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Gerard Debreu: the general equilibrium model (1921-2005) in memoriam.

The world-renowned Nobel Laureate mathematical economist Gerard Debreu passed away on December 31, 2004 at the age of 83. The economic profession will long grieve the loss of a genius of his caliber, who was well known for bringing the rigor of mathematics to economics.

I. Introduction and Background

Debreu, the son of Camille Debreu and Fernande (nee Decharne) Debreu, was born on July 4, 1921, in Calais, France. In 1941, he attended the Ecole Normale Superieure, where he studied and lived until the spring of 1944. He studied mathematics under the famous Henri Cartan and became attracted to Walrasian economics through the 1943 work of the future Nobel Laureate Maurice Allais, A la Recherche d'une Discipline Economique. In the summer of 1948 he was further influenced by a seminar given by another future Nobel Laureate Wassily Leontief. In 1949, he visited Harvard, Berkeley, Chicago, and Columbia Universities on a Rockefeller Fellowship. In the same year, he joined the Cowles Commission, which was attached to the University of Chicago at the time, where he gained a decade of experience working on Pareto optima, existence of a general economic equilibrium, and utility theory. He followed the commission when it moved to Yale University in 1955. He spent 1960 to 1961 at Stanford University, and then moved to the University of California, Berkeley in 1962, where he remained until his retirement in 1991. In 1975, he became a citizen of the U. S., with all the credits and renown of a learned person. While at Berkeley, Debreu lived in Walnut Creek, CA. He is survived by his wife, Francoise Debreu of Walnut Creek, and his two daughters, Chantal De Soto of Aptos, California, and Florence Tetrault of Vancouver, British Columbia.

Debreu's fame began when he turned his critical mind from pure mathematics toward its application to economics. The area where he made his mark is called General Equilibrium (GE), which, according to several modern economic methodologists, is "the most prestigious economics of all and it has absorbed an entire generation of some of the finest minds in modern economics" (Marchi and Blaug 1991, 508-509). GE models come in different versions, but we will focus on the popular Arrow-Debreu model. Debreu's work on that model earned him the Nobel Prize in Economics in 1983 for introducing analytical and mathematical rigor in the reformulation of the theory of general equilibrium.

The determination of price and output for the economy goes back to Adam Smith, who did not give his system a mathematical treatment. Many well-known mathematical economists have tried their hands at a GE system that would determine price and output. The best known mathematical treatment occurred around the marginal revolution in 1870, with the works of Carl Menger, the least mathematical of the marginal school; Leon Walras, the most mathematical; and Stanley Jevons, who proposed that "economics, if it is to be a science at all must be a mathematical science" (Jevons 1970, 78). For Walras, "Pure economics is, in essence, the theory of the determination of prices under a hypothetical regime of perfectly free competition" (Walras 1954, 40). He held that "This whole theory is mathematical.... Nor can we understand without mathematics why or how current equilibrium prices are arrived at not only in exchange, but also in production" (Ibid., 43).

Debreu improved upon the old Walrasian approach. Walras' system had total demand and total supply of factors (2m variables), and the demand for commodities and their prices (2n variables), for a total of 2n+2m variables and equations. Walras used a numeraire system, setting the price of one commodity equal to one. He also estimated the value of one variable from what is called Walras' Law. In the end, he had 2m+2n-1 independent equations and variables. We know that a "theory would be determinate if its equilibria could be expressed as the zeroes of a system having the same number of equations as of unknowns" (Mas-Colell 1989, 175). Paul A. Samuelson illustrated this with a system in which many individuals each have one apple and three oranges, and a taste that dictates they spend half their income on apples and half on oranges. "Having lived as a scholar in both the pre-and post-Debreu era, I can testify that the modern proofs are better than what used to pass muster for demonstration of determinate economic equilibrium. Here is how our wave-of-the-hand expositions used to go. We used to count our number of unknowns--in this case one unknown price ratio. And then we counted our number of independent equations--in this case the function for [p.sub.a]/[p.sub.o] representing aggregate supply of apples be equated to the function representing aggregate demand for apples" (Samuelson 1986, 839).

The Walrasian system had all the necessary conditions for a solution, ruling out the trivial solution of zero. However, it could lead to negative values, which do not have any meaning in economics. Also, Walras relied on a tatonnement (search) process to eliminate excess demand and supply by at least three ways--through an auctioneer, through contracting and re-contracting, and through the nullification of old contracts (Negishi 1989, 281). According to Knut Wicksell, "he clothed in a mathematical formula the very arguments which he considered insufficient when they were expressed in ordinary language" (Robinson 1964, 55-56). Mathematics began to have a bad name in the profession. "[Alfred] Marshall in his own way also rather pooh-poohed the use of mathematics. But he regarded it as a way of arriving at the truths, but not as a good way of communicating such truths" (Samuelson 1966, 1755). In the 1930s, however, mathematics started to make a comeback in the hands of stalwarts such as John R. Hicks, Samuelson, and others.

