Ever since Adam Smith's evocation of an invisible hand, market equilibrium has been supposed not only to clear markets but also to achieve an efficient allocation of resources. This view is embodied in Chapter 19 in a definition and two major results. We define a very general efficiency concept, Pareto efficiency. We then state and prove the two major results relating market equilibrium to efficient allocation, which are the two most important results in welfare economics.

The First Fundamental Theorem of Welfare Economics agrees with Adam Smith: A market equilibrium allocation is Pareto efficient. This result can be demonstrated in a surprisingly elementary fashion. It requires very little mathematical structure, and it does not require any assumption of convexity. If, despite nonconvexity, the economy has a market equilibrium, that equilibrium allocation is Pareto efficient.

The Second Fundamental Theorem of Welfare Economics requires more mathematical structure. It is a more surprising and deeper result. It says – assuming convexity of tastes and technology – that any efficient allocation can be supported as a competitive equilibrium. Find an efficient allocation. Then there are prices and a distribution of resource endowments of goods and share ownership that will allow the efficient allocation to be an equilibrium allocation at those prices and endowments. Market allocation is compatible with any efficient allocation subject to a redistribution of income.

The models treated here can be interpreted to treat allocation over time and under uncertainty. To do so, the space of commodities traded needs to be interpreted to include intertemporal trade and trade in insurance or event-contingent goods.

The Robinson Crusoe model in Chapter 2 describes the price system of a simple economy as a means of making efficient decentralized choices. That model focuses on the relationship of the production side of the market to the consumption side. The market in equilibrium allocates resources between competing productive uses (consumption and leisure) so as to use the available production technology to efficiently satisfy consumer demands. It is a model of the decentralized market arranging the allocation of resources in production to satisfy households. Another aspect of efficient allocation is to arrange efficient allocation of goods among consumers. Efficient allocation of resources requires both an efficient mix of outputs and an efficient allocation among consumers. In this section, we'll ignore the production decision and concentrate on the interpersonal allocation of a fixed mix of available goods. The production and consumption sides are considered together in Chapter 4.

The modeling technique we will use for this allocation decision is the brilliant and brilliantly simple device due to F. Y. Edgeworth, known as the Edgeworth box. Suppose we have fixed positive quantities of two goods, X and Y, and two households, 1 and 2. We would like to know how to allocate the fixed supplies of X and Y between the two households. Three allocation schemes will be developed: efficient allocation, a bilateral bargaining allocation, and a market equilibrium allocation.

The material presented in Chapters 10–25 represents fulfillment of the research agenda in Arrow and Debreu (1954). It represents most of the state of the general equilibrium theory (for economies with a finite number of households) through the 1960s. The next steps in the analysis of the field have used rather more sophisticated mathematics to develop a more refined class of results. Some of those implications are briefly illustrated in Chapter 26. In addition, the computational approach has meant an applied aspect to the general equilibrium theory, an applicability that would have surprised readers of the original article, Arrow and Debreu (1954), when it appeared.

What have we learned? The mathematical method formalizing economic concepts is immensely powerful. It gives form and generality to economic ideas and specifies the scope and limits of their application. Chapter 27 puts the results in perspective.

DefinitionA set of points S in RN is said to be convex if the line segment between any two points of the set is completely included in the set, that is, S is convex if x, y ∈ S implies {z | z = αx + (1 - α)y, 0 ≤ α ≤ 1} ⊆ S.

S is said to be strictly convex if x, y ∈ S, x ≠ y, 0 < α ≤ 1 implies αx + (1 - α)y ∈ interior S.

The notion of convexity is that a set is convex if it is connected, has no holes on the inside, and has no indentations on the boundary. Figure 8.1 displays convex and nonconvex sets. A set is strictly convex if it is convex and has a continuous strict curvature (no flat segments) on the boundary.

Properties of convex setsLet C1and C2be convex subsets of RN. Then

C1 ∩ C2is convex,

C1 + C2is convex,

C1is convex.

Proof See Exercise 8.1. QED

The concept of convexity of a set in RN is essential in mathematical economic analysis. This reflects the importance of continuous point-valued optimizing behavior. To understand the importance of convexity, consider for a moment what will happen when it is absent. Suppose widgets are consumed only in discrete lots of 100. The insistence on discrete lots is a nonconvexity. Suppose a typical widget eater at some prices to be indifferent between buying a lot of 100 and buying 0.

U-shaped cost curves and concentrated preferences

In intermediate microeconomic theory, a firm's cost function is often described as U-shaped. The notion is that firms producing at low volume have high marginal costs. The marginal costs decline as volume increases and then start to rise again. There is a region of declining marginal costs. But declining marginal costs are inconsistent with convexity of technology, and convex technology is one of the assumptions used to show the existence of general equilibrium in Chapters 14, 18, and 24. Can we reconcile the elementary U-shaped cost curve model with the existence of general equilibrium?

