Aggregate Excess Demand Function Research Articles

An Algebraic Proof of the Existence of a Competitive Equilibrium in Exchange Economies

The standard excess demand argument for existence of competitive equilibria in exchange economies invokes maximizing the market value of the aggregate excess demand function and thereby adjusting the prices toward equilibrium. By exploiting the Perron-Frobenius theorem on stochastic matrices, we offer an algebraic proof of the existence of a competitive equilibrium without resorting to such a device of excess demand.

PRICE DYNAMICS IN GENERAL EQUILIBRIUM EXPERIMENTS

AbstractGeneral equilibrium (GE) theory faces several related challenges. Classical theories of out‐of‐equilibrium adjustment are intuitive but implausibly centralized. Further, standard restrictions on individual preferences place little structure on aggregate excess demand functions. Amongst other issues, this fact implies that economies can be unstable under classical dynamics, and thus instability appears to be an unavoidable feature of GE theory. This paper recounts laboratory studies of GE environments which reveal that classical theories of dynamics do, in fact, organize market activity surprisingly well. These experiments suggest that the Walrasian auctioneer, for example, appears to be a constructive abstraction, and that instability should be accommodated within theory rather than circumscribed. Laboratory applications of GE theory focused on issues beyond stability and dynamics are also briefly surveyed.

Material needs and aggregate demand

A central conclusion of the standard theory of consumption is that consumers’ preferences can be taken as theoretical primitives. Special categories of consumption, such as “basic needs”, or of goods, such as “subsistence goods” are seen as extra theoretical baggage that add few, if any, insights. This theoretical orientation has been absorbed into the theory of aggregate demand, but the aggregate theory has a serious problem that is not shared by the individual-level theory: no matter how well-behaved the individual-level demand functions may be, the aggregate-level function can take on almost any form. This result follows from the SMD theorem, named after Sonnenschein, Mantel, and Debreu, who developed the theory; Kirman and Koch strengthened the results, and the SMD–KK theorem poses a fundamental challenge to models linking micro and macroeconomics. A standard response to the aggregation problem is to introduce a representative agent, but this merely sidesteps the problem. We argue that the aggregation problem arises, in part, because of the exclusion of needs from the theory. Specifically, we argue that material needs—such as basic needs for energy, water, food, and shelter—must be included as theoretical primitives because both the needs and the satisfiers of those needs are universal. We construct a microeconomic model with material needs and show that the form of the aggregate excess demand function is not completely arbitrary, so the SMD–KK theorem does not apply. We discuss the implications of this result.

WHAT A DIFFERENCE A SUM (∑) MAKES: SUCCESS AND FAILURE IN THE RATIONALIZATION OF DEMAND

This paper discusses the Sonnenschein–Mantel–Debreu (SMD) theorems in general equilibrium theory. It argues that the SMD results were related to the previous literature on the integrability of demand. The integrability question involved rationalizing individual demand functions, and the SMD theorems asked the same question about aggregate (market) excess demand functions. The paper’s two goals are to demonstrate how the SMD results followed naturally from the earlier work on integrability, and to point out that the profession’s reception was quite different; the integrability results were considered a success story, while the SMD results were quite negative.

What a Difference a Sum Makes: Agony and Ecstasy in the Rationalization of Demand

Most historical discussions of Arrow-Debreu general equilibrium theory discuss the negative impact of the so-called Sonnenschein-Mantel-Debreu (SMD) results on aggregate excess demand functions. This paper argues that the SMD results were a natural extension of the integrability literature of the period 1959-71 - something that has not been recognized in the historical literature - and considers some of the implications of this for the history of modern economic thought.

Direct perturbations of aggregate excess demand

We complement the Sonnenschein–Mantel–Debreu results by establishing that an exchange economy, i.e., preferences and endowments, that generates a given aggregate excess demand (AED) function is close to the economy that generates a perturbation of this AED. As a consequence, genericity and determinacy results obtained by direct perturbation of an AED are as strong as results obtained by perturbing preferences and endowments.

Excess demand function around critical prices in incomplete markets

We show that the aggregate excess demand function in an economy with incomplete real asset markets can be characterized by Walras’ law, homogeneity, and continuity around critical prices that cause one-dimensional drop of the dimension of the budget set.

Excess demand functions when new assets are introduced

We demonstrate that information on the aggregate excess demand function of an incomplete real asset market economy is not helpful toward predicting the equilibrium of a new economy to which unredundant assets are added. An innovation of assets can completely change the aggregate excess demand functions.