In his Nobel Laureate lecture and other sources, Debreu said: "To some-body trained in the uncompromising rigor of Bourbaki, counting equations and unknowns in the Walrasian system could not be satisfactory, and the nagging question of existence was posed" (Debreu 1992, 89). Then, "In the summer of 1950, [Kenneth] Arrow, at the Second Berkeley Symposium on Mathematical Statistics and Probability, and I, at a meeting of the Econometric Society at Harvard, separately treated the same problem by means of the theory of convex sets" (Ibid.).

II. Debreu's Description of the GE Model

A GE model starts with proper care in the definition of a commodity. A commodity has spatio-temporal and physically differentiated characteristics, which can be represented by a symbol, 1, in a Euclidian space, [R.sup.1]. As Debreu puts it, "By focusing attention on changes of dates one obtains ... a theory of savings, investment, capital, and interest. Similarly by focusing attention on changes of locations one obtains ... a theory of location, transportation, international trade and exchange" (Debreu 1959, 32). He later explained that Arrow's uncertainty assumption has shifted the paradigm away "from a simple reinterpretation of a primitive concept" to "a novel interpretation of the same primitive concept" that allows for "unknown choices that nature will make from the set of possible states of the world" (Hildenbrand 1989, 5).

Debreu then specified the agents, consumers and producers, and characteristics. "For any economic agent a complete plan of action ... is a specification for each commodity of the quantity that he will make available or that will be made available to him, i.e., a complete listing of the quantities of his inputs and of his outputs" (Debreu 1959, 32). "A complete description of an economy ... consists of: For each consumer, his consumption set, ... and his preference ordering.... For each producer, his production set.... The total resources" (Ibid., 75). A state of the economy is "a specification of the action of each agent," and net demand is formed from canceling out "all commodity transfers between agents of the economy" (Ibid., 75). If one also subtracts the agent's resources from net demand, one gets excess demand.

"A state is called a market equilibrium if its excess demand is 0" (Ibid., 76). More generally, equilibrium exists if "(a) ... every consumer has chosen in his consumption set a consumption that satisfies his preferences best under his budget constraint, ... (b) ... every producer has maximized his profit in his production set ... (c) ... for every commodity the excess of demand over supply is zero. The equilibrium defined by conditions (a), (b), and (c) is competitive in the sense that every agent behaves as if he had no influence on prices and considers them as given when choosing his own action" (Debreu 1982, 704).

III. Existence of Equilibrium

In one of his frequent surveys, Debreu noted that "four distinct, but closely related, approaches to the existence problem can be recognized. (1) At first, proofs of existence of an economic equilibrium were uniformly obtained by application of a fixed-point theorem of the [L. E. J.] Brouwer type or of the [Shizuo] Kakutani type or by analogous arguments ... (2) In the last decade, efficient algorithms of a combinatorial nature for the computation of an approximate economic equilibrium were developed ... (3) More recently, the theory of the fixed-point index of a map and the degree theory of maps were used.... (4) Finally, in 1976 [Stephen] Smale proposed a differential process whose generic convergence to an economics equilibrium provides an alternative constructive solution of the existence problem" (Debreu 1982, 677-678). A discussion with Professor Robert Anderson, UC Berkeley Mathematical Economics Department, underscores that these four stages have more or less remained intact, covering the popular areas of existence of equilibrium, namely, fixed points, core theory, Debreu-Scarf works, and welfare theorems. We will be concerned with mainly the first one, for, to paraphrase Professor Anderson, Debreu's work graduated progressively from undergraduate level to more entrenched graduate level research. Even with this restriction, Professor Anderson warns us about the difficulty of entering the dimensions higher than two.

Debreu explained how he got into the fixed-point solution from the mathematical side, including game theory. "I learned about the Lemma in von Neumann's article of 1937 on growth theory that Shizuo Kakutani reformulated in 1941 as a fixed point theorem. I also learned about the applications of Kakutani's theorem made by John Nash in his one-page note of 1950 on 'Equilibrium Points in N-Person Games' and by Morton Slater in his unpublished paper, also of 1950, on Lagrange multipliers. Again there was an ideal tool, this time Kakutani's theorem, for the proof that I gave in 1952 of the existence of a social equilibrium generalizing Nash's result. Since the transposition from the case of two agents to the case of n agents is immediate, we shall consider only the former which lends itself to a diagrammatic representation. Let the first agent choose an action [a.sub.1] in the a priori given set [A.sub.1], and the second agent choose an action [a.sub.2] in the a priori given set [A.sub.2]. Knowing [a.sub.2], the first agent has a set [[mu].sub.1]([a.sub.2]) of equivalent reactions. Similarly, knowing [a.sub.1], the second agent has a set [[mu].sub.2]([a.sub.1]) of equivalent reactions. [[mu].sub.1]([a.sub.2]) and [[mu].sub.2]([a.sub.1]) may be one-element sets, but in the important case of an economy with some producers operating under constant returns to scale, they will not be. The state a = ([a.sub.1],[a.sub.2]) is an equilibrium if and only if [a.sub.1] [member of] [[mu].sub.1]([a.sub.2]) and as [a.sub.2] [member of] [[mu].sub.2]([a.sub.1]), that is if and only if a [member of] [mu](a) = [[mu].sub.1]([a.sub.2]) x [[mu].sub.2]([a.sub.1])" (Debreu 1983, 90-91).

