Foundations Of The Theory Of General Equilibrium (Second Edition)

Yves Balasko

• Éditeur: WSPC
• Année: 2016
• ISBN:
  9789814651721
• Édition: 2
• Sujet: Economics
• Sous-sujet: Economic Theory

The economic theory of general equilibrium underpins the most important models used in economic theory in general and in its more specialized areas such as macroeconomics, international trade, environmental economics, growth theory, and developmental economics. In Foundations of the Theory of General Equilibrium, leading academic scholar, Yves Balasko offers a good introduction to the economic theory of general equilibrium and makes use of various mathematical tools as intuitive and easy as possible. The second half of the book addresses properties of the general equilibrium model that are still at the frontier of current research. These properties deal with the characterization of economies with a unique equilibrium and, more generally, with the relationships between the number of equilibria and the fundamentals of an economy.

Contents:

• The Exchange Model
• A Simple Linear Version of the Exchange Model
• The Exchange Model with Two Goods and Two Consumers
• Consumer Theory
• The Equilibrium Manifold
• The Natural Projection
• Equilibrium Analysis for Fixed Total Resources
• The Natural Projection and Envelope Theory
• A Duality Theory
• Several Extensions of the General Equilibrium Model

Readership: Graduate students in mathematics who want to specialize in economics and mathematical economics; researchers and professionals who will find in this book a detailed account of some of the most current developments of a difficult but essential theory.
Key Features:

• This second edition has been fully revised and augmented to include the most recent advances in the subject area, with two new chapters facilitating the introduction of the main concepts of equilibrium manifold and natural projection, along with exercises for the various chapters of the book
• Important features of this book that were already present in the first edition are the application of envelope theory to equilibrium analysis and highly advanced developments on the dual approach to equilibrium analysis. The addition of exercises in this second edition makes this material that lies at the frontier of current research in general equilibrium theory much more accessible
• Written by noted economic scholar and major contributor to the theory of general equilibrium, Yves Balasko

Chapter 1

The Exchange Model

1.1.Introduction

The theory of general economic equilibrium consists of a collection of mathematical models whose goal is to represent the main determinants and forms of economic activity. Unlike the so-called partial equilibrium models, general equilibrium models do not assume away the many complex interactions that are typical of modern economies. The first general equilibrium model is due to Walras and appeared in the Eléments d’Economie Politique Pure whose first edition was published in 1874 [41]. This model went through some evolution in the subsequent works of Pareto, von Neumann, Hicks, McKenzie, Arrow, Debreu and a few others to culminate in Debreu’s Theory of Value [12]. The formulation adopted by Debreu and the emphasis placed on work made by Arrow and Debreu in particular made such a big impression that this model quickly became known as the Arrow–Debreu model. More complex models that explicitly addressed time and uncertainty in particular, the big absentees of the Arrow–Debreu model, were then developed to fill in some gaps appearing in the Arrow–Debreu model.

The exchange model is the simplest model of all general equilibrium models because there is no specific structure in this model on the commodity space and there is no production. There is in particular no time nor uncertainty. Only exchange is considered. Despite the relative simplicity of exchange in that model, its study quickly becomes quite challenging.

The more complex models include production, public goods, externalities, financial assets, time and uncertainty. All these models are built on the simpler exchange model. This evidently reflects the economic importance of markets in modern economies that affect all forms of economic activity. This therefore explains that the properties of the exchange model are going to play a decisive role in the study of those more complex general equilibrium models. The exchange model plays in economics a role that is similar in many respects to the role played by linear algebra, calculus and point-set topology in mathematics. This reason alone would amply justify a full book devoted to the study of the exchange model.

The study of a general equilibrium model is essentially a study of the properties of the equation defined by the equality of supply and demand in that model, an equation known as the equilibrium equation. There is more than one way to skin a cat and, unsurprisingly, to study an equation. Algebra is one of them and may seem to many to be the most appropriate. But geometry can also be helpful because many properties can have nice geometric formulations. For example, squaring the circle is a question about the nature of the non-trivial roots of the equation sin x = 0. The study of equations that are differentiable in the sense that it is possible to compute their derivatives can take advantage of the global perspective and tools that have been developed in Differential Topology [1, 19, 20, 29, 35]. Of particular interest is Morse Theory that relates singularities of smooth mappings and global properties of smooth manifolds [28]. It is even more remarkable that the study of the equilibrium equation can be rephrased as the study of the natural projection, a smooth mapping between two smooth manifolds introduced by Balasko in [8]. The natural projection provides the exact setup for the application of the Arnold–Thom–Zeeman theory of singularities of smooth mappings, a theory also known under the flashy name of Catastrophe Theory [2,3,18], to the study of the properties of the equilibrium equation of the exchange model.

In this book, three different approaches are developed. This introductory chapter is devoted to a minimalist but very general presentation of the exchange model, and Chapter 2 looks at the study of the equilibrium equation from an algebraic perspective. The model is general in the sense that the number of consumers and goods are arbitrary or, in other words, larger than or equal to two. Nevertheless, the exchange model considered in this chapter is not the most general one because consumers’ preferences are assumed to be represented by log-linear utility functions. Under this restrictive assumption, solving the equilibrium equation is equivalent to finding the eigenvector associated with the Perron–Frobenius root (or eigenvalue) of some positive matrix, in other words a linear algebra problem. Chapter 3 is devoted to the special case of an exchange model with two consumers, two goods and fixed total resources. These assumptions simplify the model to such an extent that it is possible to study many of its properties with the tools of elementary analysis without compromising rigor. Geometry also plays an important role in this chapter. Incidentally, the material in the first part of that chapter is standard fare and can be found in almost every graduate textbook. This systematic treatment of the exchange model first appeared in Bowley’s remarkable little book [11] and is recalled here mainly for reader’s convenience.¹ A third-dimension, the price dimension, is added to the Edgeworth–Bowley box in order to “see” what the equilibrium manifold looks like in a three-dimensional space. Several important properties of the natural projection, the map from the equilibrium manifold into the parameter space, can then be established by just adopting this perspective. These properties are general in the sense that they remain true for an arbitrary number of goods and consumers as will be seen in later chapters. This part has no textbook equivalent. Chapter 4 prepares the study in full generality of the exchange model by developing consumer theory for an arbitrary number of goods. In this chapter, consumers are equipped with preference relations that can be represented by smooth, smoothly monotone and smoothly quasi-concave utility functions. Under these assumptions, consumers’ demand functions satisfy a number of properties that go from the Weak Axiom of Revealed Preferences (WARP) to the properties of the Slutsky matrix that involves the first-order derivatives of individual demand functions. Again, the material in this chapter is well known and can be found in any graduate textbook. It is included in this book for reader’s convenience.

From Chapter 5 on, the stage is set for the study of the equilibrium equation through the equilibrium manifold and the natural projection with the help of the concepts and tools developed in Differential Topology for the study of smooth mappings and their singularities. Chapter 5 is devoted to t...

Table des matières

• Cover Page
• Title
• Copyright
• Contents
• 1. The Exchange Model
• 2. A Simple Linear Version of the Exchange Model
• 3. The Exchange Model with Two Goods and Two Consumers
• 4. Consumer Theory
• 5. The Equilibrium Manifold
• 6. The Natural Projection
• 7. Equilibrium Analysis for Fixed Total Resources
• 8. The Natural Projection and Envelope Theory
• 9. A Duality Theory
• 10. Several Extensions of the General Equilibrium Model
• Mathematical Appendix
• Bibliography
• Index