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"Indeed, the influence of an individual participant on the economy cannot be mathematically negligible, as long as there are only finitely many participants. Thus a mathematical model appropriate to the intuitive notion of perfect competition must contain infinitely many participants. We submit that the most natural model for this purpose contains a continuum of participants, similar to the continuum of points on a line or the continuum of particles in a fluid."

(Robert J. Aumann, 1964, "Markets with a Continuum of Traders", Econometrica).

"One day I received in the mail an article written by Milnor and Shapley -- an analysis of voting in a situation in which there are some large voters and what they called an "ocean" of small voters...Then in the summer of 1961 there was a conference on Recent Advances in Game Theory at Princeton University. Herb Scarf gave a paper there that was a forerunner of the Debreu-Scarf paper on the core of an economy, and an outgrowth of previous work by Shubik (and by Edgeworth). Scarf's model had a denumerable infinity of traders, divided into a finite number of types, and he got an equivalence theorem between the core and the competitive equilibrium. However the model had various defects...I remembered the paper by Milnor and Shapley about "oceanic" games when hearing Scarf's model and said to myself, "surely, the continuum just has to be the right way of doing that.""

(Robert J. Aumann, "Interview with Feiwel", 1987, Arrow and the Ascent of Modern Economic Theory)

"A fanciful extension [of core theory] by R.J. Aumann, Karl Vind, Debreu, W. Hildenbrand, Arrow and Hahn and others utilizes a concept of an economy with an infinity of agents, whose characteristics are continuously distributed in a measure space...It seems to me that [these] lines of work will either remain standing as essentially unused but brilliant interpretational or mathematical feats -- or...may possibly be the beginning of a long development in which institutional detail is introduced piece by piece, to represent ... the various barriers on coalition formation existing in society."

(T.C. Koopmans, "Is the Theory of Competitive Equilibrium With It?", American Economic Review, 1974).

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(1) Introduction
(2) The Continuum Economy: Beginner's guide
(3) The Continuum Economy: Formal Statement
(4) Aumann's Core Equivalence Theorem
(5) Existence of Equilibrium
(6) Restricted Cores: the Epsilon equivalence theorem
(7) Atoms Revisited: the Oligopoly equivalence theorem

Edgeworth's conjecture, recall, states that as the number of agents in an economy increases, the set of core allocations "collapses" to the set of Walrasian equilibrium. Presumably, in the limit, when there are an infinite number of agents, the two will be identical. This is the "core equivalence" theorem.

But how exactly is one to model the "limiting" case? The Debreu-Scarf (1963) solution was to "replicate" the economy and show, that the Edgeworthian core "converges" to the Walrasian equilibrium. Or, more precisely, that the Walrasian equilibrium cannot be blocked by coalitions of any size, while all non-equilibrium allocations in the core can be blocked by sufficiently large coalitions.

Naturally, this is an "approximation" argument. It does not actually answer part b of Edgeworth's (1881: p.20) conjecture conclusively, namely that "Contract with perfect competition is perfectly determinate" (Edgeworth,. 1881: p.20) -- or, in modern interpretation, the core with an infinite number of agents is equivalent to the set of Walrasian equilibrium.

Now, one can argue that the Debreu-Scarf result is "close enough" to the limit case to merit no further discussion. This was Edgeworth's argument:

It is not necessary for this purpose to attack the general problem of Contract qualified by Competition, which is much more difficult than the general problem of unqualified contract already treated. It is not necessary to resolve analytically the composite mechanism of a competitive field. It will suffice to proceed synthetically, observing in a simple typical case the effect of continually introducing into the field additional competitors." (Edgeworth, 1881: p.34)

In other words, we can ignore the "limiting" case of an infinite number of agents ("Contract qualified by Competition") and proceed merely by proving core convergence.

