Excess demand functions, equilibrium prices, and existence of equilibrium

Summary. For continuous aggregate excess demand functions of economies, the existing literature (e.g. Sonnenschein (1972, 1973), Mantel (1974), Debreu (1974), Mas-Colell (1977), etc.) achieves a complete characterization only when the functions are defined on special subsets of positive prices. In this paper, we allow the functions to be defined on a larger class of price sets, (allowing, for example the closed unit simplex, including its boundary). Besides characterizing excess demands for a larger class of economies, our extension provides a useful tool for proving other results. It allows us to characterize the equilibrium price set for a larger class of economies. It also permits extending Uzawa's observation (1962), by showing that Brouwer's Fixed-Point Theorem is implied by the Arrow-Debreu Equilibrium Existence Theorem (1954, Theorem I). Received: October 18, 1995; revised version June 28, 1996     

石油与黄金产业价格联动关系研究

2002-2006年石油与黄金产业价格变动之间的关系可以用格兰杰因果关系进行检验,并根据格兰杰表示定理建立石油与黄金价格联动之间的误差修正模型来考察二者之间的长期关系和动态关系.本文认为石油价格和黄金价格上涨之间是单向的,从石油价格上涨到黄金价格上涨的因果关系,而且这种关系是长期稳定的,并没有随时间而发生结构性变化.     

Recent Challenges to the Classical Gains–from–Trade Proposition

The classical gains–from–trade conjecture was finally given a thorough Arrow–Debreu proof in 1972. Since then, the proposition has been repeatedly challenged. Some of the challenges rest on grounds which in no respect violate Arrow–Debreu assumptions. Those challenges are therefore without foundation. The remaining challenges rest on the assertion that some of the Arrow–Debreu assumptions are unnatural and unnecessarily restrictive or on the imposition of artificial restrictions on the set of permissible compensatory transfers. We conclude that none of the challenges can be sustained.     

Evenly rotating economy: A new modeling technique for an old equilibrium construct

In the middle of the twentieth century, just five years before Arrow and Debreu proved the existence of an equilibrium for a competitive economy in the Walrasian system, Ludwig von Mises introduced the English-speaking world to his alternative equilibrium construct: the evenly rotating economy. In contrast to Arrow and Debreu, which characterizes equilibrium as a unique vector of prices and quantities, Mises depicts equilibrium as a pattern of behavior. After reviewing the Misesian conception of equilibrium and its failure to take hold in the profession, I turn to the modern literature. I contend that the evenly rotating economy is a special case of the now-prevalent class of search-theoretic exchange models. As such, I argue that this class of models is particularly well suited for applications considered by economists working in the Austrian tradition.     

Limited arbitrage is necessary and sufficient for the existence of a competitive equilibrium with or without short sales

Summary A condition oflimited arbitrage is defined on the endowments and the preferences of the traders in an Arrow-Debreu economy. Theorem 1 establishes thatlimited arbitrage is necessary and sufficient for the existence of a competitive equilibrium in markets with or without short sales. Limited arbitrage bounds utility arbitrages, the diversity of the traders in the economy, and the gains from trade which they can afford from initial endowments (Proposition 2); it is related to but nonetheless different from the no-arbitrage condition used in finance. Theorem 2 establishes that an Arrow — Debreu economy has a competitive equilibrium if and only if every one of its subeconomies withN + 1 traders does, whereN is the number of commodities. Limited arbitrage has been shown elsewhere to be equivalent to the existence of the core [16], to the contractibility of spaces of preferences and to the existence of continuous anonymous social choice rules which respect unanimity [10], [14], [15], [16].This paper was circulated in December 1991 as a Working Paper of the Department of Economics, Columbia University, New York, and presented at seminars at Mathematics, Economics, and Operations Research Departments at Columbia, Harvard, Stanford University of California at Berkeley, University of Bonn and the University of Siena, at an invited presentation at the European Congress of Mathematicians, July 1992, and the Winter Meetings of the Econometric Society in Boston, January 1994. Valuable comments and suggestions from Roko Aliprantis, Masahiko Aoki, Kenneth Arrow, Duncan Foley, Geoffrey Heal, Lionel McKenzie, Paul Milgrom and two anonymous referees, and research support from NSF Grant No. 92-16028 and the Stanford Institute for Theoretical Economics are gratefully acknowledged.     

