Robert M. Anderson

The core in perfectly competitive economies

Publisher Summary This chapter presents the results on the cores of perfectly competitive exchange economies, that is economies in which the endowment of each agent is negligible on the scale of the whole economy. In the contributions of Edgeworth, Debreu and Scarf, and Aumann, the conclusion is: the core (in Aumanns case) or the intersection of the cores of all replicas (in the other cases) coincides with the set of Walrasian equilibria. One of the key elements of the Debreu and Scarf argument, the equal treatment property that permitted one to collapse the cores of all the different replicas into the same space, does not generalize even to sequences with different numbers of traders of the various types. The strong statement that the core (in Aumanns continuum setting) or the intersection of the cores (in the Debreu and Scarf replica setting) coincides with the set of Walrasian equilibria is simply not true in the case of general sequences of finite economies. Weaker forms of convergence must be substituted. Convexity of preferences, which plays no role whatever in Aumanns theorem, is seen to make a crucial difference in the form ofconvergence in large finite economies. The type of convergence that holds depends greatly on the assumptions on the sequence of economies. The various possibilities can best be thought of as lying on four largely (but not completely) independent axes: the type of convergence of individual consumptions to demands, the equilibrium nature of the price at which the demands are calculated, the degree to which the convergence is uniform over individuals, and the rate at which convergence occurs.

Non-standard analysis with applications to economics

Publisher Summary This chapter presents an introduction to nonstandard analysis and surveys its applications in mathematical economics. Nonstandard analysis is a mathematical technique, which has been widely used in diverse areas in pure and applied mathematics, including probability theory, mathematical physics, and functional analysis. It is used to formalize most areas of modern mathematics, including real and complex analysis, measure theory, probability theory, functional analysis, and point set topology; algebra is less amenable to nonstandard treatments, but even there significant applications have been found. The primary goal is to provide a careful development of nonstandard methodology in sufficient detail to allow using it in diverse areas in mathematical economics. This requires a careful study of the nonstandard treatment of real analysis, measure theory, and topological spaces. To accommodate this extended treatment of methodology the survey of work to date using nonstandard methods in mathematical economics is briefly reviewed in the chapter.

A nonstandard characterization of weak convergence

Let X be any topological space, and C(X) the space of bounded continuous functions on X. We give a nonstandard characterization of weak convergence of a net of bounded linear functionals on C(X) to a tight Baire measure on X. This characterization applies whether or not the net or the individual functionals in the net are tight. Moreover, the characterization is expressed in terms of the values of an associated net of countably additive measures on all Baire sets of X; no distinguished family, such as the family of continuity sets of the limit, is involved. As a corollary, we obtain a new proof that a tight set of measures is relatively weakly compact.

“Almost” implies “near”

We formulate a formal language in which it is meaningful to say that an object almost satisfies a property. We then show that any object which almost satisfies a property is near an object which exactly satisfies the property. We show how this principle can be used to prove existence theorems. We give an example showing how one may strengthen the statement to give information about the relationship between the amount by which the object fails to satisfy the property and the distance to the nearest object which satisfies the property. Examples are given concerning commuting matrices, additive sequences, Brouwer fixed points, competitive equilibria, and differential

Core Theory with Strongly Convex Preferences

We consider economies with preferences drawn from a very general class of strongly convex preferences, closely related to the class of convex (but intransitive and incomplete) preferences for which Mas-Colell proved the existence of competitive equilibria [13]. We prove a strong core limit theorem for sequences of such economies with a mild assumption on endowments (the largest endowment is small compared to the total endowment) and a uniform convexity condition. The results extend corresponding results in Hildenbrands book [8]. The proof, which is based on our earlier result for economies with more general preferences [2], is elementary.

Approximate Equilibria with Bounds Independent of Preferences

We prove the existence of approximate equilibria in exchange economies, giving bounds on the excess demand in terms of the number of traders and norms of the endowments, but independent of the preferences.

Edgeworth's Conjecture with Infinitely Many Commodities

Equivalence of the core and the set of Walrasian allocations has long been taken as one of the basic tests of perfect competition. The present paper examines this basic test of perfect competition in economies with an infinite dimensional space of commodities and a large finite number of agents. In this context we cannot expect equality of the core and the set of Walrasian allocations; rather, as in the finite dimensional context, we look for theorems establishing core convergence (that is, approximate decentralization of core allocations in economies with a large finite number of agents). Previous work in this area has established that core convergence for replica economies and core equivalence for economies with a continuum of agents continue to be valid under assumptions much the same as those usual in the finite dimensional context. For general large finite economies, however, we present here a sequence of examples of the failure of core convergence. These examples point to a serious disconnection between replica economies and continuum economies on the one hand an general large finite economies on the other hand. We identify the source of this disconnection as the measurability requirements that are implicit in the continuum model, and which correspond to compactness requirements that have especially serious economic content in the infinite dimensional context. We also obtain positive results. When the commodity space is a Riesz space, we show that familiar assumptions lead to a kind of local core convergence; strong assumptions lead to global core convergence. In the differentiated commodities context, we obtain core convergence results that are quite parallel to known equivalence results for continuum economies. Our positive results depend on infinite dimensional versions of the Shapley-Folkman theorem.

Rational Expectations Equilibrium with Econometric Models

We prove the existence of general economic equilibrium under uncertainty when agents form econometric models of the relationship among their private information, prices, and the state of the environment. The functional form of each agents model is specified in advance, with a finite number of parameters to be determined. Agents are then thought of as performing linear least squares estimation of the parameters. Equilibrium requires not only that markets clear, but also that each agent be using the vector of parameter values which, within a compact convex set of parameters, gives the least squares best fit to the data that is generated by the working of the economy when agents adhere to their models.

Gap-minimizing prices and quadratic core convergence

Abstract We study the decentralization of core allocations by price vectors chosen, essentially, to minimize the competitive gap. In the setting of Debreu (1975) and Grodal (1975), the competitive gap with respect to these gap-minimizing prices goes to zero as the inverse of the square of the number of agents; in Debreus setting, the convergence is uniform over agents while in Grodals setting, we show convergence on average.

A Market Value Approach to Approximate Equilibria

We consider the market value of excess demand as a measure of disequilibrium. We show that, in a fixed exchange economy, there exist approximate equilibria whose measures of disequilibrium depend only on the endowments and not on the preferences. A related bound on the norm of excess demand, depending on the endowments and the approximate equilibrium price, is also obtained. We show the existence of allocations which are nearly competitive, as measured by the largest proportion of demand given up at the allocation by any trader. We use these results to obtain, for very general sequences of exchange economies, allocations giving all traders bundles close to norm to their demands. This result includes a O(1/n) rate of convergence in the case of uniformly bounded endowments.