However, even if we admit this, it remains questionable whether the limiting case of Debreu-Scarf theorem is what we really want anyway. Specifically, in the Debreu-Scarf limit case, there are (countably) infinite number of agents of each type, but the number of types is finite and fixed. Now, although this is precisely what Edgeworth did, infinite replication is perhaps not really what Edgeworth meant when he set out his conjecture. As he writes, "The theorem admits of being extended to the general case of unequal numbers and natures" (Edgeworth, 1881: p.43). His conjecture must be considered for arbitrary number of types (i.e. allowing anybody to be their own individual type).

This is, as economists quickly realized, was quite difficult to prove. However, if the limiting case can be proved, then half the battle is won. In other words, if we can at least prove core equivalence for an infinite collection of people of arbitrary type, then we are at least assured that we have somewhere to go when trying to develop more general core convergence theorems. Even if we are not assured that the core converges to the Walrasian equilibrium, we know it is not hopeless to try to prove it does -- because we know the limiting case, core equivalence, works.

Robert J. Aumann (1964) broached the equivalence question and provided the proof. Karl Vind (1964), almost simultaneously, proved the same result, albeit via different means. However, their theorems employed a mathematical method previously (largely) unseen in economics: measure theory. This required a translation of traditional economic concepts -- agents, coalitions, allocations, equilibrium, etc. -- into this new mathematical language.

Not everyone has been happy with this transformation. Koopmans (1974) considered it a "fanciful extension". The discomfort was caused by a crucial feature of Aumann's model: the assumption that there was an uncountably infinite number of agents in the economy. Intuitively, this means that Aumann assumed that there as many agents as there are points on a line, a continuum of people. It is a matter of simple observation to conclude that there is no such economy -- and there can never be such an economy -- in the real world. What, then, is Aumann trying to accomplish?

To those familiar with Edgeworthian exchange, it is obvious that anything less than an infinite number of agents will not do to prove equivalence. Now, Aumann's use of a continuum of agents may sound like overkill -- why not merely countably infinite or "large but finite"? Firstly, as Aumann would perhaps be the first to admit, there is mathematical convenience in a continuum. There are ready-made theorems and tools we can use here. Secondly, the concept of an "atomless" economy -- an economy without "big players" -- can be made precise in a continuum economy. This, he argues, gives precision to the meaning of "perfect" competition in manner which any other number of agents cannot.

[Note: We should note here that Donald J. Brown and Abraham Robinson (1973), via the technique of non-standard analysis, made the core equivalence theorem somewhat more palatable. In particular, they demonstrated that it is "approximately" true in an economy with "large but finite" number of agents -- where the terms in quotations have a rather precise meaning. We shall consider this later.]

[Note: The following is an attempt at an intuitive, if verbose, step-by-step guide to the elements of the continuum economy for those with little mathematical background; it can otherwise be skipped.]

The first step in going to Aumann's theorem is to translate traditional economic concepts into measure theory. To those unfamiliar with this theory, there are four really novel concepts that need to be understood: s -algebra, measure, measurable function and integration. We recommend consulting the more formal treatment of these concepts in our mathematical appendix and the references contained therein. Here we will only attempt a short intuitive introduction in an economic context.

We have an infinite number of households/agents. Let us define H as the set of households. A "coalition", as we know, is a group of agents, and thus a subset of H. A s -algebra is merely a collection of subsets of H which have certain properties. Thus, we can think of a s -algebra as the "collection of coalitions". Let us call this collection Á . Thus, a coalition S Í H can be thought of as a member of this s -algebra, i.e. S Î Á .

But "how big" is a coalition? We would like to measure the size of a coalition, and this is exactly what the concept of a "measure" is. Our measure, m , assigns a number to every set that is contained in the s -algebra, so we say m (S) is the "measure" of coalition S.

So, our measure is like any measure. It assigns numbers to things -- just like inches measure distance, litres measure liquid volume, minutes measure time, etc. In our case, we want a number to represent the "size" of a coalition. If there are two coalitions, S and T, and S is "bigger" than T, then we would like it that m (S) > m (T), i.e. the measure of S is greater than the measure of T.

