Charalambos D. Aliprantis

Existence and optimality of competitive equilibria

1: The Arrow-Debreu Model.- 1.1. Preferences and Utility Functions.- 1.2. Maximal Elements.- 1.3. Demand Functions.- 1.4. Exchange Economies.- 1.5. Optimality in Exchange Economies.- 1.6. Optimality and Decentralization.- 1.7. Production Economies.- 2: Riesz Spaces of Commodities and Prices.- 2.1. Partially Ordered Vector Spaces.- 2.2. Positive Linear Functionals.- 2.3. Topological Riesz Spaces.- 2.4. Banach Lattices.- 3: Markets with Infinitely Many Commodities.- 3.1. The Economic Models.- 3.2. Proper and Myopic Preferences.- 3.3. Edgeworth Equilibria and the Core.- 3.4. Walrasian Equilibria and Quasiequilibria.- 3.5. Pareto Optimality.- 3.6. Examples of Exchange Economies.- 4: Production with Infinitely Many Commodities.- 4.1. The Model of a Production Economy.- 4.2. Edgeworth Equilibria and the Core.- 4.3. Walrasian Equilibria and Quasiequilibria.- 4.4. Approximate Supportability.- 4.5. Properness and the Welfare Theorems.- 5: The Overlapping Generations Model.- 5.1. The Setting of the OLG Model.- 5.2. The OLG Commodity-Price Duality.- 5.3. Malinvaud Optimality.- 5.4. Existence of Competitive Equilibria.- References.

Locally solid Riesz spaces with applications to economics

The lattice structure of Riesz spaces Locally solid topologies Lebesgue topologies Fatou topologies Metrizability Weak compactness in Riesz spaces Lateral completeness Market economies Solutions to the exercises Bibliography Index.

Equilibria in markets with a Riesz space of commodities

Abstract We present a new proof of the existence of competitive equilibrium for an economy with an infinite dimensional space of commodities.

The Overlapping Generations Model

In this final chapter, we turn to the other major paradigm in general equilibrium theory: the overlapping generations (OLG) model. The OLG models are extensions and elaborations of P. A. Samuelson’s celebrated pure consumption loan model [59]. Unlike the Arrow—Debreu model which has its genesis in the work of L. Walras [67], Samuelson’s model derives from I. Fisher’s classic monograph The Theory of Interest [28]. As such it shares its origins with the models of T. F. Bewley [16] and B. Peleg and M. E. Yaari [53].

A Theory of Value with Non-linear Prices: Equilibrium Analysis beyond Vector Lattices

This paper presents a new theory of value with a personalized pricing system that naturally induces a family of non-linear prices. This affords a coordinate free theory of value in which the analysis is without any lattice theoretic considerations. When commodity bundles are perfectly decomposable the generalized prices become linear and the analysis specializes to the Walrasian model. This happens, for instance, whenever the commodity space is a vector lattice and consumption sets coincide with the positive cone. Our approach affords theorems on the existence of equilibrium and provides a value-based characterization of Pareto optimality and Edgeworth equilibrium where the Walrasian linear price-based characterization fails. The analysis has applications in the finite as well as the infinite dimensional setting. Journal of Economic Literature Classification Numbers: C62, C71, D46, D51, D61.

The Daugavet equation in uniformly convex Banach spaces

Abstract It is shown that a continuous operator T : X → X on a uniformly convex Banach space satisfies the Daugavet equation ∥ I + T ∥ = 1 + ∥ T ∥ if and only if the norm ∥ T ∥ of the operator lies in the spectrum of T . Specializing this result to compact operators, we see that a compact operator on a uniformly convex Banach space satisfies the Daugavet equation if and only if its norm is an eigenvalue. The latter conclusion is in sharp contrast with the standard facts on the Daugavet equation for the spaces L 1 ( μ ) and L ∞ ( μ ). A discussion of the Daugavet property in the latter spaces is also included in the paper.

Minimum-cost portfolio insurance ☆

Minimum-cost portfolio insurance is an investment strategy that enables an investor to avoid losses while still capturing gains of a payoff of a portfolio at minimum cost. If derivative markets are complete, then holding a put option in conjunction with the reference portfolio provides minimum-cost insurance at arbitrary arbitrage-free security prices. We derive a characterization of incomplete derivative markets in which the minimum-cost portfolio insurance is independent of arbitrage-free security prices. Our characterization relies on the theory of lattice-subspaces. We establish that a necessary and sufficient condition for price-independent minimum-cost portfolio insurance is that the asset span is a lattice-subspace of the space of contingent claims. If the asset span is a lattice-subspace, then the minimum-cost portfolio insurance can be easily calculated as a portfolio that replicates the targeted payoff in a subset of states which is the same for every reference portfolio.

On weakly compact operators on Banach lattices

Consider a Banach lattice E and an order bounded weakly compact operator T: E -E. The purpose of this note is to study the weak compactness of operators that are related with T in some order sense. The main results are the following. (1) If T is a positive weakly compact operator and an operator S: E -+ E satisfies O < S 6 T, then S2 is weakly compact. (Examples show that S need not be weakly compact.) (2) If T and S are as in (1) and either S is an orthomorphism or E has an order continuous norm, then S is weakly compact. (3) If E is an abstract L-space and T is weakly compact, then the modulus I TI is weakly compact. For notation and terminology concerning Banach lattices we follow [1] and [5]. Consider a Banach lattice E and a positive weakly compact operator T: E -E. Now, if S: E -* E is an operator such that 0 < S < T holds, then what effect does the weak compactness of T have on S? Before giving some positive answers to this question, we shall present two examples to show that (in general) under these conditions S need not be a weakly compact operator. EXAMPLE 1. Let { r,J denote the sequence of Rademacher functions on [0, 1]. That is, rn(t) = Sgn sin(27st). Consider S, T: L,[O, 1] -* 1I defined by S(f) = (fof(x)r (x) dx) and T(f) = (fAf(x) dx, f f(x) dx, . . ). Then T is compact (it has rank one), and 0 < S < T holds. On the other hand, S is not a weakly compact operator. To see this, start by observing that the sequence

Problems in Operator Theory

Odds and ends Basic operator theory Operators on

Equilibria in exchange economies with a countable number of agents

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