Core convergence with asymmetric information

We analyze the ex ante incentive compatible core for replicated private information economies. We show that any allocation in the core when the economy is replicated sufficiently often is approximately Walrasian for the associated Arrow–Debreu economy.     

The measurement of waste with nonconvex technology

The measurement of waste problem is reconsidered with the generalization that nonconvex production technologies are allowed. A general equilibrium measure of waste allowing for nonconvexity in production is derived by extending the contributions by Debreu (1954) and Diewert (1981, 1985).     

Utility theory and continuous monotonic functions

Summary We examine the continuity of maps between linearly ordered spaces and apply the results to utility functions representing preference relations. In particular, we show that the continuous function constructed by Debreu is, in essence, unique.     

Arrow-Debreu equilibria with asymptotically heterogeneous expectations exist

Summary. In infinite horizon economies only local equivalence of beliefs is needed to ensure the existence of an Arrow–Debreu equilibrium. In fact, agents can even disagree completely in the long run in the sense that asymptotically, their beliefs are singular. Received: November 3, 2000; revised version: February 13, 2002     

Utility representations for partial orders

The main result of this paper is that for a continuous partial order on a second countable space there exist both an upper- and a lower-semicontinuous utility representation (but in general not jointly). This is the counterpart of the classical result of Debreu for complete preorders. The proof is based on a probabilistic approach which generalizes a theorem due to Neuefeind.     

The Arrow-Debreu model: How math can hide a fatal conceptual error

I argue that math, like love, can cover a multitude of sins, and I use the neoclassical object of adoration, the Arrow-Debreu model, as the case in point. It is commonplace that the Arrow-Debreu (AD) model of general equilibrium does not describe the real world, but it is equally commonplace to accept it as representing the pure logic of the competitive capitalist economy in an idealized world free of transactions costs. I show that the AD model fails even as an idealized model; it actually mistakes the logic of pure capitalism. Unlike McKenzie’s model of idealized general equilibrium under constant returns to scale, Arrow and Debreu claim to have shown the existence of competitive equilibrium under decreasing returns to scale and positive pure profits. The AD model (again unlike the McKinzie model) needs to assign the profits to individuals and this is done using the notion of “ownership of the production set.” But this notion suffers from a fatal ambiguity. If Arrow and Debreu interpret it to mean “ownership of a corporation” then a simple argument in the form “labor can hire capital or capital can hire labor” defeats the alleged necessity of assigning residual claimancy to the corporation. A given corporation may or may not end up exploiting a set of production opportunities (represented by a production set) depending on whether it hires in labor and undertakes production or hires out its capital to others (all by assumption at the parametrically given prices). In the latter case, residual claimancy is elsewhere. There is no such property right as “ownership of a production set” in a private property market economy. The legal party which purchases or already owns all the inputs used up in production has the defensible legal claim on the outputs: there is no need to also “purchase the production set.” At any set of prices that allow positive pure profits, anyone in the idealized AD model could bid up the price of the inputs and thus try to reap a smaller but still positive profit. Therefore,pace Arrow and Debreu, there could be no equilibrium with positive pure profits. In the Appendix, the property rights fallacy that afflicts the AD model is shown to also afflict orthodox capital theory and corporate finance theory.     

Mathematical Needs and Economic Interpretations

This paper discusses why the disconnection between formal structuresand theoretical content in economic model building can leadto undesirable consequences. By contrast, linking formal andverbal contents by means of coherent and relevant interpretationsis worth the effort: it is a relatively simple way of helpingto improve mathematical economic theorization. By way of illustration,the interpretation that Arrow and Debreu put on the inclusionof free goods in their proof of existence of general equilibriumis also discussed.     

Comparative statics for state-contingent technologies

The implications of supermodularity conditions in comparative-static analysis are analyzed for a generalized version of the separable-effort representation of a firm facing stochastic prices and a stochastic technology. Previous analysis is generalized in two ways. General risk-averse, as opposed to expected-utility, preferences are considered. The stochastic technology is represented by an Arrow–Debreu state-space representation. It is shown that results familiar from the theory of the price-taking firm in the absence of risk generalize to the uncertain case.      