But what do we mean by "size"? Quite straightforwardly, the size of the coalition S is the proportion of all people in the economy who are members of this coalition. To illustrate this, suppose that H is a finite set (i.e. there are a finite number of households); then an appropriate measure of the size of coalition S would be the number of agents in the coalition relative to the total number of agents in the economy, i.e.

m (S) = #S/#H

Of course, if the number of agents is uncountably infinite (H is an infinite set), then this measure is not quite appropriate as we cannot divide by the number of total households. A different measure must be used, which nonetheless captures the idea that relatively "big" coalitions have larger measures than relatively "small" coalitions.

The easiest way to do so is to take all the people in H and line them up one after the other. As there are a continuum of people, then the easiest thing to do is to just put each person on a point in a line segment. Let the interval [0, 1] be our "set of people", H. Note that this is a continuum: there are as many people in the economy as there are points on a line in the [0, 1] segment, an uncountably infinite number of people.

Now, it is important to note that H = [0, 1] is just a set of "names" with no numerical meaning. We just take each household and place it on the line somewhere between 0 and 1 (See Figure 1). There is no significance to being the 1/3th or 1/4th, or 7/8th household. Just because Mr. 3/4 is closer to Ms. 3/5 than he is to Mr. 1/8 in the [0, 1] interval , does not mean he is "closer" in any economic sense. It is a mere index.

Fig. 1 - Set of Agents and a Coalition

 Then a coalition is merely a "subset" of [0, 1] and the natural "measure" of this subset is length. This is what is called a Lebesgue measure. Thus the "measure" of all the people in the economy is m (H) = (1 - 0) = 1. Suppose all the people on the line between 1/4 and 1/2 form a coalition S (see Figure 1). Then this coalition S is of "length" (1/2 - 1/4) = 1/4, or simply m (S) = 1/4. What if the coalition is formed by people scattered in [0, 1], e.g. suppose all the people between 1/4 and 1/2 and also all the people between 3/4 and 1 form a single coalition, S? No problem: we just add up the lengths, i.e. m (S) = (1/2-1/4) + (1-3/4) = 1/2.

What if a coalition has no "length"? For instance, what happens when there is a coalition S formed by five people, each in a different place on the [0, 1] segment? Well, here we begin the see the magic of the continuum: a person or any finite group of people have no length and thus are of measure zero, i.e. m (S) = 0. In economic language, this translates into the following: an individual person or a finite group of people are insignificant.

It is useful to reflect on this for a moment. Suppose there is only ten people in a coalition. Can they "block" a proposed allocation? Intuitively, they can't. Recall, that there are an infinite number of people in the economy so that, relatively speaking, two people are quite "insignificant". They are ten points on a line; take them away, and the line still looks pretty much like a line. Their share in the allocation and endowments are infinitesimal relative to the whole economy. If they "withdrew" and took their endowments with them, nobody would really notice they were gone: there are still an uncountable infinity of people left over. Thus, we say that a coalition composed of only ten individuals has "measure zero", m (S) = 0, i.e. it is insignificant.

So, for a coalition to "matter" it must be made up of a "lot" of people, where by a "lot", we mean simply that if all the coalition members withdrew, then the economy would lose a significant portion of its endowment. The results would matter for everybody else. How much is "a lot"? Well it can't be less than infinity! The way to say "a lot" is to think of them as a "sub-segment" of H. Suppose all the agents are on line between between 0 and 1, i.e. H = [0, 1]. Then a "significant" coalition would be one composed of all the agents between 1/4 and 1/2, i.e. S = [1/4, 1/2]. Obviously, this coalition consists of a quarter of the total population, so it must matter! It is also a "segment" of the line, thus the coalition itself contains an uncountably infinite number of agents. But it is "smaller" than the total. The way we'd say it, then, is simply that this coalition has "positive measure", m (S) > 0, although that measure is smaller than the measure of the total population, i.e. 1/4 = m (S) < m (H) = 1.

