Beth E Allen

Bertrand-edgeworth oligopoly in large markets

The relation between perfectly competitive and monopolistically competitive equilibria is analysed for a Bertrand-Edgeworth model of a single market in which capacity constrained firms choose prices as strategies. The market always has a Nash equilibrium in pure or mixed strategies. As the number of firms increases, the corresponding equilibria converge in distribution to a perfectly competitive price. This result provides a justification for perfect competition that is based on an explicit account of price formation. However, monopoly prices persist with a positive but vanishing probability. Regularity or well defined inverse demand functions are not required.

Some Stochastic Processes of Interdependent Demand and Technological Diffusion of an Innovation Exhibiting Externalities among Adopters

One frequently hears the opinion that as twentieth century society becomes more complex, technological change assumes more significance and occurs at a more r apid pace. Investment in researclh and development results in a myriad of potential innovations. Yet, as the frontiers of technology are pLshed outward, uncertainity about the usefulniess of innovations becomes an increasingly important factor in the decision whether or not to adopt the new innovation. Furthermore, the decisions of individual economic agents tend to become increasingly interrelated. Many recent innovations are such that whether others adopt the device influences the benefits of adoption which accrue to a given consumer or firm. Such externalities may be positive (for example, consider a nations decision to use the metric system, the wearing of a style by a fashion leader, a merchants decision to accept a certain credit card, or the purchase of a coinmunication device by someone with whom one desires to communicate) or negative (for instance, imitation in the production of a faddish consumption item for which the potential market is limited, or the puLrchase of a telephone by an obscene caller). Our goal is to analyze techn-ological diffLsioll with interdependencies when uncertainty and informational considerationis are an inherent part of the theoretical model. The uncertainty may be of two types: (a) technological uLncertainty the usefulness or quality of the innovation is at least partially unlknown, and (b) adoption uncertainity incomplete or inaccurate knowledge about who is usinlg the innovation. In our stochastic model, adoptioni is reversible the decision to use the innovation is not permanent, but rather, agents may change their minds. Agents are not able to learn about the characteristics (quality, profitability, usefulness, etc.) of the innovation. This assumption limits the applicability of

Neighboring information and distributions of agents' characteristics under uncertainty

Abstract This paper proposes a topological structure for information. Specifically, information is considered an arbitrary sub-σ-field of the σ-field (of measurable subsets of states of the world) which represents events. A complete metric is defined on the space of all equivalence classes of sub-σ-fields; its uniformity is the same for all probability measures (on the abstract measurable space of states of the world and events) which are uniformly absolutely continuous. With this topology, economic behavior depends continuously on the agents information; demand functions under uncertainty are jointly continuous in utilities, endowments, and information. Moreover, the value of information is a continuous function of the information sub-σ-field. Technical properties of the topology are also given.

Utility perturbations and the equilibrium price set

Abstract For pure exchange economies in which agents are described by a compact smooth manifold of smooth strictly monotonic and strictly concave utilities, it is shown that, at least generically, the equilibrium price set is a smooth manifold of the same dimension. Given any smooth selection from the equilibrium price manifold and any sufficiently close smooth function, the function is a selection from the equilibrium price correspondence for some manifold of economies close to the original one. In particular, the set of equilibria corresponding to any open neighborhood of an economy contains an open subset of the price simplex.

Differential information economies: Introduction

Since the Cournot-Nash and Arrow-Debreu-McKenzie contributions to equilibrium theory with complete information, two main advances in equilibrium theory with differential information appeared. The first one is by Harsanyi (1967), who introduces differential information into the Cournot-Nash model, and the second one is by Radner (1968), who introduces differential information into the Arrow-Debreu-McKenzie model. Those two papers generated a literature on Bayesian Cournot-Nash equilibrium and on Walrasian expectations equilibrium respectively. In the seventies and eighties there was a growing literature on Rational Expectations Equilibrium (REE), which is a natural extension of the Arrow-DebreuMcKenzie deterministic model of Walrasian equilibrium to a differential information framework. This literature on REE (a non-cooperative equilibrium concept) didn’t provide an adequate explanation as to how prices reveal the same information to agents who are differentially informed, or to put it differently, how agents who have different information obtain the same information from the equilibrium prices. In other words, prices don’t reflect the informational asymmetries of agents and this can be a major criticism of the REE concept under full revelation. A new literature emerges from two early seminal works on cooperative equilibrium concepts with differential information. The first one was by Wilson (1979), who considers the core of an economy with differential information,

A Stochastic Interactive Model for the Diffusion of Information

Two important generalizations of information diffusion models are the presence of stochastic effects and the possibility of arbitrary patterns of influence among individuals. A Markov random fields model includes both of these features. Under very weak assumptions, there is a unique equilibrium distribution of information patterns for given stochastic (local) interactions among a finite population. This has implications for policies to influence the transmission of information. The dynamic behavior of a special and simple case of the model tends to approximate the standard (logistic) diffusion curve. For an infinite population, uniqueness of equilibrium distributions may fail; some sufficient conditions to ensure uniqueness are given.

Smooth preferences and the approximate expected utility hypothesis

Abstract Mild smoothness conditions on continuous complete preorders over lotteries imply that various local versions of the expected utility hypothesis are satisfied, i.e., for small deviations, there is a utility (representing the individuals preferences) that is linear in probabilities. For these results, notions of smooth preferences over an infinite dimensional set of probability measures are developed.

The existence of rational expectations equilibrium: a retrospective

This paper provides a selective review of theoretical research on the consistency of rational expectations equilibrium and its properties in microeconomic models. The general equilibrium framework is emphasized throughout the paper. After defining rational expectations equilibrium for a pure exchange economy, the paper presents a simple counterexample to illustrate that rational expectations equilibria need not exist. Results are summarized for the generic existence of fully revealing rational expectations equilibria in smooth economies satisfying additional dimensionality assumptions. Then the rational expectations equilibrium existence problem is related to earlier analysis of informationally decentralized allocation mechanisms. Next the efficiency properties of rational expectations equilibrium allocations are examined. Finally, the possibilities for partially revealing rational expectations equilibria are discussed.

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

Abstract 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.

The demand for (Differentiated) information

A framework for distinguishing between the quantity of information and its quality or type is presented in which information is an indivisible differentiated commodity for which satiation occurs at one unit. Uncountably many types of information are possible which can be costlessly combined by agents. Similarity of information is expressed by a metric which reflects substitution possibilities among different information structures. In the model, traders desire information only because it helps them to maximize state dependent utilities under uncertainty. Then the individual demand for information is well defined, but possibly nonconvex valued because of the indivisibilities.