Linear exchange economies with a continuum of agents

The purpose of this paper is to study how the equilibrium prices vary with respect to the initial endowments in a linear exchange economy with a continuum of agents. We first state the model and give conditions of an increasing strength for existence, uniqueness and continuity of equilibrium prices. Then, if we restrict ourselves to economies with essentially bounded initial endowments and if we assume that there is, from the point of view of preferences, only a finite number of types of agents, we show that, on an open dense subset of the space of initial endowments, the equilibrium price vector is an infinitely differentiable function of the initial endowments. The proof of this claim is based on a formula allowing to compute the equilibrium price vector around a so-called “regular” endowment where it is known.     

Observable restrictions of general equilibrium models with financial markets

This paper examines whether general equilibrium models of exchange economies with incomplete financial markets impose restrictions on prices of commodities and assets given the stochastic processes of dividends and aggregate endowments. We show that the assumption of time-separable expected utility implies restriction on the cross-section of asset prices as well as on spot commodity prices. However, a relaxation of the assumption of time separability will generally destroy these restriction.     

The testable implications of competitive equilibrium in economies with externalities

Suppose that one has a data set consisting of prices and individual endowments for some economy. Brown and Matzkin (Econometrica 64:1249–1262, 1996) have shown that there are conditions that the data have to satisfy, if the observed prices are determined by the competitive equilibrium process, given the observed endowments, when there are no external effects in the economy’s interactions. The results here show that the same conclusion does not apply, in general, if the economy exhibits externalities. On the other hand: (i) some restrictions exist if there exist at least two commodities on which the individuals’ preferences are weakly separable; (ii) although extremely mild, restrictions exist too if one observed individual consumption for the economy that causes the external effects; and (iii) importantly, even if the previous two cases do not apply, restrictions exist when the externalities that exist are in the form of a public good.     

The determinacy of equilibrium in economies of overlapping generations

We study the problem of uniqueness of equilibrium paths in the overlapping generations model. We show that, despite local calculation based on counting equations and unknowns, the equilibrium path may be unique. We do this by constucting an example of an economy of overlapping generations with just one equilibrium up to time shift, beside the steady states. Time is either discrete or continuous; in either case, it extend into the infinite future and, possibly, the infinite past. There is one, non-storable commodity at each date. The economy is stationary; intertemporal preferences are logarithmic; the endowments and discount factors of individuals need not depend continuously on time. With continuous time, equilibrium paths of prices are smooth; this, even for endowments and discount factors of individuals that do not depend continuously on time. With discrete time, as the number of periods in the life-span of individuals increases, equilibrium paths converge to the continuous time solutions. If time extends infinitely into the infinite past as well as into the infinite future, in continuous time, all non-stationary equilibrium paths of prices are time-shifts of a single path; in addition, there are two stationary solutions; in discrete time, there is a one dimensional family of non-stationary solutions, up to time-shift; however, the indeterminacy vanishes as the number of periods in the life span of individuals tends to infinity. If, alternatively, time has a finite starting point, in discrete time the degree of indeterminacy increases with the life-span of individuals, and, in continuous time, it is infinite; however, these are families of exponentially decreasing oscillations which, although they may exhibit pseudo-chaotic behaviour for a while, as time tends to infinity they all get damped, and asymptotic behaviour is that of the economy that originates in the infinite past. Stefano Lovo made interesting comments.     

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     

Taste-homogeneity of optimal jurisdictions in a Tiebout economy with crowding types and endogenous educational investment choices

We examine a local public goods economy with differentiated crowding. The main innovation is that we assume that the crowding effects of agents are a result of choices that agents make. For example, agents may be crowded (positively or negatively) by the skills that other members of their jurisdiction possess and these skills may be acquired through utility maximizing educational investment choices made in response to equilibrium wages and educational costs. In such an environment, we show that taste-homogeneous jurisdictions are optimal. This contrasts with results for both the standard differentiated crowding model and the crowding types model. We also show that the core and equilibrium are equivalent, and that decentralization is possible through anonymous prices having a structure similar to cost–share equilibrium prices.     