But is this appropriate? Well, yes. According to Cournot (1838), an economy is "perfectly competitive" if the exit or entry of any one agent into the economy does not make a difference. In short, an agent cannot, by his actions, influence the outcome. In a continuum economy, this is true: a single agent has measure zero. He does not, by himself, "count". If he leaves, it doesn't matter. What about two agents? Or three? Or a hundred? Or a billion? Or a quadrillion? It doesn't matter: a single person or a quadrillion people just as insignificant relative the the uncountably infinite number of people that remain. Any coalition composed of a finite number of agents is insignificant relative to the total size of the economy -- because no matter how (finitely) many you remove, the remaining infinite number of people is so absolutely overwhelming that nobody will notice the coalition has left.

This is how Aumann (1964) envisaged the concept of "perfect competition" can be made precise: the entry or exit of any finite coalition does not matter. There is a technical condition that makes this true: atomlessness. What "atomless" implies is that "nobody matters". An "atom" is a person who "matters", i.e. a person who has positive measure. So, if m (h) > 0, then we say household h is "an atom". His exit will affect things.

Now, do not fall into the common mistake of confusing "atom" with "small". Atom, in original Greek, means "indivisible". So, a subset A of X is an "atom" if it has positive measure and cannot be further subdivided; formally:

Atom: A subset A of H is called an atom if m (A) > 0 and for any B Ì A, then either m (B) = m (A) or m (B) = 0.

Intuitively, people in an economy are things that cannot be subdivided. So think of the H = [0, 1] interval again, but suppose that the whole portion A = [1/4, 1/2] is made up of one person, call him "Mr. Atom", who is, of course, unsubdividable (see Figure 2). Suppose the rest of the line, H\A = [0, 1/4] È [1/2, 1] is made up of everybody else, one on each point of that segment. Thus, we still have an uncountably infinite number of people -- they are all just "crowded" in the segments [0, 1/4] and [1/2, 1]. By himself, Mr. Atom has measure m (A) = m (1/4, 1/2) = 1/4 > 0; while the rest of the people put together have measure m (H\A) = m (0, 1/4) + m (1/2, 1) = 3/4 > 0. Thus Mr. Atom counts as much as one-quarter of the total population. He is an enormous person (think about it).

Fig. 2 - An Atom

So, in order to ensure that the economy is "perfectly competitive", then Aumann required that we have an atomless economy. More precisely, it is defined as follows:

Atomless: H is "atomless", if for any subset S Í H, we can find a strict subset of that, T Ì S, where 0 < m (T) < m (S).

What this means is that if we have a coalition of positive measure, we can take a smaller coalition from it -- and it also has positive but strictly smaller measure. Why this excludes atoms can be made clear. Consider our enormous Mr. Atom. Suppose he forms a coalition with ten other (regularly-sized) people. This coalition, S, will have positive measure because, as we know, Mr. Atom has positive measure and he's in it. Now, take a subset of this coalition, T Ì S, say remove Mr. Atom from it. The ten remaining people, together, have zero measure. Thus, our T Ì S has now zero measure, m (T) = 0. Thus, m (S) > m (T) = 0. But "atomlessness" also requires that m (T) > 0, so this does not fit the bill. What if, in constructing T, we didn't remove Mr. Atom but, say, we removed five of the other people. Then, m (T) > 0, but as the removed people would have measure zero, then it remains that m (T) = m (S). Again this violates atomlessness. Finally, what if there were two atoms, say Mr. Atom and Mrs. Atom, as well as ten other people in a coalition S. Removing Mrs. Atom reduces the measure of the coalition which (because it still retain Mr. Atom) is still positive. This seems to fulfill our condition, m (S) > m (T) > 0. However, notice that the "atomless" property must apply to all coalitions -- thus including the remaining coalition T -- the one without Mrs. Atom -- and we know, from before, that this does not fulfill the atomlessness condition.

