In Chapters 10–18, we developed the theory of firm and household behavior, concentrating on the case of strictly convex preferences and strictly convex production technology sets. Using strict convexity allowed us to use point-valued supply and demand functions. There are many settings, however, where this mathematically simple formulation seems inappropriate economically; for example, when there are perfect substitutes in consumption or when production technologies are linear. In these cases, where weak rather than strict convexity holds, supply and demand relations appear to be set valued. Figure G.1 presents the example of a firm with a linear production technology and the resulting set-valued supply function. Figure G.2 shows the case of a consumer choosing between perfect substitutes with the resulting set-valued demand behavior. It is important in these examples that preferences and technology be convex, even though they are not strictly convex. That assures us that a household demand or a firm supply at given prices can be characterized as a convex set. Figure G.3 depicts, in partial equilibrium, typical resulting supply and demand curves and possible market equilibria.

We need a mathematical treatment that will allow us to deal with this additional complexity. Fortunately, there is an available theory of continuous point-to-set mappings that fully parallels the theory of continuous functions. We will develop concepts of continuity and a fixed-point theorem that will allow us to duplicate, in the more general setting of set-valued supply and demand, the results on existence of equilibrium we developed for point-valued supply and demand in Chapters 11–18.

Commodities and prices

We've seen examples of general equilibrium economic systems characterized by N commodities for N = 2 (Robinson Crusoe in Chapter 2; Edgeworth box in Chapter 3), N = 4 (2 × 2 × 2) in Chapter 4, and arbitrary positive integer N in Chapter 5. Chapters 6 through 9 summarized the mathematics suitable for analyzing these economies using RN as the commodity space. To represent a list of quantities of N goods, we'll use a point in RN. The expression x = (x1, x2, x3, …, xN) represents a commodity bundle. That is, x is a shopping list: x1 of good 1, x2 of good 2, and so forth through xN of good N. The coordinates xn (n = 1, 2, …, N) may be either positive or negative (subject to interpretation).

The price system consists of an N-tuple p = (p1, p2, …, pN). Let pn ≥ 0 for all n = 1, …, N. The value of a bundle x ∈ RN at prices p is p · x.

What are these N commodities? That turns out to be rather a deeper question than it appears, so a full discussion will be postponed until Chapter 20.

The formal structure of pure economic theory

The plan for the rest of this book is to develop a formal mathematical model of a market economy. Professor Debreu describes below some of the strengths of this approach.

We have covered a classic array of topics in this volume: formulation of the general equilibrium model, existence of general equilibrium, efficiency of general equilibrium, the core of a market economy, and futures and contingent commodity markets.

Overview and summary

The ideas treated in this volume have focused on a single unifying idea as a framework for analyzing economic activity: the general equilibrium of a competitive market. General equilibrium – treating all markets and their interactions simultaneously – is thought here to be the appropriate model to decide whether there are well-defined solutions to the economic decision-making mechanism and whether they have efficiency properties making them desirable. This class of questions goes back well over 200 years in scientific economics. The way we answer them here is the Arrow-Debreu version of the Walrasian economic model (Arrow and Debreu [1954], Arrow [1951]).

We have come to several principal conclusions:

1. (i) The models have well-defined solutions; equilibria exist. Sufficient conditions for this result are scarcity, continuity, and convexity on both the consumer and producer sides. This is true both in settings where demand and supply are characterized as point-valued functions and optimizing behavior is uniquely well defined (Theorems 5.1, 14.1, and 18.1) and where they are characterized as set-valued mappings (Theorem 24.7) recognizing the multiplicity of equally profitable production plans a firm may have or the variety of equally satisfactory affordable consumption plans households may consider.

2. […]

The simplest general economic equilibrium system we can consider consists of a single household, usually named Robinson Crusoe. This one-person economy has many of the usual problems of any economy: production and consumption choices. The simple structure of the economy allows us fully to model a single centralized family of efficient allocation decisions. We can then, somewhat artificially, decompose the one-person economy into separate production and consumption sectors interacting through a market mechanism. This is a common classroom exercise, designed to illustrate the concepts of efficient allocation, general equilibrium, and decentralization through a market mechanism. In the one-person economy it is particularly easy to present the concept of efficient allocation. Because there is only one agent, there is a unique maximand (the utility function of the lone household/person/agent). The efficiency concept is simply to maximize Robinson's utility subject to the available resources and technology. Problems of distribution among individuals (regarding both considerations of efficiency and fairness) do not arise because there is only one household.

The exercise we perform in the Robinson Crusoe model is to solve two apparently quite separate problems and then show that they are nearly identical. First, we will solve for an efficient allocation in the Robinson Crusoe economy. This is a centralized solution concept because it treats the consumption and production decision in a single unified fashion. That is, we find a production and consumption plan that maximizes Robinson's utility subject to the constraints of available resources and technology.