A characterization of the demand for land

In a model of an economy where land is modelled explicitly as heterogeneous subsets of the plane and utility takes an integral form, demand can be characterized as a specific map from endowments and prices into subsets of the plane. A modified form of the Neyman-Pearson lemma is used to obtain this result, the modification being necessary because the analog of randomized tests must be excluded from this model.     

Factor Substitution and Relative Factor Prices

This paper examines the effects of factor endowments on factor prices in a three‐factor, two‐commodity general‐equilibrium model with endogenous commodity demand and prices. Unlike the conventional small open‐economy model that assumes constant commodity prices, factor substitution influences the direction of these effects. When a factor endowment increases, complementarity with the expanding factor benefits an unchanged factor, but substitutability harms it. If the unchanged factors are complements, there is a possibility of a rise in the expanding factor's price. A comparison of this closed‐economy model with the small open‐economy model reveals the role of international trade, which dampens the effect on the expanding factor's price.     

Market equilibrium with heterogeneous recursive-utility-maximizing agents

Summary This paper considers a heterogeneous agent Lucas style exchange economy. For a class of recursive utility functions containing the standard additive expected utility functions, I demonstrate that there exist market equilibria characterized by stationary (ergodic) Markov processes for consumption, portfolio holdings, asset prices and the unobserved utilities. No assumptions about market completeness are made, and there are no restrictions on the underlying information filtration.Other contributions of this paper include: (i) an existence and uniqueness theorem of intertemporal utility for the general class of recursive generators; (ii) the optimum principle as well as its corresponding Euler equation derived for the agent's consumption and portfolio choice problem under recursive utility, and (iii) a single-agent equilibrium asset pricing formula which generalizes that of Epstein and Zin (1989).This paper is a part of my PhD dissertation at the University of Toronto. I would like to thank Larry Epstein for his enthusiastic supervision, helpful discussion and valuable comments. Thanks also to Tan Wang and especially Darrell Duffie for valuable comments.     

Ambiguity,uncertainty aversion and equilibrium welfare

Summary. In order to analyse the effect of ambiguity and uncertainty aversion on equilibrium welfare, a two period, pure exchange one good economy is considered. Agents are Choquet-expected-utility maximizers with same convex capacity and strictly concave utility index. It is proven that equilibrium is indeterminate whenever several probabilities in the core of the capacity minimize the expected value of aggregate endowment and not all agents have same expected endowments under those probabilities. It is further shown that small changes in aggregate endowment may have drastic welfare implications. A more general model is considered in the case of no aggregate uncertainty: agents have a set of priors and are uncertainty averse as modelled by Gilboa-Schmeidler [1989]. In the case of complete markets, it is shown that assets have a spread of equilibrium prices similar to the spread of no-arbitrage prices compatible with absence of arbitrage in markets with imperfections.Received: 2 June 2000, Revised: 27 March 2003, JEL Classification Numbers: D46, D59,D60, G12.I have benefited from conversations with L. Epstein, F. Magnien and J. M. Tallon.     

Incomplete financial markets and differential information

We provide a model of an incomplete markets economy where private restrictions on consumption are interpreted as lack of information. We prove existence of an equilibrium where agents are unable to infer any additional information from prices. When assets are nominal, these non-enlightening equilibrium prices result in a reduction of the degree of real indeterminacy.     

Consumption-loan type models the Samuelson case

This is a further study of the overlapping generations model with fixed amount of fiat money. We show that when the economy is Samuelsonian then there exists some Pareto optimal stationary competitive equilibrium from some initial money holdings. Moreover, if the steady-state interest factor is less than unity, then a non-monetary Pareto optimal equilibrium exists from any initial money endowments.     

General equilibrium and the new neoclassical synthesis

We present a general equilibrium model of the new neoclassical synthesis that has the same level of generality as the Arrow–Debreu model. This involves a stochastic multi-period economy with a monetary sector and sticky commodity prices. We formulate the notion of a sticky price equilibrium where all agents form rational expectations on prices for commodities and assets, interest rates, and rationing. We present a general result showing that monetary policy imposes no restrictions whatsoever on nominal equilibrium price levels and that the set of sticky price equilibria has a dimension equal to the number of terminal date-events. Stickiness of prices implies that this indeterminacy is real.     

