Wayne Shafer

Equilibrium in abstract economies without ordered preferences

Abstract We extend a theorem of G. Debreu on the existence of equilibrium in a generalized N -person game (‘A Social Equilibrium Existence Theorem’). Applying standard techniques the result can be used to prove the existence of equilibrium in economies with interdependent preferences, price-dependent preferences and preferences which may be both non-transitive and non-complete. This generalizes a recent theorem of A. Mas-Colell.

Equilibrium in incomplete markets: I: A basic model of generic existence

Abstract This paper demonstrates the generic existence of general equilibria in incomplete markets. Our economy is a model of two periods, with uncertainty over the state of nature to be revealed in the second period. Securities are claims to commodity bundles in the second period that are contingent on the state of nature, and are insufficient in number to span all state contingent claims to value, regardless of the announced spot commodity prices. Under smooth preference assumptions, equilibria exist except for an exceptional set of endowments and securities, a closed set of measure zero. The paper includes partial results for fixed securities, showing the existence of equilibria except for an exceptional set of endowments.

Market demand and excess demand functions

Publisher Summary Market demand and market excess demand functions are defined by summing the preference-maximizing actions of consumers. These functions are central to most treatments of value theory. It discusses that the demand theory of the individual consumer has been well developed. Just as consumer demand theory is the study of properties common to demand functions that are generated by the preference-maximization of an individual consumer, market demand theory investigates properties that are shared by market demand functions. As market demand is often the observable and relevant variable for analysis, it is important to see how it is restricted by the utility hypothesis. When preferences are homothetic and the distribution of income is independent of prices, the market demand function has all the properties of a consumer demand function. The chapter discusses conditions under which market demand and market excess demand functions are consumer demand functions. It also provides an overview of Debreus theorem, which explains that for every continuous function F that satisfies homogeneity and Walras Law and for every compact set of positive prices K, there exists an economy with market excess demand function F on K.

Equilibrium in incomplete markets: II: Generic existence in stochastic economies

Abstract This paper demonstrates the generic existence of equilibria with incomplete markets in a stochastic economy. Uncertainty is represented by an event tree. Securities that are claims to bundles of commodities and units of account at future dates, contingent on states of the world, are sold at each date and state. There is an insufficient number of securities to complete markets, even with the spanning advantages of repeated trade through time. Preferences are ‘smooth’. We include partial results for linear restrictions on trading strategies, and show that security prices may be assigned to be the conditional expected sum of the future spot market values of dividends.

Some Results on Comparative Statics under Uncertainty

This paper presents results on comparative statics for a class of decision problems under uncertainty. Necessary and sufficient conditions are derived for parameter changes and stochastically dominant shifts in the return in the two-asset portfolio problem. These results give conditions for the demand for money to be inversely related to the rate of interest. Further applications include the qualitive behavior of aggregate savings under uncertainty and the production decision of a firm facing price uncertainty. Copyright 1987 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Equilibrium in economies without ordered preferences or free disposal

Abstract Recent work in competitive equilibrium analysis, pioneered by Mas-Colell (1974), ‘An equil- brium existence theorem without complete or transitive preferences’, has established existence of equilibrium in an economy in which preferences are not orderings. The purpose of this note is to show that the assumption of free disposal is not necessary for existence.

Ergodic fluctuations in deterministic economic models

Abstract Nonperiodic business fluctuations are shown to exist with positive measure generically and to behave like stationary stochastic processes in the standard dynamic macro model. The implication is that so called ‘chaotic’ fluctuations are probable for random initial conditions in a large class of recursive economies.

Solving systems of simultaneous equations in economics

We show that there is a broad range of systems of simultaneous equations that arise in economics as descriptions of equilibrium that can be solved in elementary fashion via degree theory. Some of these systems are not susceptible to analysis by standard Brouwer fixed point methods. Two of our applications are to general equilibrium with incomplete markets, and to Nonconvex production with noncompetitive pricing rules.

Characterisation of generically complete real asset structures

This paper compares the equilibrium allocations of a stochastic economy induced by two market structures: contingent markets and a sequential system of spot and real asset markets. We exhibit a regularity condition on the asset structure which we prove to be a necessary and sufficient condition for the equilibrium allocations for the two market structures to coincide generically.

Some theorems on the existence of competitive equilibrium

Abstract The purpose of this paper is to point out a relationship between theorems on the existence of competitive equilibrium in economies with externalities, and recent results (pioneered by A. Mas-Colell) on the existence of equilibrium for economies in which consumer preferences are neither complete nor transitive. This observation leads both to a substantial strengthening of the theorem on the existence of equilibrium with externalities, and at the same time to a revealing perspective on the Mas-Colell theorem.