Wealth Inequality and Asset Pricing

In an Arrow–Debreu exchange economy with identical agents except for their initial endowment, we examine how wealth inequality affects the equilibrium level of the equity premium and the risk-free rate. We first show that wealth inequality raises the equity premium if and only if the inverse of absolute risk aversion is concave in wealth. We then show that the equilibrium risk-free rate is reduced by wealth inequality if the inverse of the coefficient of absolute prudence is concave. We also prove that the combination of a small uninsurable background risk with wealth inequality biases asset pricing towards a larger equity premium and a smaller risk-free rate.     

Pricing rules and Arrow–Debreu ambiguous valuation

This paper considers pricing rules of single-period securities markets with finitely many states. Our main result characterizes those pricing rules C that are super-replication prices of a frictionless and arbitrage-free incomplete asset structure with a bond. This characterization relies on the equivalence between the sets of frictionless securities and securities priced by C. The former captures securities without bid-ask spreads, while the second captures the class of securities where, if some of its delivers is replaced by a higher payoff, then the resulting security is characterized by a higher value priced by C. We also analyze the special case of pricing rules associated with securities markets admitting a structure of basic assets paying one in some event and nothing otherwise. In this case, we show that the pricing rule can be characterized in terms of capacities. This Arrow–Debreu ambiguous state price can be viewed as a generalization for incomplete markets of Arrow–Debreu state price valuation. Also, some interesting cases are given by pricing rules determined by an integral w.r.t. a risk-neutral capacity. For instance, incomplete markets of Arrow securities and a bond are revealed by a Choquet integral w.r.t. a special risk-neutral capacity.     

Existence of competitive equilibrium when incomes may be zero

Summary In a pure exchange economy, there exists a price vector which is a quasi-equilibrium (Debreu 1962), but this may not be a competitive equilibrium if some individuals' demand functions are discontinuous because their incomes may be zero. We show nonetheless, in a pure exchange economy with free disposal, that there is asequence of prices approaching the quasi-equilibrium along which total excess demands tend to a non-positive limit.Comments on earlier drafts from K. J. Arrow, D. Duffie, H. E. Scarf, and D. Schmeidler are gratefully acknowledged.     

Least convex capacities

Summary. Debreu proposed the notion of `least concave utility' as a way to disentangle risk attitudes from the certainty preferences embedded in a von-Neumann Morgenstern index. This paper studies preferences under uncertainty, as opposed to risk, and examines a corresponding decomposition of preference. The analysis is carried out within the Choquet expected utility model of preference and is centered on the notion of a least convex capacity. Received: May 7, 1997; revised version: November 5, 1997     

An elementary proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley Theorem

Summary This note provides an elementary short proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley (K-K-M-S) Theorem based on Brouwer's fixed point theorem. The usefulness of the K-K-M-S Theorem lies in the fact that it can be applied to prove directly Scarf's (1967) Theorem, i.e. any balanced game has a non-empty core. We also show that the K-K-M-S Theorem and the Gale-Nikaido-Debreu Theorem can be proved by the same arguments.We wish to thank Roko Aliprantis for useful comments.     

A simple proof of K-K-M-S theorem

Summary We give a simple proof of the K-K-M-S theorem based on the Kakutani fixed point theorem, the separation theorem for convex sets and the Berge maximum theorem.The author is grateful to Gerard Debreu for his advice and to Tatsuro Ichiishi, Michael Todd, Rajiv Vohra, Nicholas Yannelis, Lin Zhou and two anonymous referees for their comments. The original version of this note was written in the fall 1992 while the author was visiting the Department of Economics, University of California at Berkeley.     

Equilibrium for Overlapping Generations in Continuous Time

We prove the existence of general equilibrium for continuous-time overlapping-generations models. Previous theorems exclude all non-linear C.E.S. and von Neumann–Morgenstern preferences and exclude production. Our primitive assumptions are satisfied by such preferences and by all Markovian production technologies satisfying Bewley's assumptions for Arrow–Debreu models provided that storage is possible, at some finite rate of depreciation and some positive capacity. A non-existence example shows our Markovian and storage assumptions cannot be dropped.Journal of Economic LiteratureClassification Numbers: C62, D51, D90.