Convexity of preferences was one of the assumptions used to demonstrate continuity or convexity of demand behavior needed for the proofs of existence of general equilibrium in Chapters 14, 18, and 24. But surely there are instances where convexity does not hold. A household might be equally pleased with a blue suit and a gray suit but half a blue suit and half a gray suit is not so satisfactory. A resident may be equally satisfied with an apartment in San Francisco or one in Boston; half time in each is less satisfactory. The household has concentrated preferences (or a preference for concentrating consumption). Can these preferences be reconciled with the existence of general economic equilibrium?

We'll argue in this chapter that the answer is “yes.” Using the Shapley-Folkman theorem we'll establish the existence of approximate equilibrium in these settings.

This chapter surveys very briefly developments in the general equilibrium theory of the last several decades. There is no room here for the richness and detail that these topics merit. They each have a population of books and articles of their own. Nevertheless, even a beginning student of general equilibrium theory can appreciate a notion of the scope of the generalizations.

Large economies

Chapters 22 and 25 emphasized the importance of large numbers of households in the economy. As the economy becomes large, the core converges to the competitive equilibrium allocations (Theorem 22.2), and, indeed, this result is true even without the assumption of convex preferences (Theorem 22.3). Further – concentrating on the limiting behavior of the economy as the economy becomes large – the assumptions of convex technology and convex preferences are no longer required for existence of competitive equilibrium (Theorem 25.1).

These results are stated as limiting behavior as the economy becomes large. The alternative is to state the results directly for a large economy – an economy with an infinite number of households. One way to do this is to think of the set of households as the points on the unit interval [0, 1], an uncountable infinity of households. Then, instead of summing the demands of the households to find total demand, it is appropriate to integrate the demand function over the interval. It is important to emphasize that each point in the interval is negligible (has wealth infinitesimally small compared to the total).

Correspondences

We will call a point-to-set mapping a correspondence. A function maps points into points. A correspondence (or point-to-set mapping) maps points into sets of points. Let A and B be sets. We would like to describe a correspondence from A to B. For each x ∈ A we associate a nonempty set β ⊂ B by a rule ϕ. Then we say β = ϕ(x), and ϕ is a correspondence. The notation to designate this mapping is ϕ : A → B. For example, suppose A and B are both the set of human population. Then we could let ϕ be the cousin correspondence ϕ(x) = {y | y is x's cousin}. Note that if x ∈ A and y ∈ B, it is meaningless or false to say y = ϕ(x), rather we say y ∈ ϕ(x). The graph of the correspondence is a subset of A × B : {(x, y) | x ∈ A, y ∈ B and y ∈ ϕ(x)}.

For example, let A = B = R. We might consider ϕ(x) = {y | x - 1 ≤ y ≤ x + 1}. The graph of ϕ(·) appears in Figure 23.1.

Upper hemicontinuity (also known as upper semicontinuity)

In the balance of this chapter and the next, we concentrate on mappings from one real Euclidean space into another, from RN into RK, for N ≥ 1 and K ≥ 1.

Pareto efficiency

The purpose of economic activity is to allocate scarce resources to promote the welfare of households in their consumption of goods and services. There is a very large number of possible allocations of resources (typically, an uncountable infinity), but most of them are wasteful – we can do better. Some wasteful allocations are those that do not make effective use of productive resources (corresponding to points inside the production frontier in the Robinson Crusoe economy). An alternative form of inefficiency occurs in allocations that allocate the mix of outputs among consumers without equating marginal rates of substitution (subject to boundary conditions), leaving room for improvement in the mix of consumption across households (wasteful points corresponding to those off the locus of tangencies in the Edgeworth box).

Economic theory does not give us precise guidance as to the desirable distribution of income and wealth across households. The theory is agnostic on the distribution of income between Smith and Jones and between Rockefeller and Micawber. We are led then to posit a criterion of nonwastefulness as a standard for the effective utilization of scarce resources, while avoiding the moral question of the desirable distribution of income. The nonwastefulness criterion is Pareto efficiency, and it is fundamentally a simple idea. A (Pareto) improvement in allocation is a reallocation that increases some household's utility (moves higher in the preference quasi-ordering) while reducing no household's utility. An allocation is Pareto efficient if there is no further room among attainable allocations for (Pareto) improvement.

Now we need to take one further step, to bring the production decision and the interpersonal allocation decision together. The Edgeworth box model, presented in Chapter 3, treats efficient allocation of consumption among households but doesn't treat production. The Robinson Crusoe model, developed in Chapter 2, treats efficient choice of production outputs but doesn't treat consumption allocation between households. Neither treats explicitly the efficient allocation of inputs to production. We'll integrate all of these disparate elements in this chapter, by introducing a 2 factor × 2 commodity × 2 household general equilibrium model.