"In our article of 1954, Arrow and I cast a competitive economy in the form of a social system of the preceding type.... In this manner a proof of existence, resting ultimately on Kakutani's theorem, was obtained for an equilibrium of an economy made up of interacting consumers and producers.... In the early fifties, the time had undoubtedly come for solutions of the existence problem. In addition to the work of Arrow and me, begun independently and completed jointly, Lionel McKenzie at Duke University proved the existence of an 'Equilibrium in Graham's Model of World Trade and Other Competitive Systems' [1954], also using Kakutani's theorem. A different approach taken independently by David Gale ... in Copenhagen, Hukukane Nikaido [1956] in Tokyo, and Debreu [1956] in Chicago permitted the substantial simplification given in my Theory of Value [1959] of the complex proof of Arrow and Debreu" (Ibid., 91).

It is clear that a substantial mastery of mathematical tools is needed before one can get a feel for Debreu's contribution to GE. In order to lay the groundwork for the readers to get an intuitive feeling of Debreu's contribution, we take the following approach in the rest of this memoriam: Part IV takes up some disagreements over GE, and Part V speculates about the future of GE.

IIIa. Debreu's Tools of the Trade

In his Presentation of the Nobel Laureate Speech, Professor Karl-Goran Maler of the Royal Academy of Sciences said: "In the development of the general equilibrium theory, Professor Debreu has not merely given us information about the price mechanism, but also introduced new analytical techniques, new tools in the toolbox of economists. Gerard Debreu symbolizes the use of a new mathematical apparatus, an apparatus comprehended by most economists only abstractly. Nevertheless, his work has given us an improved intuitive understanding of the underlying economic relevance. His clarity and analytical rigor, as well as the distinction drawn by him between an economic theory and its interpretation, have given his work important bearing on the choice of methods and analytical techniques within economic theory on a par with any other living economist" (Maler, 1992).

Samuelson has said that "Debreu is known for his unpretentious no-nonsense approach to the subject" (Weintraub 2002, 113). It is common to mention that Debreu is from the Bourbaki school of mathematics, and that sends a shiver down the spine of any aspiring economist. Debreu revealed in an interview to E. Roy Weintraub that his early high school education in geometry "called for imagination, intuition, experimentation, and I think that it gave me an excellent education, and a very good preparation for the geometric viewpoint that I have often taken up in my work since" (Weintraub 2002, 127). In 1990, at a public lecture, one of the present authors asked Debreu what his contribution to the tools of his trade had been, and what his unique mathematical contribution was. He replied that he mainly relied on other people's tools. It is difficult to list them all, but his early works are sprinkled with convex set theory, fixed point theorem, and partial ordering; and his later works made use of topology, measure theory, and non-standard analysis. We present in this section some of the intuitive tools that are unique to him, tools that he contributed to economics and that are understandable at an undergraduate level.

IIIb. Convex Sets

A cursory check of the literature reveals that convex analysis goes back to ancient Greece: "This is what Archimedes said 'I give the name convex on one side of those surfaces for which the line joining any two of its points ... will lie on one side of the surface.' This definition has remained virtually unchanged up to the present day" (Gamkrelidze 1980, 3). A popularization of this concept states that "A convex set has the property that a collection that contains two items also contains an average of these two items" (Kay 2004, 179). This is not an unrealistic assumption. "It probably isn't an exaggeration to say that the behavior of market economies depends on how convex the world is. To get a sense of why this is so, imagine dropping a ball into a bowl: it circles round, slows down, and eventually arrives at some sort of equilibrium" because the bowl is convex. Now if the bowl is turned upside down, the points in the bowl are no longer convex, and the position of the ball is unpredictable (Kay 2004, 180).

The space we live in (3-dimensional) is convex because you can join any two points in it by a line. By the same logic, a plane (2-dimensional), such as a sheet of paper, can be separated into two halves, two convex sets, and we can join any two points in each half plane by a line. From that point, it is easy to show that the intersection of convex sets are convex, and to define other convex properties such as convex hull, one-dimensional convex figures which can be a line, a line segment, or a ray.

III.b.1 Consumption Set

An example of a convex set is the consumption set. If X is a set of all possible consumption, then X is not zero, and it lives in the space [R.sup.1] mentioned previously. One concern is to represent a utility function that takes on many variables by a number: U([x.sub.1], [x.sub.2]) = [x.sub.1 x [x.sub.2]. Or more formally: U:X [right arrow] R, where R represents a real number. We can build our intuition of such a utility function by asking a person to rank his or her preference of a bundle of one commodity with a bundle of another. An example of a bundle is [x.sub.1] = (2 grapes, 1 banana), and another would be [x.sub.2] = (5 grapes, 4 bananas). Normally these bundles can be plotted on a (Grapes, Banana, Utility), i.e., a ([x.sub.1], [x.sub.2], U), coordinate system, and utilities are ranked in an ordinal way. Our objective is to assign a number to those rankings, which would make the ranking of utility cardinal. Originally, the concept of utility was first thought out cardinally. For example, when we eat an apple we say we get so many utils. This way we can add up all the utils we get from consuming many different commodities. Debreu, therefore, stands on the shoulders of others in wanting to strengthen the foundation of the old cardinal utility concept. Such as concept is useful for instance in mathematical programming (Mas-Colell et al. 1995, 46).