Construction of a state space for interrelated securities with an application to temporary equilibrium theory

Summary We construct an endogenous state space in an exchange economy with possibly infinite horizon. Every period agents trade securities whose payoffs depend on future dividends and asset prices. We reject the perfect foresight assumption on the ground that agents have not only limited knowledge of other individuals' endowments and preferences, but also limited capacity to compute equilibria. We choose instead absence of arbitrage as the principle which allows agents to determine if a system of future prices is possible. We give an alogrithm to compute the set of nonarbitrage prices every period, with both finite and infinite horizon. We then apply this endogenous structure of uncertainty to an infinite horizon temporary equilibrium model.I would like to thank Professor Donald Brown for his constant help and guidance. I have also greatly benefited from helpful discussions with Professors Jacques Drèze, Bernard Dumas, Mordecai Kurz, Carsten Nielsen, Jan Werner, and Ho-Mou Wu.     

The Identification of Preferences from Equilibrium Prices under Uncertainty

The competitive equilibrium correspondence, which associates equilibrium prices of commodities and assets with allocations of endowments, identifies the preferences and beliefs of individuals under uncertainty; this is the case even if the asset market is incomplete. Journal of Economic Literature Classification Numbers: D52, D80.     

Competitive equilibria in semi-algebraic economies

This paper develops a method to compute the equilibrium correspondence for exchange economies with semi-algebraic preferences. Given a class of semi-algebraic exchange economies parameterized by individual endowments and possibly other exogenous variables such as preference parameters or asset payoffs, there exists a semi-algebraic correspondence that maps parameters to positive numbers such that for generic parameters each competitive equilibrium can be associated with an element of the correspondence and each endogenous variable (i.e. prices and consumptions) is a rational function of that value of the correspondence and the parameters.This correspondence can be characterized as zeros of a univariate polynomial equation that satisfy additional polynomial inequalities. This polynomial as well as the rational functions that determine equilibrium can be computed using versions of Buchberger's algorithm which is part of most computer algebra systems. The computation is exact whenever the input data (i.e. preference parameters etc.) are rational. Therefore, the result provides theoretical foundations for a systematic analysis of multiplicity in applied general equilibrium.     

A Simple Model of Privately Profitable But Socially Useless Speculation

The paper presents a general equilibrium model of a pure exchange economy with stochastic endowments, in which speculation in the forward market is profitable and stabilizes prices but is useless from a welfare point of view. Reconciling the Siegel paradox with the theory of incomplete markets, we show that banning speculation by closing the forward market may increase social welfare. We also show that the addition of a market might reduce the gains from international trade for all participating countries.
JEL Classification Numbers: F10, F11, F19.     

Rational expectations in the overlapping generations model

This paper reports results on the character of the rational expectations equilibria of a stochastic overlapping generations model with heterogenous markets. The model considered is a stationary overlapping generations model in which the endowments of young agents are subject to i.i.d. random shocks. The main result shown is that if there are l > 1 commodities traded in every period, then for most preferences, the rational expectations equilibrium stochastic process of prices and allocations necessarily exhibits serial correlation. This is in marked contrast to the one commodity model in which there always exists an equilibrium which is measure isomorphic to the endowment process.     

Dynamic effects of changes to Japanese immigration policy

This paper uses a single-sector dynamic stochastic general equilibrium model with heterogeneous households to analyse Japanese immigration policy. We examine the effects on output, consumption, factor prices, and utility. We do this for both steady states and transition paths. We find that: (a) aggregate output, investment, and consumption in Japan are likely to rise with any sort of loosening of immigration restrictions; (b) allowing more skilled immigration generates greater aggregate changes; (c) raising skilled immigration relative to unskilled immigration drives down skilled workers’ wages, consumption, and utility, while cutting the skilled to unskilled immigration share has the opposite effects; and (d) such immigration policy changes have small effects compared to those that occur naturally due to business cycle fluctuations     

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.