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 Rn. 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.
Competitive equilibria in economies of overlapping generations are different from competitive equilibria in economies that have extended over finitely many periods, finite economies for short. These differences concern the properties of competitive equilibria, such as existence, optimality and determinacy or local uniqueness; and the phenomena compatible with competitive equilibria, such as net aggregate debt or fiat money with a positive price. The chapter discusses that all the variants of the model of overlapping generations allow for infinity of time periods and hence for infinitely many commodities. This infinity of time periods and commodities is not a mathematical curiosity, but rather is central to the economic significance of the model. Consider a pay-as-you-go system of social security. In economic models in which the rationality of individuals is presumed to be unlimited, the fact that no new generation would appear at some point in the very distant future would also lead to the immediate break down of the social security system. In an economy of overlapping generations, the temporal and demographic structure is explicit, which evidently enriches the study of problems such as the transfer of value over time. It also allows the claim that the inability of individuals to trade directly with individuals whose consumption and endowment spans commence after they have perished is the distinguishing feature of economies of overlapping generations.
This chapter aims to (1) set enough of the mathematics of dynamical systems from the perspective of chaos theory so that the endogenous emergence of chaotic dynamics and other highly complex dynamics are understood, (ii) measures of instability and complexity are explained, (iii) how one tests for the presence of chaos and other complex nonlinear dynamics in time series data is discussed; and (iv) how these concepts have been applied in economics is explained. The chapter focuses on discussing the fabric and unity of this subject from the point of view of economic theory and econometrics. Basic mathematics has been developed so that ideas and the notion of chaotic dynamics are defined precisely. In computer science, there is a great deal of interest in the properties of random number generators. Measures that are designed to detect chaos can also be used to test random number generators. It raises the issue of what criteria should be used in judging whether or not finite length sequences are random. In statistical hypothesis testing, certain measures that are used to detect chaotic dynamics have been used to test for stochastic stationarity and independence.
This chapter surveys the state of the art regarding the Arrow–Debreu model of a Walrasian economy, consisting of a finite number of agents and commodities where one assumes perfect information, complete markets, no market imperfections, such as externalities, public goods, or non-convexities in consumption or production; firms are price-taking profit maximizers, and households are price-taking utility maximizers. In such a world, the basic properties of the classical Arrow–Debreu model consist of the existence of competitive equilibria, the first and second welfare theorems, the computation of equilibria, and the local uniqueness and finiteness of equilibria. The chapter discusses that it is useful to adopt Walras' original conception of a competitive equilibrium as a solution to a (nonlinear) system of equations. Convexity plays an essential role in Scarf's analysis, both in the derivation of the market excess demand function from optimizing behavior on the part of agents and in the existence of a fixed-point that follows from Brouwer's theorem. Scarf's algorithm and its generalizations are the primary means of doing comparative statics in general equilibrium models. Computable general equilibrium models have replaced activity analysis and input–output analysis as the basic method of analyzing tax policy in national economies or trade policies among nations.
General equilibrium theory is applied wholesale to obtain a theory of value for security markets. The modern theory of value for security markets extends general equilibrium theory in various ways: (1) it explicitly treats general multiperiod trading opportunities under uncertainty and in incomplete markets; (2) it investigates, in remarkable depth, implications of the law of one price—that is, of arbitrage-free prices; (3) to represent security returns in convenient and testable ways, it places strong restrictions on preferences and exploits a great deal of probability theory, especially the theories of Markov processes and stochastic integration, separately and together. The chapter discusses that as financial market theory grows, it laps over the boundaries of the general equilibrium paradigm to focus on the process of price formation. The microstructure of security markets has come under increasing scrutiny; the theory of specialist market makers, for example, is gradually being filled out. The need to address asymmetric information, in particular, has led to strategic models of investment behavior. Despite the diverse aims of financial economic theory, the chapter summarizes the developments in finance that rest or build on general equilibrium theory, emphasizing the valuation of financial assets.
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.
Monopolistic competition is defined as a situation of imperfect competition with some important features: (a) the products sold are differentiated; (b) firms themselves set the price of these goods; (c) the number of sellers is large and each firm disregards the effects of its price decisions on the actions of its competitors; (d) entry is unrestricted and proceeds until profits are reduced to zero or the smallest possible number consistent with the fact that the number of firms is an integer. The chapter discusses that the theory of monopolistic competition poses important and difficult conceptual problems. It focuses on models with a generally large number of price setters and indicates in the conclusion a number of alternative presentations. It introduces a basic model and studies problems of existence of an equilibrium and endogenous product differentiation and finally discusses macroeconomic applications.
Este é o primeiro número da nova série da editora SpringerVerlag, "Studies in Economic Theory", editada por C.D. Aliprantis (Departamento de Matemática, IUPUI, Indianápolis, IN, EUA) e N.C. Yannelis (Departamento de Economia, Universidade de Illinois, Champaign, IL, EUA). Mais uma vez essa editora demonstra que está na vanguarda da divulgação científica na área de teoria econômica avançada.