The cardinal utility concept is therefore born out of a preference relationship. On X, the commodity space, we want a binary relation. A binary relation allows a pair-wise comparison of commodities. To get a feel for this, if we wish to say that the weight of Albert, w(A), is not greater than the weight of Ben, w(B), then we can write w(A) [less than or equal to] w(B). This would imply that Albert, A, is not heavier than Ben, B. Consumers are therefore comparing (ordering or ranking) their bundles, [x.sub.1] and [x.sub.2], in a pair-wise or binary manner. If [x.sub.1] is "at least as good as" [x.sub.2], we will use the symbol [??], and if they are indifferent, we will use the symbol ~. A utility function U:X [right arrow] R, where R represents real numbers, represents a preference relation of the set X if certain conditions hold, i.e. [x.sub.1], [x.sub.2] [member of] R, [x.sub.1] [??] [x.sub.2] [??] U([x.sub.1]) [greater than or equal to] U([x.sub.2]). What do we need to construct such a utility function? First, on the preference side we require that the agents be rational, which means that the consumers should make a complete ranking of all bundles, and the ranking should be transitive. Additionally, because it was found that lexicographic preference relations are discontinuous, it is necessary to add that the utility function be continuous as well. Continuity means that if we are told that a series of consumption bundles are "at least as good as" a given consumption, then the limit of the sequence will have that property. For convenience, we can make some more assumptions: (1) assume that "more is better"; this will imply that people are non-satiated, (2) assume free disposal, i.e. consumers can get rid of goods costlessly, and (3) assume convexity, implying that a mixture of two extreme bundles is preferred to the extremes, and that consumption is divisible. With these assumptions we can have an intuitive feel of how to construct a utility function.

The feel goes like this: let us decide to assign utility to only bundles that are on an indifference curve. We can find these indifference curves by traveling on a line from the origin of a diagram, 0, or the worst case consumption set, outwards to meet an indifference curve. If we do not use the convexity assumption, then the indifference curve can be winding, but that is not germane to the proof.

Take a two dimensional coordinate system, ([x.sub.1], [x.sub.2]). Plot the point e = (1,1), a point on the diagonal. Now, we can move on the diagonal by multiplying e by a number, say t, to get t x e. The number t can be less than or greater than one, which will make us move on the diagonal line e. Now draw in an indifference curve to meet the t x e line. We have decided to call this meeting our utility function, i.e. U(x) = t x e.

We need to check that the preference assumption holds. Given that t(x.sub.1]) [greater than or equal to] t([x.sub.2]) iff [x.sub.1] [??] [x.sub.2]. First, we can substitute t([x.sub.1]) x e and t([x.sub.2]) x e for t([x.sub.1]) and t([x.sub.2]). Second, since t([x.sub.1]) x e ~ [x.sub.1] and t([x.sub.2]) x e ~ [x.sub.2], we can substitute [x.sub.1], and [x.sub.2] in to get [x.sub.1] [??] [x.sub.2].

Conversely, let's start with [x.sub.1] [??] [x.sub.2]. First, since t([x.sub.1]) x e ~ [x.sub.1] and t([x.sub.2]) x e ~ [x.sub.2], substitute t([x.sub.1]) x e and t([x.sub.2])- e for x and [x.sub.2]. Second, dropping the e's in the expressions yields t([x.sub.1]) [greater than or equal to] t([x.sub.2]). So, we have shown that t([x.sub.1]) [greater than or equal to] t([x.sub.2]) iff [x.sub.1] [??] [x.sub.2].

Check that the continuity assumption holds. This is much more difficult to do. But, intuitively, we need to show that distance is a continuous concept. We indicate distance from the origin to t x e by the bi-directional arrow in Figure 1. In symbolic form, we can write [parallel]t(x) x e[parallel] for distance. So, U(x) = [parallel]t(x) x e[parallel]. We need to show that a series of x's and distances converge to the same point x on the indifference curve. Since points above and below the indifference curve includes the indifference curve, then the distance on the ray is closed. Therefore, the limit points of the series are closed. Closeness and limit points of the sequences, therefore, imply continuity.

[FIGURE 1 OMITTED]

IIIc. Separating Planes

In economics, we encounter disjointed convex sets, such as consumption and production sets, to be separated by a plane, representing the budget line in 2-dimensions. Debreu defined separation of planes as follows: "A hyperplane H is separating for two sets A, B if A is contained in one of the closed half-spaces determined by H and B in the other" (Debreu 1959a, 95). Debreu proceeded to give an intuitive proof of separation using Figure 2.

