability. The empirical results are such that the importance of including the rational expectations variable is evident even though the overall fit of the equation is comparable to one of the existing adaptive expectations models.     

Mean-variance analysis in temporary equilibrium

In this paper we take the first few steps towards a new theory of portfolio choice in the spirit of conventional mean-variance analysis but without strong assumptions on preferences or the distributions for returns. In this model agents form beliefs about returns based on conjectures about finitely many moments. In temporary equilibrium all current markets clear and conjectures about moments are correct. We prove the existence of a steady-state sequence of temporary equilibria and identify conditions on the structure of beliefs that ensure that the steady-state temporary equilibrium beliefs are in some sense accurate and closely approximate rational expectations.     

Existence of equilibria in the overlapping generations model: the nontransitive case

Summary This paper investigates the existence of competitive equilibria in dynamic exchange models with countably many periods and countably many agents. At each period the commodity space can be finite or infinite dimensional. The preferences of agents are not assumed to be transitive or complete. A first equilibrium existence theorem is established under the classical assumption that there exists a finite set of non-negligible agents. In the particular case of an overlapping generations model, a second existence theorem allows simultaneously for finite-lived assets and infinite-lived assets and limits the previous assumption to infinite-lived assets. This theorem covers obviously the standard case of an overlapping generations model where the agents have no endowment outside their lifetime.     

Notes on optimal transportation

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

Bayesian–Walrasian equilibria: beyond the rational expectations equilibrium

In general rational expectations equilibrium (REE), as introduced in Radner (Econometrica 47:655–678, 1978) in an Arrow–Debreu–McKenzie setting with uncertainty, does not exist. Moreover, it fails to be fully Pareto optimal and incentive compatible and is also not implementable as a perfect Bayesian equilibrium of an extensive form game (Glycopantis et al. in Econ Theory 26:765–791, 2005). The lack of all the above properties is mainly due to the fact that the agents are supposed to predict the equilibrium market clearing price (as agent’s expected maximized utility is conditioned on the information that equilibrium prices reveal), which leads inevitably to the presumption that agents know all the primitives in the economy, i.e., random initial endowments, random utility functions and private information sets. To get around this problematic equilibrium notion, we introduce a new concept called Bayesian–Walrasian equilibrium (BWE) which has Bayesian features. In particular, agents try to predict the market-clearing prices using Bayesian updating and evaluate their consumption in terms of Bayesian price estimates, which are different for each individual. In this framework agents maximize expected utility conditioned on their own private information about the state of nature, subject to a Bayesian estimated budget constraint. Market clearing is not an intrinsic part of the definition of BWE. However, both in the case of perfect foresight and in the case of symmetric information BWE leads to a statewise market clearing; it then becomes an ex post Walrasian equilibrium allocation. This new BWE exists under standard assumptions, in contrast to the REE. In particular, we show that our new BWE exists in the well-known example in Kreps (J Econ Theory 14:32–43, 1977), where REE fails to exist. This work was done in the Spring of 2005, when EJB was a visiting professor at the University of Illinois.     

A Lagrangean approach to the limit theorem on the core of an economy

Conclusion It is shown that an allocation which remains in the core under replication gives rise to an utility allocation with the following property: The utility level of no class can be increased without decreasing the utility level of at least one other class even if the number of consumers is allowed to vary continuously. An examination of the first order conditions the corresponding maximum problem with constraints leads to the conclusion that the allocation can be sustained by a price vector as a competitive equilibrium. This is the limit theorem on the core, proved in this paper with a simple Lagrangean technique.The notion of Pareto efficiency with a variable number of consumers proves useful for other purposes as well. For an extensive discussion of that point the reader is referred to Schweizer (1981).Ours, of course, is not the first calculus approach to the limit theorem [see e. g. L. Johansen (1978)]. But, as far as I know, the method of differentiating the Lagrangean simply with respect to the number of consumers is novel and leads to a straightforward proof of the theorem which can be reproduced in class-rooms even at the undergraduate level.The author wants to thank Joseph Greenberg and Werner Hildenbrand as well as an anonymous referee for helpful comments.     

The existence of budget-constrained pareto-efficient allocations

The existence of a budget constrained Pareto efficient allocation is proved. Preferences are assumed to be strongly monotonic and continuous pre-orderings. In addition, it is assumed that the set of Pareto efficient allocations, corresponding to a utility distribution, is a convex set. The mathematical tool is Brouwer's (or Kakutani's) fixed point theorem.     

Sufficient conditions for the existence of sunspot equilibria

This paper reports on conditions on agents' preferences and endowments sufficient to guarantee the existence of sunspot equilibria in a simple overlapping generations model of pure exchange. Sunspot equilibria are those in which uncertainty extrinsic to the economy operates through expectations to yield a fulfilled expectations competitive equilibrium in which the extrinsic randomness has real effects on prices and allocations. The paper also provides necessary and sufficient conditions for these equilibria to have agents trading in a fixed stock of valued fiat money. The condition derived can be interpreted as requiring that intertemporal income effects appropriately dominate substitution effects.     

Saddlepath learning

Saddlepath learning occurs when agents learn adaptively using a perceived law of motion that has the same form as the saddlepath relationship in rational expectations equilibrium. Under saddlepath learning, we obtain a completely general relationship between determinacy and e-stability, and generalise minimum state variable results previously derived only under full information. When the system is determinate, we show that a learning process based on the saddlepath is always e-stable. When the system is indeterminate, we find there is a unique MSV solution that is iteratively e-stable. However, in this case there is a sunspot solution that is learnable as well. We conclude by demonstrating that our results hold for any information set.     

Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

Does the Lucas critique apply during hyperinflation?: empirical evidence from four hyperinflationary episodes

Farmer (Review of Economic Studies, 58, 321–32, 1991) suggests that in a model in which there are multiple rational expectations equilibria agents may find it useful to coordinate their expectations in a unique rational expectations equilibrium which is supported by a self-fulfilling forecast rule having the property of being immune to the Lucas Critique. In this paper, Farmer's hypothesis is tested using data from hyperinflationary episodes. It is believed that those episodes are suitable for testing this hypothesis because an agent who lives in a hyperinflationary environment usually faces frequent changes in policy regime. The agent may thus choose a selffulfilling forecast rule which is immune to the Lucas Critique as a way of hedging against unanticipated policy regime switches. The empirical results show mixed evidence on Farmer's hypothesis during the hyperinflationary episodes studied.