Über die Autorin bzw. den Autor

Yves Balasko is professor of economics at the University of York. He is the author of Foundations of the Theory of General Equilibrium and The Equilibrium Manifold.

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"Balasko provides a clear, thorough, and completely up-to-date look at the basic economic model of supply and demand, in which values are determined through the interaction of individual preferences, endowments, and technologies. This will be an excellent text for advanced graduate students in economic theory, an introduction for mathematics students with an interest in applications of differential topology, and a must-have reference for researchers in general equilibrium theory."--Karl Shell, Cornell University

"This book performs a valuable pedagogical service for the economics profession by providing a detailed development of general equilibrium theory that will be accessible to graduate students of economics."--Stephen E. Spear, Carnegie Mellon University

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General Equilibrium Theory of Value

PRINCETON UNIVERSITY PRESS

Contents

Preface....................................................................xiChapter 1 Goods and Prices.................................................1Chapter 2 Preferences and Utility..........................................5Chapter 3 Demand Functions.................................................19Chapter 4 The Exchange Model...............................................37Chapter 5 The Equilibrium Manifold.........................................47Chapter 6 Applications of the Global Coordinate System.....................56Chapter 7 The Broad Picture................................................62Chapter 8 The Fine Picture.................................................74Chapter 9 Production with Decreasing Returns...............................82Chapter 10 Equilibrium with Decreasing Returns.............................96Chapter 11 Production with Constant Returns................................108Chapter 12 Equilibrium with Constant Returns...............................124Postscript.................................................................145Appendix A Notation........................................................149Appendix B Point-set Topology..............................................151Appendix C Smooth Manifolds................................................152Appendix D Singularities of Smooth Maps....................................155Appendix E Convexity.......................................................159Appendix F Miscellany......................................................166References.................................................................167Index......................................................................171

Chapter One

1.1 Introduction

The aim of this chapter is to develop the main aspects of the economic environment in which economic agents operate. There are two categories of economic agents, consumers and firms. Consumers buy and sell goods with the ultimate goal of consuming those goods. Firms buy goods that they transform into other goods that they later sell. An economy is made up of these consumers and firms. After having developed models of the consumers and firms, we will combine them into a model of an economy with private ownership of production. Before developing these models, it is necessary to be somewhat more explicit about the economic goods and their prices that define the economic environment.

1.2 GOODS

THE CHARACTERISTICS OF AN ECONOMIC GOOD

Economic goods are defined by their "physical" characteristics or properties. These physical characteristics, which have to be taken in a very broad sense, may include all forms of services. But economic goods often feature other aspects than their physical properties. For example, the location and the date of delivery of an economic good are sufficiently important to be specified for each economic good. The conditions under which delivery takes place is also specified in many general equilibrium models, especially those that involve uncertainty. In these models, the delivery may depend on the realization of some state of nature. Such goods are known as contingent goods. To sum up, economic goods are defined by their characteristics, which may include many more things than just the physical properties of the goods.

THE MEASURABILITY REQUIREMENT

Goods that have a price like all the goods traded in markets have to be measurable. Measurability is a concept that comes from physics. For an economic good to be measurable, it is necessary to have a meaningful definition of the equality and the sum of two quantities of that good. In that regard, not all economic goods are measurable. For example, some public goods like national defense are typically not measurable because it is almost impossible to define the equality of two levels of national defense. Needless to say, the sum of two levels of national defense is even harder to conceive.

The goods considered in general equilibrium models are measurable. They are also divisible. Units are defined once and for all for each good in the economy.

THE COMMODITY SPACE

The number of goods is finite and denoted by l, a number greater than or equal to 2. The commodity bundle x = (x¹, ..., x^l) consists of x¹ units of the first good, x² units of the second good, up to x^l units of the l-th good. The commodity space is the Euclidean space R^l. Quantities of goods in a commodity bundle like x can be positive, negative, or equal to zero.

1.3 Prices

We associate with every commodity j a price p_j. Prices are strictly positive. The price p_j of commodity j is actually the price of one unit of commodity j.

THE PRICE VECTOR

The price vector p = (p₁, ..., p_j, ..., p_l) [member of] R^l has for coordinates the prices p_j > 0 of the various goods.

VALUE OF A COMMODITY BUNDLE FOR A GIVEN PRICE SYSTEM

Let x = (x¹, ... , x^l) [member of] R^l be a commodity bundle and p = (p¹, ..., pl) [member of] R^l₊₊ some arbitrary price vector. The value of the commodity bundle x [member of] R^l given the price vector p [member of] R^l₊₊ is equal to the inner product

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1.4 Relative Prices

Given the price vector p = (p¹, ..., pl) [member of] R^l₊₊, the price of good h relative to the price of good k is defined by the ratio p_h/p_k.

Relative prices depend only on the direction defined by the price vector p = (p¹, ..., p^h, ..., p_k, ..., p_l) [member of] R^l₊₊, not on its length. An important assumption that will be made in future chapters will be that consumption and production decisions depend only on relative prices or, in other words, on the direction defined by the price vector. It has therefore become customary to use some kind of normalization for the price vectors p [member of] Rl₊₊.

1.5 Price Normalization

There are quite a few ways to normalize the price vector p [member of] Rl₊₊. Mathematicians tend to choose some norm on the vector space R^l and to set that norm equal to one.

THE EUCLIDEAN NORMALIZATION

One example of such a normalization is to set the Euclidean norm |p| = ((p₁)² + ··· + (p_l)²)^1/2 to one. Perfectly satisfactory from a mathematical perspective, this normalization has little economic appeal. We will not use this normalization in the book.

THE SIMPLEX NORMALIZATION

Another normalization that is particularly handy when dealing with the behavior of consumers and firms when some prices tend to zero is the simplex normalization.

Definition 1.1. The price...