[FIGURE 2 OMITTED]

We give an intuitive explanation of Debreu's diagram that, while not fit for a mathematician, may be worthwhile for a learner. Figure 2 uses the technique of proof by contradiction. The hyperplane H separates the halfplane A, above, from the halfplane B, below. Keep in mind that we want to prove that "the hyperplane H through x', perpendicular to [x.sup.o] x' is separating for A and [x.sup.o]" (Debreu 1959a, 95).

We start out with the assumption that the distance from [x.sup.o] to x', d([x.sup.o],x'), is the shortest distance you can find from of all points x in A, to [x.sup.o] in B. Thus defined, any other point, x, in A will be further[x.sup.o] from [x.sup.o] in B than x' in A. Let us make the argument that there is a point x" in A, but that it lies below H, i.e., it lies on the same side of H with [x.sup.o]. Another way of saying this is that the distance from [x.sup.o] to x", d([x.sup.o],x"), is greater than or equal to, [greater than or equal to], the distance from [x.sup.o] to x', d([x.sup.o],x'). How would this be a contradiction to the assumption that x' is the closest point in A to [x.sup.o] in B? It is a contradiction because we can now see that [bar.x] would be in A and closer to [x.sup.o] than x' is away from [x.sup.o]. In other words, d([x.sup.o], x') [greater than or equal to] d([x.sup.o], [bar.x]). As we have started with the assumption that d([x.sup.o],x') was the shortest distance from [x.sup.o] in B to x' in A, we have reached a contradiction.

What is Debreu's proof intuitive to? It is intuitive to, for instance, an earlier proof, still prevalent in textbooks, that was introduced by John von Neumann and Oskar Morgenstern (Neumann et al. 1944). Neumann et al. state that "We must find a correspondence between utilities and numbers which carries the relation u > v and the operations [alpha]u + (1 - [alpha])v for utilities into the synonymous concepts for numbers.... Denote the correspondence by ... u [right arrow] p = v(u) ... u being the utility and v(u) the number which the correspondence attaches to it. Our requirements are then: ... u > v implies v(u) > v(v)" (Neumann et al. 1944, 24).

A modern attempt to generalize Debreu's intuitive concept goes as follows: Consider the stick person in Figure 3. The head (H) is compact, i.e. closed and bounded. The body (B) is closed. Let the minimum point in H at the arrowhead be h', and the maximum point in B at the arrow tail be b'. The separation theorem says, for instance, that there is a vector orthogonal to the point b' on the arm such that for any h in H, ph' [less than or equal to] ph, and for any b in B, pb' [greater than or equal to] pb. If we denote the length of the neck by p = h'-b', the distance of the neck from the head to where it meets the shoulder, then we get p.p = p(h'-b'). This yields ph' > pb' because distance is positive. We want to show that for any point b in B, pb' [greater than or equal to] pb, and for any point h in H, ph' [less than or equal to] ph. Because the head and body are convex, we can join any points in them by a line. In the case of the body, join b and b' this way: [b.sup.*] = (1 - a)b' + ab, where a is a percent between 0 and 1. This line will be contained in the body by the convexity assumption. Now, if you form [[absolute value of h'- [b.sup.*]].sup.2], then, with some manipulation, you get pb' [greater than or equal to] pb. A similar analysis will yield ph' [less than or equal to] ph in the head area. Then it's possible to slip a hyperplane between these sets.

[FIGURE 3 OMITTED]

IIId. Fixed Point Theorems

A story was told by Sir Karl Popper to give the feeling that absolute truth exists. Popper was in the company of Alfred Tarski, and they were on a mountain-climbing mission. The day was foggy, with the fog completely covering the mountain peak. Tarski took advantage of the situation to indicate the existence of absolute truth, at least his brand of it. He said that, just because you cannot see the peak, you nevertheless feel intuitively that the peak exists. Similarly, Debreu has used Brouwer's and Kukutani's fixed point theorems to show that GE prices exist. This was a fruitful exercise because, later on, researchers such as Herbert E. Scarf gave algorithms to compute those prices. Today, by the method of complementarity, the world is swamped with computational general equilibrium analysis (CGE). For instance, the International Trade Council has developed a CGE model, which it used to advise President Clinton about the welfare gains from trade to be attained from the U.S. joining NAFFA. Also, the WTO has its brand of CGE models to determine the potential benefits from forming the FTAA on world economies. Private researchers and institutions have developed databases and computer software to make this kind of analysis routine, even though GE models are still being pursued in a rigorous manner.