The Robinson Crusoe model treated the consumption/production interaction with only one household. We can now combine the Robinson Crusoe production decision with the Edgeworth box consumption allocation to portray the production/interpersonal allocation decision at one shot. The joint equilibrium of production and interpersonal allocation is depicted in Figure 4.1. For each price ratio, the production sector chooses the profit-maximizing output mix. The Edgeworth box then depicts the allocation of these outputs between households. The budget line in the box shows how households react to prevailing prices. The figure shows the production decision as profit maximization subject to prevailing prices, technology, and resources, just as in the Robinson Crusoe model. The slopes of the isoprofit line and of the budget line are identical.

Introduction

We have already demonstrated the existence and efficiency of general equilibrium in an economy of N goods with active markets for trading them. But what are these N goods? The answer is that they could be anything. This generality reflects the distinctive power of mathematical modeling. The model and its interpretation are separate. We have a mathematical model that provides a general family of results based on mathematical relations among the variables. How we label the variables and interpret the results is now up to us. The model could apply to trading mineral samples at annual meetings of an amateur gemologists society. It can apply to the trading and production of a small closed economy. It can apply to trading and production of an entire world economy. In each case, of course, it applies only if the assumptions of the model are fulfilled. What we know in each instance is that if the assumptions of the model are fulfilled then the conclusions follow: There will be market clearing prices that lead to a Pareto-efficient allocation. This is true whether the prices and allocations are for rock samples, the goods available in a small economy, or those available throughout the world. We have left until now a more complete discussion of the range of goods to be allocated by the market mechanism.

The foundations of modern economic general equilibrium theory are contained in a surprisingly short list of references. For primary sources, it is sufficient to master Arrow and Debreu (1954), Arrow (1951), Arrow (1953), and Debreu and Scarf (1963). An even shorter list is comprehensive; Debreu (1959) and Debreu and Scarf (1963) cover the topic admirably. Why should anyone write (or read!) a secondary source, a textbook? Because, unfortunately, this body of material is extremely difficult for most students to read and comprehend. Professor Hahn described Debreu's (1959) book as “very short, but it may well take as long to read as many works three times as long. This is not due to faulty exposition but to the demands rigorous analysis makes on the reader. It is to be hoped that no one will be put off by this, for the … return … is very high indeed” (Hahn [1961]). Unfortunately, in teaching economic theory we find that many capable students are indeed put off by the mathematical abstraction of the above works. What theorists regard as elegantly terse expression, students may find inaccessible formality. The focus of this textbook is to overcome this barrier and to make this body of work accessible to a wider audience of advanced undergraduate and graduate students in economics.

This book presents the theory of general economic equilibrium incrementally, from elementary to more sophisticated treatments. Part A (Chapters 1 through 5) presents an elementary introduction.

Let us review some basic elements of set theory:

Logical inference In mathematical logic the word implies means “leads to the logical inference that” and can be represented by the symbol of the double shaft arrow, ⇒. This represents a strong causal relation.

Definition of a set We think of a set as a group or collection, defined by the items in the collection. A typical set might consist of all UCSD freshmen, all surfers in Southern California (there is obviously some overlap here), or the positive integers between 1 and 10. We might call a set by another name, such as a collection, a family, a class, an aggregate, or an ensemble. We use the notation of a pair of braces, { }, to denote a set. We can use a description of elements of the set to define the set. Thus, the entity denoted {x | x has property P} is the set of all things with property P (whatever that is). The set of positive integers between 1 and 10 can be expressed then as {1, 2, …, 9, 10} or, equivalently, as {x | x is an integer, 1 ≤ x ≤ 10}.

Elements of a set The elements of a set are the things in the collection. If x is an element of the set A, we write x ∈ A.

Like the first edition of this work, the second edition begins with a celebration. In 2005 at the University of California at Berkeley there was an enthusiastic conference celebrating the life and work of our late colleague Gerard Debreu. Professors, researchers, and students gathered literally from all over the world. For three days and nights, papers were presented, reminiscences shared, testimonials and tributes spoken. Gerard Debreu – half of the Arrow-Debreu team – had reshaped our field and created the specialty we loved. Prof. Hugo Sonnenschein remarked:

That's why the present volume appeared: to make Theory of Value more easily accessible to a wide audience, because the Arrow-Debreu model is the standard of the field. We who work the field should understand it well.

For the past decade, students and colleagues have remarked on the first edition of this book: appreciating, criticizing, suggesting revisions and corrections.

It is a pleasure to acknowledge two distinctive contributions. Colleagues at the University of Copenhagen have been extraordinarily helpful. Professor Peter Sørensen and the late Professor Birgit Grodal both went over the entire volume, making immensely useful suggestions and corrections.