Thom-Boardman Stratification of Aggregate Excess Demand and Finiteness of Equilibria

The paper establishes that the Thom-Boardman stratification result of singularity theory applies generically to the world of aggregate excess demand (AED) functions. In addition, if AED is analytic, then the number of Walrasian equilibria - including non-regular ones - is finite.

Direct Perturbations of Aggregate Excess Demand

We complement the Sonnenschein-Mantel-Debreu results by establishing that an exchange economy, i.e., preferences and endowments, that generates a given aggregate excess demand (AED) function is close to the economy that generates a perturbation of this AED. As a consequence, genericity and determinacy results obtained by direct perturbation of an AED are as strong as results obtained by perturbing preferences and endowments.

Stabilizing Consumer Choice: The Role of 'True Dynamic' Stability and Related Concepts in the History of Consumer Choice Theory

It is often argued that the inability of Arrow-Debreu general theory to produce an adequate proof of the of the Walrasian price adjustment mechanism was one of the program's most significant failures (a failure that is directly related to the infamous Sonnenschein-Mantel-Debreu results on aggregate excess demand functions). This paper will not question this standard interpretation of the history of general theory, but makes the case that characterizing the stability question in terms of market - in particular the of the price vector in Walrasian general model - actually helped to stabilize the standard model of consumer choice in general theory and elsewhere within microeconomics. The problem of the of consumer's equilibrium was much discussed early in the twentieth century and it has recently re-emerged in a different guise as the endowment effects and path dependencies of contemporary experimental and behavioral economics, and yet it disappeared from mainstream discussion during the period 1950-1980. This paper argues that shifting the discussion from the intra-agent of the individual consumer to the inter-agent of the competitive market contributed - despite its ultimately negative impact on general theory - to the long period of stable normal science consumer choice theory enjoyed during the middle of the 20th century. The paper ends by considering some of the implications of this interpretation of the history of the literature.

The Sonnenschein-Mantel-Debreu Results after Thirty Years

The properties of aggregate excess demand or market demand play a key role in the existence, uniqueness, stability, and comparative statics of competitive general equilibrium. They are also important for the theory of imperfectly competitive general equilibrium and the microfoundations of macroeconomics. Because observations of demand are likely to occur at the aggregate level, whereas economic theory has been largely concerned with individual behavior, the properties of aggregate excess or market demand are central to econometric identification. The importance of these projects for the development of general equilibrium theory can be seen in Kenneth Arrow and Frank Hahn’s General Competitive Analysis (1971). Soon after their book was published, though, Hugo Sonnenschein (1973) posed the following problem. Consider a function that maps prices into quantities. What conditions must this function satisfy if it is to be the aggregate excess demand function of a well-behaved exchange economy? The existence theory highlighted the importance of the function’s satisfying continuity, homogeneity of degree zero, and Walras’s law. Were these all of the restrictions on aggregate excess demand implied by the usual

An Analysis of General Equilibrium with Positive Transaction Costs and Non-Convex Preference

After taking into account positive transaction costs via tackling static nonlinear programming with dynamic programming, this paper reaches three main conclusions: 1) The aggregate excess demand function of each market will be discontinuous at many price points, but the transfiguration of the graph of aggregate excess demand of each commodity will be continuous with respect to prices of other commodities, and at those price points incurring discontinuity, the corresponding traders' utility or profit will be indifferent between different choices. 2) Via a new approach able to tackle the discontinuous aggregate excess demand, it is proved that, there must exist a market equilibrium price vector in a perfectly competitive economy with positive transaction costs even if each consumer's preference isn't convex. 3) Transaction cost probably spoils the satisfaction of those two fundamental theorems of welfare economics, as well as it ruins the social aggregate welfare through three approaches.

Characterizations of pseudomonotone maps and economic equilibrium

The paper surveys some recent contributions to the analysis of pseudomonotone maps and their application to economics. It is shown that the concept of pseudomonotonicity is strongly related to a notion of rationality of consumer behaviour which is well known in economics as a weak version of the Weak Axiom of Revealed Preference. In economic models of general equilibrium, a pseudomonotone aggregate excess demand function is seen to guarantee convexity of the set of price equilibria and, in the case of so-called regular economies, even the uniqueness of equilibrium. Some general characterizations of differentiable pseudomonotone maps, which reduce to necessary and sufficient first order conditions under the assumption of regularity, are provided. These conditions are employed in order to prove pseudomonotonicity of aggregate excess demand in economies displaying a particular kind of heterogeneous consumer behaviour.

