General Equilibrium Theory of Value

Index

Preface

The publication of Debreu’s Theory of Value dates back to more than 50 years ago. In that book, Debreu elegantly combined axiomatic rigor and advanced mathematics to prove two major properties of economies with private ownership of production: the existence of equilibrium and the two theorems of welfare economics. Existence is a necessary step. The two theorems of welfare economics are more interesting from an economic perspective. The first theorem states that every equilibrium allocation is Pareto efficient, a theorem understood as evidence of the efficiency of competition in market economies. The second theorem states that every Pareto efficient allocation can be realized as the equilibrium allocation of some competitive economy after suitable redistribution of individual resources, a property expressing the neutrality of competitive markets.

The proofs worked out by Arrow, Debreu, and McKenzie in the 1950s were considered to be such breakthroughs that many textbooks still limit themselves to the existence and welfare theorems and ignore all the developments that have been going on in the theory of general equilibrium since the publication of Debreu’s book.

My goal is to give a much needed update on the properties of economies with private ownership of production. The initial impetus for the study of the general equilibrium model from the differentiable point of view came from Debreu’s 1970 landmark paper. The results of that paper, which were limited to the exchange model, were rapidly extended to more general versions of the equilibrium model. Production was dealt with by Fuchs, T. Kehoe, Mas-Colell, and Smale in particular. Further results were also proved for the exchange model by Delbaen, Dierker, and myself. My own work started with the study of the equilibrium manifold. It rapidly evolved into a program focused on the study of the projection map from the equilibrium manifold onto the parameter space. That approach proved to be highly fruitful. That line of research was also pursued by Bonnisseau, Crès, Ghiglino, Jofre, Jouini, Keiding, Rivera, and Tvede among others for the standard version of the general equilibrium model. I have extended it with K. Shell and D. Cass in research on the overlapping-generations model and the general equilibrium model with incomplete asset markets.

This book is limited to the general equilibrium model with private ownership of production, the very model considered in Debreu’s Theory of Value. Independently of its intrinsic interest, this model is crucial for the study of the more specialized versions of the general equilibrium model that address overlapping-generation or incomplete asset markets for example.

The existence and welfare theorems can be proved with point-set topology and elementary convex analysis as they are in Debreu’s book. Most other properties of the equilibrium equation, however, exploit the differentiability of that equation. The appropriate mathematical tools are then the implicit function theorem, Sard’s theorem about the set of singular values of smooth maps, and the stratified structure of semi-algebraic sets. Except for the implicit function theorem, none of these theorems are taught at the undergraduate level. In addition, these theorems have the reputation of being difficult. The consequence is that very few economists feel comfortable with the differentiable approach. Properties like the multiplicity and discontinuities of market prices in competitive economies are all too often ignored at both theoretical and applied levels.

One goal of this book is therefore to facilitate the access to the differentiable point of view in the study of the general equilibrium model. Mathematics are simplified as much as possible. A real effort has been made to explain the key concepts so that prior knowledge is not necessary. The mathematical prerequisites for this book are on a par with those required by most other advanced economic books: linear algebra, multivariate calculus, and point-set topology.

This book should convince its readers that our understanding of the general equilibrium model with private ownership of production has significantly progressed during the last 50 years.

This book can serve as a modern introduction to general equilibrium theory. It is based on the experience of teaching general equilibrium theory at various levels. Its content is appropriate for a set of approximately 20 one-hour lectures for first-year graduate students.

CHAPTER 1

Goods and Prices

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, . . . , xℓ) consists of x¹ units of the first good, x² units of the second good, up to xℓ units of the ℓ-th good. The commodity space 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 pj. Prices are strictly positive. The price pj of commodity j is actually the price of one unit of commodity j.

THE PRICE VECTOR

The price vector p = (p1, . . . , pj, . . . , pℓℓ has for coordinates the prices pj > 0 of the various goods.

VALUE OF A COMMODITY BUNDLE FOR A GIVEN PRICE SYSTEM

Let x = (x¹, . . . , xℓℓ be a commodity bundle and p = (p1, . . . , pℓsome arbitrary price vector. The value of the commodity bundle x ℓ given the price vector p is equal to the inner product

1.4 RELATIVE PRICES

Given the price vector p = (p1, . . . , pℓ, the price of good h relative to the price of good k is defined by the ratio ph/pk.

Relative prices depend only on the direction defined by the price vector p = (p1, . . . , ph, . . . , pk, . . . , pℓ, 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 .

1.5 PRICE NORMALIZATION

There are quite a few ways to normalize the price vector p 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 + . . . + (pℓ)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 vector p = (p1, . . . , pℓis simplex normalized if Σk pk = 1. We denote by SΣ the set {p X | Σk pk = 1} of simplex normalized prices.

Note that all prices are strictly positive for p Sconsists of the price vector p = (p1, . . . , pℓ) where some coordinates can be equal to zero. The boundary ∂S\ SΣ consists of the price vectors p that have at least one coordinate equal to zero.

THE NUMERAIRE NORMALIZATION

Another price normalization has the favor of many economists. It consists in giving to some good the role played by money for expressing the prices of the other goods in quantities of that good. Such a good is known as the numeraire.