Professor Debreu, in one of his last set of lectures, explained fixed points this way: We are concerned with mapping points on, say, a rubber sheet to other points on the rubber sheet. Stretching the rubber sheet, thereby pulling points from where they were to other positions, can do this but we should not pull in a manner that would tear the rubber sheet. The idea of a fixed point is that, in the process of stretching the rubber sheet, one point will not move. It will be mapped onto itself. For those of you familiar with demand and supply curves, here is another intuitive feel: Imagine the demand and supply curves are two separate graphs. Mark off two quantities, a supply quantity, [Q.sub.s], and a demand quantity, [Q.sub.d]. Then trace up from the two quantity axes to the demand and the supply curves to find two prices, say a supply price, Ps, and a demand price, Pd. If we can find a function, a map, of f([P.sub.s]) = [P.sub.d] or f([Q.sub.s]) = Qd, then we have a fixed point mapping for the equilibrium problem (Baumol 1965, 494). (Baumol claimed that this is the direction Lionel Mackenzie took when he originally proposed his contribution in 1954.) For higher dimensions, the concept of the "hairy-ball" problem gives a similar intuition. Loosely speaking, try to move every hair on you head from its current position to the position of another hair. The fixed-point theorems say that one hair cannot be so moved. It will have to stay fixed.

In writing about Debreu, Samuelson was drawn into the fixed point solution from the two dimensional point-of-view as well. From a simple function, Samuelson explained the intuitive concept of a fixed point as follows: "draw a square and pencil in the 45-degree diagonal connecting its southwest and northeast comers. Is there any way to draw a curve that goes from the square's east side to its west side without taking pencil off the paper, such that the curve and the diagonal have no single point in common? Brower's fixed-point theorem in one dimension proves that there is indeed no possible way. Similarly, under specified conditions about people's tastes and goods endowments, there is no way for curves of supply and demand to be drawn without having at least one intersection point in common" (Samuelson 1986, 839).

IIIe. Continuity

Continuity is generally defined from the point of view that f(x) [right arrow] f([x.sub.0]) as x [right arrow] + [x.sub.0]. This method is used for Brouwer's fixed point theorem, which is defined as follows: If a theorem says that if fix) is a continuous point-to-point mapping of a closed set, S into itself, then there exists a point x in S such that x = f(x). However, we will be dealing with point-to-set mapping where f(x) and f([x.sub.0]) take on sets of values, as in the Kukutani fixed point theorem, which generalizes Brouwer's theorem to a point-to-set mapping. Kukutani's theorem states that if S is closed, and [phi] is an upper semicontinuous mapping from within S to a closed convex subset of x, then there exists a point x in S such that x is in [phi](x) (Karlin 1959, 408-409).

In order to get a feel for upper semicontinuity, let us consider the point-to-set mapping: Y = {y| 1/3x [less than or equal to] y [less than or equal to] x}. We are pulling elements of x from a domain set, say B, and they go to a subset of image points y in A. If we restrict the mapping to the interval [0,1], we can imagine the image of the mapping to be the area between the equation y = x, that is the diagonal example that Samuelson used above, and the equation y = 1/3 x. Now, our focus is on what happens as x [right arrow] [x.sub.0]. There will be an image set Y([x.sub.0]), and a set S of limit points indicating all the approach paths of sequences in the image set as x [right arrow]) [x.sub.0]. If S is a subset of Y([x.sub.0]), the mapping is upper semicontinuous (Lancaster 1968, 347).

IIIf. Existence of Equilibrium

Hildenbrand wrote that "when Debreu began to write his Theory of Value in 1954 he based the existence proof on a result that today is called the 'fundamental lemma'.... It was proved independently of Debreu by D. Gale and H. Nikaido" (Hildenbrand 1989, 20). The description we now give is from Nikaido (1956). Start with a referee setting the price, R Consumers will maximize their utility, [U.sub.i](x), subject to PX = PA, where A is endowment and goods X is in space E = {X| 0 [less than or equal to] X [less than or equal to] C), C being an arbitrary bundle such that C > A. All acceptable bundles with respect to P are labeled [[phi].sub.i] (P), the ith individual demand function. Its sum is just [phi](P). If total demand, X, does not match total available bundles, A, the referee must make an adjustment. The difference is X-A. Its value is P(X-A). The referee's objective is to pay a person a value PX that is greater than the endowment value PA. In other words, choose a price, Q, that will maximize the price-manipulating function: [theta]X= {P | P(X-A) = max Q(X-A) for all Q in [S.sup.k]}, where X is total demand lying in [GAMMA].

We now have a demand function and a price-adjustment function.

[S.sup.k] [contains as member] P [right arrow]) [phi](P) [subset] [GAMMA] (Demand function)

[GAMMA] [contains as member] X [right arrow] [theta](X) [subset] [S.sup.k] (Price-manipulating function)

We want to choose (X,P) in this Demand and Price adjustment space, [GAMMA] x [S.sup.k], so that [phi](P) x [theta](X) is contained in [GAMMA] x [S.sup.k] This is possible because the mapping is upper semicontinuous. Therefore, the equilibrium price exists.

Upper Semicontinuity is easy to show for [theta](X). Given [P.sub.n] [right arrow] P in [S.sup.k], [X.sub.n] [right arrow] X in [GAMMA] and [P.sub.n] [member of] [theta]([X.sub.n]), then P [member of] [theta](X). The proof is that, for any Q price-constellation, [P.sub.n] ([X.sub.n] - A) [greater than or equal to] Q([X.sub.n] - A), and, for the whole sequence, n [right arrow] [infinity], p (X - A) [greater than or equal to] Q(X - A). Therefore, P [member of] [theta](X). The reader can check the original source for the rest of the continuous tests.