A note on the decomposition (at a point) of aggregate excess demand on the Grassmannian

This paper analyzes the properties of aggregate excess demand functions for economies with an arbitrary finite set of N commodities where agents face trading restrictions of a general, abstract form: their budget set is defined by K-dimensional planes in R N . It is shown that, if there are at least K agents in the economy, the only general property satisfied by the value of aggregate excess demand and its derivative, at any arbitrary point, is Walras Law. The result is established by considering an economy where agents' preferences are of a `generalized Leontief' type.

Disaggregation of excess demand functions in incomplete markets

AbstractWe are interested in general equilibrium incomplete markets, where the number ofconsumers is N, the number of goods is L, and the dimension of the space of admissibletrades is K..the case of complete markets being then s Ly1 . We prove that, ifNGK, any non-vanishing analytic function satisfying the natural extension of the Walraslaw is, locally at least, the excess demand function of such a market. To be precise,consider a map u“Fu.associating with a T-dimensional parameteru a K-dimensionallinear subspace Fu.of R L , representing the set of market transactions allowed by u.Given parameter values u 1 , ...,u T , and a non-vanishing analytic function Z defined onsome neighbourhood of u with values in R L , with X. .ugFu; , then there existconcave utility functions U n ,1FnFN and individual endowments v 1 ,...,v N , such thatthe corresponding aggregate excess demand function coincides with Z on a possiblysmaller neighbourhood of. u.IfZ vanishes at u, the disaggregation is still possible, butrequires .Kq1 agents. q1999 Elsevier Science S.A. All rights reserved.

Characterizing C 2 aggregate excess demand functions

We integrate and sharpen two characterizations of aggregate excess demand functions: we obtain Mas-Colell's (1977) equilibrium invariance, and strengthen Geanakoplos' (1984) weakly concave utility functions to strictly concave ones. Our proofs modify and extend Geanakoplos' utility-construction. We note two applications: a sharper characterization of equilibrium price sets (cf. Mas-Colell (1977)), and a basis for the studies of computable economies (Richter and Wong (1996)).

Conditions for the Existence of Market Equilibrium

most remarkable achievements of modern microeconomic theory are the proof of the existence of an equilibrium and the First and Second Theorems of Welfare Economics. . . . Understandably, therefore, much attention has been devoted to various interpretations, alternative proofs, and extensions of these basic results (Luenberger 1994, 147). In an earlier article (Bryant 1994), I discussed a common misinterpretation of the second theorem of welfare economics. This article complements that one in that I discuss here a defect in the standard graduate textbook treatment of the existence problem for market equilibrium. In particular, I draw attention to assumptions that appear, one way or another, in all known existence theorems and that play a central role in those theorems by ensuring that an economy's aggregate excess demand function never behaves discontinuously. Although these assumptions are crucial in establishing the existence of market equilibrium and often turn out to be quite restrictive, their role is not generally discussed in the literature and is almost never acknowledged in the major graduate textbooks in microeconomics. As a consequence, graduate students are often given little insight into the potentially limited range of circumstances under which market equilibrium can be expected to exist. The impression (in some cases, the explicit instruction) may be that existence theorems for market equilibrium rest on quite general foundations when the opposite is arguably the case.

The Intrinsic Limits of Modern Economic Theory: The Emperor has No Clothes

Assumptions of the uniqueness and stability of general equilibrium in a Walrasian framework have no theoretical justification. This paper argues that the key reason for this is that the Walrasian model treats people as acting independently of one another, especially in their demand behavior. This independence plays an essential role in constructing economies with arbitrary excess demand functions. Thus, making individual behavior dependent or similar may open the way to obtaining meaningful restrictions on aggregate excess demand functions. Copyright 1989 by Royal Economic Society.

Substitutability and the welfare effects of endowment increases

Mas-Colell has conjectured that if all commodities are normal goods for a certain consumer, and if the aggregate excess demand function of the other consumers satisfies the condition of gross substitutability, then an increase in his initial endowments will lead to a new equilibrium allocation which will be preferred by him. This excludes the ‘immiserizing growth’ paradox. It is shown that a sufficient condition is that commodities be also net substitutes for the consumer that has been singled out.