Robert Aumann (1964) regarded "atomlessness" as the essence of a perfectly competitive economy. Now, note a continuum does not imply atomlessness, but atomlessness does imply a continuum. Finite sets cannot be atomless. Thus, many economists who work in this field don't even bother to say that the set of agents is a continuum; they just say the economy is "atomless" and then proceed. Atomlessness is the primitive assumption.

[Note: A final point about the s -algebra, the collection of coalitions in our economy, Á . Does this include all coalitions possible? No, but it contains "practically all" coalitions. This is because we actually construct Á on the basis of the Lebesgue measure. Such a construction will yield us an enormous amount of possible coalitions, but not exactly all of them. Thus we must really say that Á is the set of "practically all" possible coalitions.]

So we have an idea of households and coalitions in an atomless (continuum) economy. But what is an allocation? An allocation assigns a bundle to every household in the economy. In a finite economy, a convenient mathematical way of representing this, of course, is as a vector x = [x¹, x², ..., x^h, ...], where x^h is the bundle of goods going to the hth household. However, with infinite agents, we would have to have a vector of infinite size -- and, indeed, with a continuum of agents, such a vector would, in fact, not be possible at all!

So, the alternative procedure is to conceive of an allocation as a mapping from households to the commodity space, i.e. x: H ® C, where C, the commodity space, is a subset of Rⁿ (n is the number of commodities). An individual household receives a particular allocation, which can be denoted x(h), i.e. the part of the mapping x that corresponds to a particular household h. Of course, x(h) Î C. Endowment is a particular allocation, and thus we can consider the mapping e: H ® C as assigning an endowment bundle to every household. The hth household's endowment is denoted e(h) Î C.

What about preferences? The preferences of a particular household are formed over allocation that that household receives, thus preferences are formed x(h) Î C. We denote the hth household's preferences by the strict binary relation >_h Í C ´ C. Thus, if x(h) >_h y(h), then household h strictly prefers allocation x(h) to allocation y(h). Every household receives a set of preferences. Under certain regular conditions, as we know, these preferences are representable by a utility function. Let u^h(.) denote the utility function of the hth household.

There is a slightly odd convention in much of general equilibrium theory that has emerged since the 1970s: namely the idea of a "set of preferences", which is often denoted à . A particular type of preference is an "element" of this set, i.e. >_h Πà . Thus, we can define an economy E as a mapping that gives to each household H a particular preference structure >_h Πà and a particular endowment, e(h) ΠC i.e. the mapping:

E: H ® Ã ´ C

is the definition of an "economy". Alternatively, we can simply define an economy as E = (>_h, e(h))_{hÎ H}, a collection of sets of preferences and endowments.

[Note: the convention of describing the economy as a "mapping" from the set of agents names to the set of preferences and endowments, etc., and to continually use mapping notation throughout, rather than extensively indexing things, has been particularly promoted by Werner Hildenbrand (e.g. 1974) and thus is sometimes referred to as the "Hildenbrand convention".]

Finally, let us go to the market. Although we have an infinite number of people, the number of different types of goods we have is finite -- specifically, let us have n goods. Thus, the set of prices can be merely conceived as a vector p = [p₁, p₂, .., p_n], where p_i is the price of the ith good. Now, a particular agent's demand for a good is just as in regular theory:

x(h) = argmax u^h(x)

s.t. 

p·x £ p·e(h)

i.e. the hth household's demand, denoted as x(h), is thus the bundle of goods he receives that maximizes his utility subject to the budget constraint formed by his endowment, e(h) and the given set of prices, p.

The next natural step is to go from individual demand x(h) to market or total demand, but here we begin running into trouble. Now, the set of demands in the economy is an allocation mapping x: H ® C, while the set of endowments is also a mapping e: H ® C. What is the total market demand and the total endowment? If we had a finite number of households, this would merely be the sums:

market demand = å _{hÎ H} x(h)

market endowment = å _{hÎ H} e(h)

Unfortunately, when we have an infinite number of households, then these sums will both be infinite (it is here that Herbert Scarf's (1962) equivalence theorem stumbled) -- and with a continuum of households, we cannot even sum at all! In non-standard analysis, the concept of an infinite number is meaningful and can be added, subtracted, etc. like regular numbers. However, in standard mathematics, "infinite" amounts are not numbers and thus cannot be compared with each other or handled like regular numbers. We need something else.

