Handbook of Mathematical Economics 

About: Handbook of Mathematical Economics is an academic journal. The journal publishes majorly in the area(s): General equilibrium theory & Equilibrium selection. It has an ISSN identifier of 1573-4382. Over the lifetime, 41 publications have been published receiving 2471 citations.

Chapter 12 Duality approaches to microeconomic theory

Abstract: Publisher Summary This chapter develops the duality between cost and production functions. The chapter derives the regularity conditions that a cost function C must have and shows how a production function is constructed from a given cost function. The chapter considers the duality between a (direct) production function F and the corresponding indirect production function G. Under certain regularity conditions, G can also completely describe the technology, and thus there is a duality between direct and indirect production functions. The duality theorems have two interpretations: one in the producer context and the other in the consumer context. The chapter discusses a variety of other duality theorems—that is, other methods for equivalently describing tastes or technology, either locally or globally, in the one-output, N-inputs context. The mathematical theorems presented in the chapter appear to be only theoretical results devoid of practical applications. However, this is not the case. The chapter also surveys some of the applications of the duality theorems developed earlier. These applications fall in two main categories: (1) the measurement of technology or preferences and (2) the derivation of comparative statics results.

Chapter 26 Optimal economic growth, turnpike theorems and comparative dynamics

Abstract: Publisher Summary This chapter is concerned with the long-term tendencies of paths of capital accumulation that maximize, in some sense, a utility sum for society over an unbounded time span However, the structure of the problem is characteristic of all economizing over time whether on the social scale or the scale of the individual or the firm The mathematical methods that are used are closely allied to the old mathematical discipline, calculus of variations The chapter discusses that the utility function depends on time, as in the standard theory of the calculus of variations Also the function to be maximized is the sum of utility functions for each period over the future It is described as a separable utility function over the sequence of future capital stocks and corresponds to the integral of calculus of variations As the consumption of one period influences the utility of later consumption, the separability assumption is not exact The treatment of utility in a period as dependent on initial and terminal stocks is not a restriction because the usual assumptions that make utility depend on consumption and consumption on production and terminal stocks implies that an equivalent utility depending on capital stocks exists The chapter also discusses that the primary sources of the optimal growth model are aggregate savings programs and capital accumulation programs for an economy, the theorems, and methods of the subject find applications in other areas with increasing frequency

Chapter 20 Equilibrium under uncertainty

Abstract: Publisher Summary This chapter reviews several concepts of equilibrium that have appeared in the recent literature on markets and uncertainty and focuses on various theories of equilibrium under uncertainty in a sequence of markets, in the absence of complete forward markets for contingent delivery. A stationary state is in full equilibrium, not merely when demands equal supplies at the currently established prices but also when the same prices continue to rule at all dates—when prices are constant over time. The test of equilibrium over time is considered to be more important than mere arithmetical sameness or difference, noting that it implies constant prices in a stationary economy but does not necessarily implies constant prices in an economy subject to change. The Arrow–Debreu model provides a solid foundation of equilibrium analysis upon which to build. Modern decision theory and probability theory enable dealing more effectively with the issues raised by uncertainty and with the conceptual problems of describing an economy that is both stationary and subject to change.

Chapter 15 Existence of competitive equilibrium

Abstract: Publisher Summary This chapter discusses that the mathematical model of a competitive economy conceived as an attempt to explain the state of equilibrium reached by a large number of small agents interacting through markets. Four distinct, but closely related, approaches to the existence problem are recognized. At first, proofs of existence of an economic equilibrium were uniformly obtained by application of a fixed-point theorem of the Brouwer type or Kakutani type or by analogous arguments. This approach, which has remained of central importance to the present, is the subject of the chapter. Second, in the past decade, efficient algorithms of a combinatorial nature for the computation of an approximate economic equilibrium were developed. Third, recently, the theory of the fixed-point index of a map and the degree theory of maps were used to establish the existence of an economic equilibrium, and finally a differential process was proposed whose generic convergence to an economic equilibrium provides an alternative constructive solution of the existence problem.

Chapter 34 Equilibrium theory in infinite dimensional spaces

Abstract: Publisher Summary This chapter summarizes the account of the extension of the classical general equilibrium model to an infinite dimensional setting. The classical finite dimensional theory, the commodity space is the canonical finite dimensional linear space R n . By contrast, there is no canonical infinite dimensional linear space. Different economic applications require models involving different infinite dimensional linear spaces. The mathematical discipline of functional analysis has already been well developed as a tool for the abstract study of linear spaces. The chapter follows the methodology of functional analysis and attacks the existence problem. Advantage of this method is that it yields general results, capable of application in a wide variety of specific models. An important line of research in classical general equilibrium theory has been the relationship of the core to the set of competitive allocations. In the infinite dimensional setting, an extensive body of work has been developed, which centers around the infinite-dimensional version of the Debreu–Scarf core convergence theorem.