Publisher Summary Equilibrium with rational expectations is a central construct of modern economic theory. The chapter studies surveys, which are primarily aimed at assessing the properties and relevance of this construct in a general equilibrium framework; in particular, they are related to the issue of multiplicity of rational expectation equilibria. The viewpoint discussed in the chapter is associated with the so-called concept of sunspot equilibrium. It focuses on aspects of general equilibrium with rational expectations that have been given a central role in recent literature. It presents the problem of sunspot equilibria in the broader perspective. It also discusses an example of sunspot equilibrium and a simple economic system—namely, a sequential economy with infinite horizon and time independent structure. Furthermore, this economy is one step forward looking and has no predetermined variable.
Publisher Summary In recent years the Walrasian general equilibrium model has become an important tool for applied work in such fields as development economics, international trade, macroeconomics, and public finance. This chapter discusses that economic equilibria are usually solutions to fixed point problems rather than solutions to convex optimization problems. This leads to two difficulties that are closely related: first, equilibria may be difficult to compute; second, a model economy may have more than one equilibria. The chapter explores these two issues for a number of stylized economies and analyzes economies with infinite numbers of goods, economies in which time and uncertainty play important roles. Studying economies of this sort is interesting not only for its own sake but also because of the insights it provides into the properties of economies with large but finite numbers of goods. Finally, the chapter extends an analysis to economies that include distortionary taxes and externalities.
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.
This chapter discusses that the problem of economic planning is best viewed as one of solving an extremely large constrained maximization problem. The objective function is identified with a measure of economic welfare, which has maximized subject to a variety of constraints, including those imposed by the details of available production processes and by the economy's endowment of economic resources. In a typical decentralized planning scheme, one might find that a central authority was responsible for ensuring that for each good supply and demand were in balance, taken across the whole economy, and that this authority was provided with the minimum information sufficient for execution of this task. Responsibility for ensuring that technological constraints were satisfied would be delegated to firms in whose processes the constraints were embodied. The chapter also explores that most planning procedures discussed have been iterative in the sense that they view the planning problem as being solved by a trial and error process, in which information exchanges among the participants in the decentralized process lead to the construction of better and better approximations to the solution of the planning problem.
This chapter discusses an assortment of recent economic studies in the same way: as steps toward characterizing those organization designs that do well, according to some measure of gross performance, with the informational and administrative resources they require. The steps turn out to be diverse and modest, but the problem is difficult. Piecing the assorted contributions together, one is still far indeed from a unified theory of efficient organization design. The main stumbling block remains the modeling of technology and cost. Some elements of cost have been studied intensively and even elegantly. The chapter examines the Shannon theory in connection with transmission, and the theory of finite-state machines in connection with the assignment of output/state pairs to input/state pairs in one-step designs with memory. Techniques have been developed for the study of a design's gross performance, for example, the computation of expected payoff for a given team information structure, and these remain useful in efficiency studies when good cost models become available. The theory, even in its present form, has already been useful in revealing how difficult it is (1) to define certain widely current terms sharply and agreeably to most usages and (2) to verify certain widely held conjectures.
This chapter surveys issues in positive second-best theory, specifically the theory of the optimal pricing of goods produced by public firms—that is, firms whose objective is the maximization of social welfare. It is assumed that these firms, characteristically, display increasing returns to scale. In these situations, first-best optima may require lump-sum taxes and subsidies. In view of the size of the public sector in most industrialized countries, it is difficult to imagine that public activities can be financed without distortionary effects elsewhere in the economy. The chapter assumes perfect possibilities for lump-sum income transfers to focus on the efficiency aspect of optimal pricing. If lump-sum redistribution is impossible, deviations from marginal costs for prices under public control may be motivated by distributional considerations—that is, the government may want to use its excise tax power to improve the income distribution. The chapter provides general formulation that forms a basis for special cases that are analyzed in more detail subsequently. It also discusses the joint decision for the optimal supply of public goods and the pricing problem. Finally, it analyzes some issues in predation and Ramsey pricing in a dynamic context.
This chapter discusses the social choice theory. There is social choice problems, which deals with methods of marshalling information, particularly those relating to the people involved, to arrive at correct social judgments or acceptable group decisions. But the natures of the possible informational inputs vary, as do the required outputs of judgments, decisions, or the required means of settlement. The balance of moral and pragmatic considerations also varies with the nature of the exercise. There are other differences, for example, whether the procedures permit the use of discretion in interpreting individual utilities or are mechanical. The nature of the exercise affects the appropriate specification of the social choice format. This relates to distinctions among structures such as social welfare functions, social decision functions, social choice functions or functional collective choice rules, or social welfare functional. It also affects the appropriateness of particular axioms within a given structure, for example, whether the social welfare function satisfies the independence condition or what types of interpersonal comparability if any is used. The relevance of the various results presented and discussed depends on the particular nature of the exercise to which application is sought.