IIIg. A Concrete Example

We wish to end with a concrete example of how GE is entering the modern textbooks of economics. We illustrate this with the following two dimensional model that is now widely used in many textbooks to build a feeling for Debreu's solution (Aliprantis et al. 1990). The economy has three agents: 1, 2, 3, and two commodities: x, y, with endowment ([w.sub.1], [w.sub.2], [w.sub.3]) = {(1,2), (1,1), (2,3)}, and utility functions [U.sub.1] = xy, [U.sub.2] = [x.sup.2] y, and [U.sub.3] = (x[y.sup.2]). Using the Lagrange Multiplier for agent 1 yields: y = [lambda][p.sub.1], x = [lambda][p.sub.2], and [p.sub.1] x + [p.sub.2]y = [p.sub.1] + 2[p.sub.2] (Aliprantis et al. 1990, 34-36). We can now find demand for agent 1: [x.sub.1](p) = {([p.sub.1] + 2[p.sub.2])/2[p.sub.1], ([p.sub.1] + 2[p.sub.2])]2[p.sub.2]}. Repeat the procedure to find [x.sub.2](p) and [x.sub.3](p). If we add the three demands, we get total demand, z(p), and by subtracting the total endowment, (4,6), we get excess demand. The equilibrium price will be [p.sup.e] = (0.45, 0.55), approximately.

IV. Disagreements about GE

Disagreements range from complete rejection of GE models to pointing out damning criticisms with the scientifically honest view of making the model a progressive research program. We look at the latter.

IVa. Paradigm vs. Theory

From a methodological perspective, Mark Blaug has told of Debreu's place in the mathematical schema somewhat differently. For him, Kenneth Arrow and Frank Hahn have committed a "category mistake" by saying that Walras wanted to make precise the Smithian doctrine (Marchi and Blaug 1991, 508). Walras' contribution is a "framework" or "paradigm" rather than a "theory." It does not have empirical testable propositions. It was almost still-born: it "went into decline almost as soon as Walras had formulated it" (Ibid., 1991, 507). It started to come out of oblivion by the works of John Hicks, Paul Samuelson, and Oskar Lange in the 1930s and "came to the very forefront of economic theorizing in the famous Arrow/Debreu articles of the 1950s" (Ibid., 1991, 507). One of the hallmarks of that framework is that it could be turned into "operationally meaningful theorems," which can make it testable, falsifiable, or verifiable in the form of, say, Computational General Equilibrium (CGE) models, Leontief's input-output form, or Maynard Keynes' IS-LM models (Ibid., 506). It is said that "Debreu and others have made significant contributions to the understanding of Keynesian economies just by describing so precisely what would have to be the case if there were to be no Keynesian problems" (Hahn 1984, 65).

IVb. The Aim of GE

What GE is about is not without controversy. Some say it is about the finding of a set of equilibrium prices that are stable and unique, given the smallest set of assumptions.

But "an Arrow-Debreu equilibrium may exist when there are increasing returns ... it is perfectly possible for an Arrow-Debreu equilibrium to exist even though the axioms of the theory are violated" (Hahn 1984, 51). In other words, one needs the "absence of significant economies of scale in production" to show that "equilibrium prices can indeed be found" (Ibid., 74). Joseph Stiglitz, for instance, evolved a parallel GE universe where information is central. He steered the research from a competitive to an information paradigm (Stiglitz 2004, 35). Others have tried to accommodate risk, irrationality, and cooperation rather than competition (Kay 2004, 207). The big question is how does one decide to give up a theory? By the realism of its assumption? By a fixed number of times that it fails? By its lack of operationally meaningful theorems? When one of its additional assumptions fails? We do not have a definitive answer. Like sophisticated falsificationists, Debreu and others have continued to work with the GE model to answer new and anomalous questions. We give a brief review of some of these attempts below.

IVc. Weakness of GE

Frank Hahn stated that it is incorrect to "claim that Debreu was looking for the 'minimum basic assumptions for establishing the existence of an equilibrium set of prices which is (a) unique, (b) stable.' Debreu did not concern himself with either" (Hahn 1984, 48). Hahn overarching point seems to be that "it is undesirable to have an equilibrium notion in which information is as perfect and as costless as it is in Arrow-Debreu" (Hahn 1984, 53). He mentioned works that were done to (1) accommodate available information, (2) examine the extent that prices are efficient information signals, (3) study transaction possibilities that may be costly in a sequential economy, (4) recognize stochastic equilibrium, (5) differentiate random preferences and endowments, and (6) to analyze multivalued short-period equilibrium (Hahn 1984, 53-53).