The trick comes by integrating with respect to measure m . What does this mean? The integral we use does not give us the total amount of endowments and demands in the economy, but rather the "average" amount. To understand this, suppose were were to ask what is the average endowment of apples in the economy? The easiest way is to add up all the apples everyone has and divide it by the number of people. Alternatively, we could ask: "who has only 1 apple?" and then count how many people raise their hands; and then ask "who has only 2 apples?" and then count how many people raise their hands, and so on. We can then figure out the "average" amount of apples by adding each category by the proportion of people who raised their hands. So if a third of the people raised their hands when asked about 1 apple and two-thirds raised their hands asked about 2 apples, then the average amount of apples in the economy is 1·(1/3) + 2·(2/3) = 5/3 apples per person.

In integration, we use precisely this last procedure, a bit jazzed up. The "measure", m , we spoke of earlier, recall, gives us a number which represents the "relative" size of a group of people. Integration, roughly speaking, is thus the "adding up" of different categories weighted by the measure of the groups of people who have that amount. To fix ideas more clearly, consider Figure 3, where we have drawn the allocation mapping x:H ® C, which gives the number of apples that each agent has. The set of households H is all the points between 0 and 1.

Now, the mapping x drawn in Figure 3 measures, as we noted, the number of apples each person has. It looks like a pretty smooth function, but as the names are "arbitrarily" placed, no meaning should be read into this "smoothness". It could be quite jagged and crazy-looking.

[Note: there is one caveat: the function x must be "measurable", or, more accurately, "integrable"; this is given a precise meaning elsewhere, but intuitively, it cannot be too radically jagged. This is a mathematical requirement, and it does not translate readily into economic meaningfulness.]

Fig. 3 - Allocation of a Good

The process of integration is simple: ask who has one apple, and Mr. a will raise his hand. He is then "measured" by m (a), which is his "size" relative to the population (we are temporarily assuming a finite number of people, so m (a) is not zero). Ask who has two apples, and Messrs. b, c and d will raise their hand. Consequently, they form a set, (b, c, d) whose relative size is measured by m (b, c, d). Ask who has three, and Messrs. e, f and g will respond, and thus we measure their relative size, m (e, f, g), and so on. We ask for every amount of apples -- including all fractions of apples -- until everyone has responded. Then, we add up the amounts weighted by the measures, i.e.

1·m (a) + 2·m (b, c, d) + 3·m (e, f, g) + 4m (h) + ....etc.

and this sum is the integral of the curve x. Notice that the old intuition of the integral of a curve x as being the area under that curve fits with our definition (with a slight adjustment that we have to restrict the agents between [0, 1] so as to retain the concept of "average"). This is, incidentally, known as the process of Lebesgue integration. So, the average allocation can be written as:

x^m = ò _H x(h) dm (h)

(we use the superscript "m" to denote mean), i.e. the average bundle x^m is the integral of all bundles over all households, with respect to the relative population measure m . The analogue with the finite case is, of course:

x^m = (1/#H)·å _{hÎ H} x(h)

the average bundle per household.

[Note: Some confusion may arise here. A bundle is a vector of goods received by a household, x(h) = [x₁(h), x₂(h), .., x_n(h)], where x_i(h) is the ith good received by household h. So what do we mean when we say "average bundle", x^m? Isn't an integral a number? When we say "average bundle", this is shorthand for n integrals. What we really do is integrate to obtain the average for each good in the economy, and then call the resulting vector of averages, the "average bundle", i.e. we really mean x^m = [x₁^m, x₂^m, .., x_n^m], where x_i^m is the average of good i, and defined as x_i^m = ò _H x_i(h) dm . It is merely a convenience to call the result an "average bundle" rather than "a bundle of averages".]