Numeraire is not money because money, paper money (also known as fiat money) is not an economic good in our sense. Paper money cannot be consumed physically like an orange or a banana. Paper money is not an argument of the utility functions considered by the classical theory of the consumer, a theory that we address in chapter 2. Similarly, paper money is not a direct input of the production process in the classical theory of the firm considered in chapter 5. If the numeraire cannot be confused with paper money, gold has many attributes of a numeraire. In the times of the Gold Exchange Standard, prices were expressed in quantities of gold, an argument of consumers’ utility functions and producers’ production processes.

In practice, having a numeraire amounts to setting the price of the numeraire commodity to one. In this book, we choose the ℓ-th commodity to be the numeraire: pℓ = 1.

Definition 1.2. The price vector p = (p1, . . . , pℓis numeraire normalized if pℓ = 1. We denote by S the set {p | pℓ = 1} of numeraire normalized prices.

Unless the contrary is specified, all price vectors from now on are numeraire normalized. The main problem with the numeraire price normalization is that it treats goods asymmetrically, which may not be very satisfactory from a mathematical perspective. This defect is more than compensated by the relative simplicity it gives to the computation of derivatives of demand and supply functions with respect to prices, operations that we will do extensively in later chapters.

1.6 NOTES AND COMMENTS

This introductory chapter is very similar to the first chapter of Debreu’s Theory of Value [23].

The necessity of measurability for the goods that claim to have market prices is often neglected. Most public goods and environmental goods are not measurable. This has not prevented Samuelson and his followers to treat public goods as if they were measurable [7, 57].

Some mathematicians use the Euclidean normalization of the price vector because the price vector then belongs to the sphere of unit radius. Then, the aggregate excess demand of an exchange economy can be viewed as defining a vector field on the sphere, which enables the application of several powerful theorems of algebraic topology to the study of this vector field [26].

CHAPTER 2

Preferences and Utility

CLASSICAL CONSUMER THEORY is essentially the theory of utility maximization under a budget constraint. This theory starts with the definition of consumers’ preferences. In classical consumer theory, preferences are assumed to be transitive, complete, monotone and convex. These preferences can then be represented by utility functions. The latter are mathematically easier to handle than preferences. Another reason for being interested in utility functions goes back to the early phases of economic theory. Then, it was thought that utility functions could be used as a measure of consumer’s satisfaction or utility. Pareto suggested the term of ophelimité instead of utility to avoid any misleading interpretation. This chapter is devoted to a presentation of the basic issues regarding preferences and their representability by utility functions.

2.1 CONSUMPTION SETS

The first item to be defined is the consumption set. It is usually defined as the set of commodity bundles that can be physically consumed. In such a case, the consumption set should be the non-negative orthant or some subset of the non-negative orthant. However, negative consumption can also be interpreted as the delivery to the market of some positive quantities of goods, in which case the consumption set can contain commodity bundles with negative coordinates. Negative consumption is explicitely mentioned by Debreu [23]. It is observed in financial markets and, more generally, in markets where short sales are permitted and when goods are not necessarily bought for immediate physical consumption.

ℓ. Furthermore, this choice is the only one that is really consistent with the existence of preferences since the commodity bundles that are feasible will always be preferred to those that are not feasible.

Assumption 2.1. The consumption set of every consumer is the full commodity space ℓ.

2.2 BINARY RELATIONS

Preferences are represented by binary relations defined on consumption sets. Being complete preorders, these relations are representable by utility functions. Other main properties of preferences are monotonicity and convexity. The goal of classical consumer theory is to derive properties of demand functions that come to play a crucial role in the general equilibrium model from the properties of consumer’s preference relations.

2.2.1 Properties of Binary Relations

ℓ.

i i is an ordered pair (x, y) where the elements x and y are by definition related to each other if the pair (x, yi. One then writes x y.

Definition 2.2. The binary relation is transitive, reflexive, symmetric, antisymmetric, and complete if it satisfies the following properties:

Transitivity: x y and y z imply x z;

Reflexivity: x x is true for every x ;

Symmetry: x y implies y x;

Antisymmetry: x y and y x imply the equality x = y;

Completeness: for any x and y in , either x y or y x is satisfied.

2.2.2 Examples of Binary Relations

is in fact the equality x = ythat are different from the equality cannot be simultaneously symmetric and antisymmetric.

PREORDERS

Definition 2.3. A complete preorder on is a binary relation that is transitive, reflexive, and complete.

ORDERS

Definition 2.4. A complete order on is a binary relation that is transitive, reflexive, antisymmetric, and complete.

EQUIVALENCE RELATIONS

Definition 2.5. An equivalence relation ~i on is a binary relation that is transitive, reflexive, and symmetric.

A useful concept is the one of equivalence class. Let x . The equivalence class Ci(xconsisting of the elements y that are equivalent to x:

by which it is meant: 1) two equivalence classes Ci(x) and Ci(y) are either disjoint (i.e., their intersection is empty) or identical depending on whether x and y are equivalent or not; 2) the union of all equivalence classes Ci(x) for x .

/ ~i and is known as the quotient set by the equivalence relation ~i.

Equality is a typical equivalence relation but its equivalence classes consist of only one element each.

A preorder differs from an equivalence relation by not being symmetric.

2.2.3 Equivalence Relation Associated with a Preorder

. The binary relation x ~i y is then defined by x y and y x.

Proposition 2.6. The binary relation ~i associated with the preorder is an equivalence relation on .

Proof. Let us prove that the binary relation ~i is reflexive, symmetric,