An early criticism of socialists includes problems with distribution. Hahn adds that "distribution of preferences of agents is not God-given, and is different for different societies ... what is needed is ... a theory of preference formation and of the way endowments come to be what they are" (Hahn 1991, 67-68). If we take the days of typewriters as an example, people who learned to type developed that type of 'human capital'; and, when technology changed, it was not adopted because of the cost of retraining. Therefore, if equilibrium were defined to ignore what happened in the past, it would not be a true equilibrium. Again, take increasing returns to scale. Suppose one of two techniques with equal possibilities were chosen by chance and yielded increasing returns perhaps by specialization. Then one technique that was not chosen would be rendered inferior only because it was not chosen (Hahn 1991, 71).

Hahn's criticism seems to have mellowed. In his 1984 version, he stated that "the theory itself, however, is likely to recede and be superseded" (Hahn 1984, 86-87). In his later 1991 version he seems to take a cautionary view: "I am keen to end on a cautionary note. It is not my view that current economic theorizing is totally misdirected or useless. Rather the reverse. I think economists have probably attained a durable understanding of many important phenomena ... To the practical economists they are not of deep concern ... What I am proposing is that theorists should catch up with them" (Hahn 1991, 74).

Another unsettled notion of the Arrow-Debreu model resides in the area of core theory. Core theory can be traced back to the Edgeworth's contract curve. The issue is how to generalize the core for large economies. One way to do so is to make replica economies by grouping or replicating similar preferences and endowments. A measurable achievement here is that a price system for a decentralized core allocation exists. The topic of equilibrium for large economies was a springboard for continuum of agents' models.

V. Future of GE

Current research seems to have balked at the point of showing the comparative static properties of GE. As Samuelson has pointed out, this is a necessary condition to move the development above the childish level of parroting demand and supply analysis. As a refresher, such analysis requires one to check that the excess demand function have certain properties. Those properties are that it (1) is single valued and bounded from below, (2) is continuous, (3) is homogenous, and (4) obeys Walras' Law. Under conditions of gross substitutability, it is possible to demonstrate comparative static properties in the sense of Hicks and Samuelson. Hicks' model predicts that prices that exceed demand will be higher for those goods whose price changes are smaller. Samuelson predicted that excess demand will be more positive when only one good in the ceteris paribus set of goods changes, and less positive if the price of a second one in the set changes, showing the Le Chatelier principle at work. Debreu, however, did not think that the excess demand functions do not have enough structure to answer this problem. This area is now receiving much attention in the literature (Hildrenbrand 1989, 26-27).

If we now add some operational content to this theorem, we can perform CGE estimates for an economy or the world. The ITC and the WTO, for instance, use CGE models to predict the effect that freer trade among integrated areas such as NAF-FA and the EU will have on the welfare of a country. CGE is a simulation type of model. Given empirical estimates of import elasticities, substitution among commodities, and inputs on the one hand, and specification for utility and production functions on the other hand, it computes benchmark prices. Counterfactual assumptions can be made to obtain effects of policy measures. In short, CGE is a way to do general equilibrium with numbers, and it is made possible by the revival of GE analysis.

Conclusion

In this memoriam, we show our appreciation for a few intuitive concepts Debreu has bequeathed to us; but we have just begun to scratch the surface of the fruits of Debreu's erudition on GE. Today, the ideas of convexity can be found in most popular introductory textbooks at the undergraduate level. However, research is continually being done, from questioning the basic assumptions of GE models to expanding the horizon of their applications. Being armed with a few of the intuitive insights, a modern student is able to begin to penetrate the modern all-pervasive subject of GE, knowing that it will be a long journey.

We have covered several important simple areas, only mentioning game theory and welfare economics. Game theory was mentioned by Debreu in his prize lecture, but that got to the heart of the problem very quickly. His way of teaching game theory is more gentle, starting with simple parlor game, going through the proofs of minimax theorem, and ending with correspondence mapping. His way of teaching the welfare side starts with simple proofs that are available for the first welfare theorem, and ends with the theory that the Walrasian is in the core, a complex subject.

In general, Debreu's teaching style was comprehensive from the simple to the abstract. It is clear that he preferred the mathematical to the economics side. You can hear him take pride and delight in saying that a certain conclusion has been reached from a purely mathematical point of view, without the use of economics, underscoring that economic concepts formulated in the language of mathematics help us to grasp things that would ordinarily not be possible. He was fond of saying that mathematicians tend to do well when they are young, while economists do well when they are old. He clinched this tale with the notion that students of mathematical economics are very lucky, for they get the best of both the mathematical and economic worlds.

Professor Debreu was very considerate of budding mathematical economists. Students could easily interrupt his lecture to clarify problem areas. When students asked unclear questions, a typical response was that he did not quite understand the question. As the student tried again, he would say "I am beginning to see your point." He would sometimes pump-prime his students, saying "I would like to see more of this diagram in the literature." Once, when Lall Ramrattan, one of the authors of this article, met Debreu on the streets of the Berkeley campus, he exclaimed in a reprimanding tone: "what have you been doing?" This was his way of encouraging the student to do more by way of contributing to the literature. Mathematical economists never had a greater ambassador than Professor Gerard Debreu.

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Lall Ramrattan * and Michael Szenberg **

* University of California, Berkeley Extension; lallram@netscape.net

** Lubin School of Business, Pace University; MSzenberg@Pace.edu
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