In Chapters 11 through 14 we will develop a version of the complete Arrow-Debreu model of the economy. The theory of the firm and production sector is presented in Chapter 11 and that of households and demand in Chapter 12. We bring them together with Walras's Law in Chapter 13 and the existence of general equilibrium in Chapter 14.

As we noted in Chapter 7, the typical characterization of economic activity of firms and households is as a maximization subject to constraint. Recall Corollary 7.2 to Theorem 7.5. For maximization to be well defined, sufficient conditions are that the maximand be a continuous function of its arguments and that the opportunity set be compact. That pretty well sets the agenda for characterizing firm and household behavior. We have to find continuous functions for them to maximize. We should find compact constraint sets for them to do it on. That will characterize firm supply and household demand behavior. Although these are not necessary conditions, they are the best generally sufficient conditions available.

Finding continuous functions for the firm and household to optimize does not pose a problem. For the firm, the obvious choice is profits. For the household, the traditional maximand is utility, though we will go to some effort to derive the continuous utility function from the more primitive assumption of a preference ordering. The obvious constraint set for the firm is a representation of the firm's technically available possibilities – the possible input-output combinations based on available technology represented as a subset of RN.

General equilibrium with set-valued supply and demand

Our plan in this chapter is to take the model of production, consumption, the economy, and market equilibrium of Chapters 15–18 and restate it for the case of set-valued demand and supply behavior. Formally this means that we dispense with assumptions of strict convexity of tastes and production technology, C.VI(SC) and P.V. We rely rather on convexity, C.VI(C) and P.I. Under the remaining assumptions on consumption and production behavior, this will allow us to characterize demand and supply behavior as upper hemicontinuous, convex-valued correspondences. In turn, excess demand will then be characterized as upper hemicontinuous and convex-valued. A model of price adjustment that is also upper hemicontinuous and convex valued completes the picture: Applying the Kakutani Fixed-Point Theorem allows us to find a fixed point in price space that achieves a market equilibrium.

Just as we did in Chapters 15–18, we treat the economy in two formats: an artificially restricted bounded economy denoted by the superscript tilde notation (~) and an unrestricted economy (representing the true model we are really interested in). The artificially restricted economy is a purely technical construct, designed to allow us to develop the properties of the underlying unrestricted economy in a more tractable setting. The technique of the proof is to note that the restricted budget, demand, supply, and profit behavior is always well defined because it represents optimizing behavior on a compact set.

The Brouwer Fixed-Point Theorem is a profound and powerful result. It turns out to be essential in proving the existence of general equilibrium. We have already seen that it is convenient (in Chapter 5), but it can be shown to be indispensable (Chapter 18).

The Brouwer Fixed-Point Theorem says that a continuous function from a compact convex set into itself has a fixed point. There is at least one point that is left unchanged by the mapping. Note that the convexity is essential. For example, the fixed point property is not true (and the theorem is inapplicable) for a function mapping the circumference of a circle into itself. Indeed, typical of well-constructed mathematical results, all of the assumptions are essential. The fixed-point property will not hold for a discontinuous function or on an open or unbounded set.

In R, the Brouwer Fixed-Point Theorem takes a particularly simple form, equivalent to the Intermediate Value Theorem. Let f map the closed interval [a, b] into itself. Then the theorem is equivalent to the assertion that every continuous curve y = f(x) from one side of the square [a, b] × [a, b] to the opposite side must intersect the diagonal (the line y = x). See Figure 9.1.

Economic applications do not require that the economist know or understand the proof of the Brouwer Theorem.

Bargaining and competition

The model we have been using so far is competitive in a rather refined sense. All agents act as price takers. They treat prices parametrically, as variables that they cannot control and to which they must adapt. The prices themselves are set by an impersonal market mechanism (idealized as the Walrasian auctioneer). The assumption that individual buyers and sellers are powerless to affect market prices reflects one idea of the notion of competition, that the market is so large that individual actions have no impact. But that makes up only half of what we mean by competition. In ordinary usage, we say competition occurs when each economic agent tries to do as well as possible by making the most advantageous deals he can. This is the idea of competition as conflict. One of the major achievements of modern general equilibrium theory is that we can demonstrate formally that these two notions of competition are equivalent. We can show mathematically that a model of bargaining and deal making where each buyer and seller tries to get the best deal possible leads to a price-taking equilibrium in a large economy. Hence, we can demonstrate the soundness of the informal notion that large economies leave individuals strategically powerless. We will present a concept of the outcome of strategic bargaining known as the core of the market economy. The core appeared in Chapter 3 as the contract curve.