[Note: there are many ways of writing the integral. The notation is not strict here. ò _S x(h) dm (h), ò _S x(h) dm , ò _S x(h), ò _S x, etc. are all alternative ways of writing the same thing : the integral of the function x with respect to the measure m over the set S. As the measure will not change, don't worry about m (which we won't even bother to write down sometimes). Pay attention only to the function in the interior, (as ò _S x, ò _S y, etc. are different things) and the set over which these things are integrated (as ò _S x, ò _H x, etc.are different things)].

As we can speak of the average bundle demanded, x^m = ò _H x(h) dm , we can also speak of the average endowment bundle, e^m = ò _H e(h) dm . We say the "market for the ith good clears" if:

ò _H x_i(h) dm = ò _H e_i(h) dm

the average demand for the ith good is equal the average endowment of that good. This may seem a peculiar definition of "market-clearing", as we are used to saying that total demand for good i must be equal to total endowment of good i. Who cares about "average" demand and "average" endowment? The suspicion is well-founded, but, alas, there is little else we can do in a continuum economy. Total demand and total endowment is infinite -- as we have an infinite number of people -- and we cannot compare infinites. So, we have no choice but to work with "average" to define market-clearing.

[Note: continuum economies are sometimes described as being composed of infinitesimal or infinitely-small agents. This is not true. A continuum economy is made up normally-sized agents, there are just an infinite number of them. That is why the "totals" -- total demand and total endowment -- are infinite. If agents were infinitesimally small, this would not be the case.]

In a Walrasian equilibrium, all markets clear, so the above condition must hold for all markets i = 1, 2, .., n. Or, more succinctly:

ò _H x(h) dm = ò _H e(h) dm

as shorthand for "all markets clear". We are sure that this is a "Walrasian equilibrium" allocation if the x(h) we are talking about is the demanded bundle by household h, so there is associated set of equilibrium prices, p.

Having some idea of equilibrium, let us return to the Edgeworthian core. A coalition, as noted, is a subset of the set of households, S Í H (or more formally, S is an element of the s -algebra Á ). We say an allocation x: H ® C is "blocked" if there is a coalition S that can block x. This is defined as it normally is, i.e. S is a blocking coalition if there is an allocation y: H ® C such that:

(i) m (S) > 0;
(ii) y(h) >_h x(h) almost everywhere in S;
(iii) ò _S y(h) dm = ò _S y(h) dm .

The odd language in these three conditions must be explained a little. The first is the statement that the coalition must have "positive measure". This means the coalition "matters", in the sense in which we spoke of before. What about (ii)? This simply says that everyone in the coalition prefers the new allocation y to the old one, x. The technical meaning of "almost everywhere" means, really, "everyone" -- because those who do not prefer it are, when combined, of measure zero. So if everybody in the coalition, except for ten people, prefers y to x, then these ten disgruntled coalition members do not matter relative to the size of the whole coalition (recall that the coalition itself is of infinite size). So, when we say "almost everyone", take it to mean "everyone" and don't worry about those few angry individuals.

[Note: this raises a curious question: are core allocations Pareto-optimal? Strictly, no -- because a person (who is negligible) can be possibly be made better off. To overcome this, the criteria of "Pareto-optimality" ought to be rewritten in "almost everyone" form, i.e. an allocation is Pareto-optimal if "almost everyone" agrees that it cannot be improved upon by another allocation. This is something of a departure from the original meaning of the concept: not only can some people can be made better off at Pareto-optimal allocations, but to call an allocation Pareto-suboptimal requires that we find an infinite number of people (a group of positive measure) that can made better off by a move. (cf. Hildenbrand, 1969, 1974: p.230).]

What about (iii)? Again, this is a feasibility restriction for the coalition. Now, recall that, in principle, for a coalition to work, then the trades they do among themselves must be met by the endowments that they have, so the total endowment of the coalition must be sufficient for them to conduct the trades for the alternative allocation y among themselves. However, as we have an infinite number of people in the coalition, then we are back to the same problem of defining "totals". We have no recourse but to return to talking about "averages". This is what condition (iii) says: for the coalition, the average endowment must be equal to the average allocation.

If all three conditions (i), (ii) and (iii) are met for some coalition S, then allocation x is "blocked". The rest follows simply: the Edgeworthian core is, of course, merely the set of allocations which are not blockable. That's all there is to it.

In sum, the atomless (and thus continuum) economy of Aumann (1964) is supposed to represent the extreme limiting case of a perfectly competitive economy. The concepts of demand, Walrasian equilibrium, blocking coalition, core, etc. are all translatable into measure-theoretic language that is necessary for a continuum. The great loss of this method, of course, is that we can't work with totals, but we must work with "averages", which is not exactly the same thing. But it is close enough. In the next section we will restate the mathematical description of the economy (a bit more formally) before going on to prove Aumann's theorem: namely, that the set of Walrasian allocations in this economy is identical the set of core allocations.

Let H be the set of households in the economy. Let Á be a s -algebra and m a measure. Then, (H, Á , m ) defines a measure space. By assumption, it is atomless, i.e. for any S Î Á , there is a T Î Á such that m (S) > m (T) > 0. For normalization, let m (H) = 1.

Let C be a commodity space, defined as C Í R₊ⁿ (there are n commodities). An allocation x: H ® C is a measurable mapping. We use x(h) to denote the bundle received by the hth household.

Let the binary relation >_h Í C ´ C describe the (strict) preferences of the hth household. Defining à as the set of preferences, then >_h Î Ã . Preferences in à can be assumed to have conventional properties of completeness, transitivity, reflexivity, continuity, monotonicity but (surprise!) not convexity. Let endowment be defined as a measurable mapping e: H ® C, so e(h) is the endowment of the hth household. Thus, we can define an economy E itself as a mapping:

E: H ® Ã ´ C

or E = (>_h, e(h))_{hÎ H}.

Blocking: Let x: H ® C be an allocation for the economy E. The allocation is "blocked" if there is a coalition S Î Á and an alternative allocation y: H ® C such that:

(i) m (S) > 0
(ii) y(h) >_h x(h) almost everywhere in S;
(iii) ò _S y dm = ò x dm

where (i) indicates that the coalition must be of positive measure, (ii) almost everyone in the coalition prefers alternative allocation y and (iii) the alternative is feasible for the coalition.

Unblockable: Let x: H ® C be an allocation. If there is no coalition S Î Á and alternative allocation y: H ® C that blocks allocation x, then x is "unblockable".

and finally:

Core: the core of economy E, denoted C(E), is the set of feasible allocations, x: H ® C, which are unblockable.

Let us now turn to the Walrasian equilibrium. Let p Î R₊ⁿ be the set of prices in the economy. Then:

Walrasian equilibrium: a (Walrasian) equilibrium is defined as a price-allocation pair, (p, x), where:

(i) x(h) = argmax u^h(x) s.t. px £ pe(h) almost everywhere in H
(ii) ò x dm = ò e dm

so (i) states that x(h), the bundle received by (almost every) household, maximizes utility subject to the budget constraint defined by the equilibrium prices, p; while (ii) states that markets clear (average demand equals average supply in every market). If this holds, then we call the allocation x the (Walrasian) "equilibrium allocation" and define W(E) as the set of (Walrasian) equilibrium allocations in the economy E.

As can be easily deduced from this definition, for any x Î W(E), it will be true that " y Î Rⁿ₊, if y(h) >_h x(h), then py^h > pe^h almost everywhere in H. In other words, if y is preferred by any agent to the equilibrium allocation x , then it is not affordable to him. This serves as an alternative definition of a Walrasian equilibrium.

We are now ready to proceed to Aumann's Core Equivalence Theorem.