Edgeworth's
"Indeterminacy of Contract"

A Blocking Coalition

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"It may be said that in pure economics there is only one theorem, but that it is a very difficult one: the theory of bargain in a wide sense."

(Francis Ysidro Edgeworth, 1889, "On the Application of Mathematics to Political Economy", Nature)

"Equilibrium is attained when the existing contracts can neither be varied without recontract with the consent of the existing parties, nor by recontract within the field of competition. The advantage of this general method is that it is applicable to the particular case of imperfect competition where the conceptions of demand and supply at a price are no longer appropriate."

(Francis Y. Edgeworth, Mathematical Psychics, 1881: p.31)

"I have been particularly impressed by one thing, and that is that economists who are mediocre mathematicians, like Jevons, have produced excellent economic theory, whereas some mathematicians who have an inadequate knowledge of economics, like Edgeworth, ...talk a lot of nonsense."

(Léon Walras, "Letter to Perozzo, March 18, 1890")

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Contents

(1) Introduction
(2) Barter Exchange
(3) Recontracting and the Core
(4) The Core: mathematical restatement

(1) Introduction

In Léon Walras's (1874) theory, exchange is price-mediated through the market. By this we mean that agents do not exchange "with each other", but rather "with the market". The "market" offers a set of prices and agents express their desired demands from the market and supplies to the market. Once that is done, the market thereupon verifies whether demands meet supplies for every good. If they do, the market then rearranges the goods so that everyone gets what they demand -- what is called a "Walrasian equilibrium". If they don't, the market then cancels everything, and offers up another set of prices and asks the agents to recompute their demands and supplies. The market will continue to adjust prices until equilibrium is reached.

We are generally familiar with the Walrasian mechanism. It is effectively what is expressed when one first encounters the familiar "theory of demand and supply". But the English economist Francis Ysidro Edgeworth noted that "[Walras] describes a way rather than the way by which economic equilibrium is reached" (Edgeworth, 1925: p.311). In his view:

"Walras's laboured description of prices set up or "cried" in the market is calculated to divert attention from a sort of higgling which may be regarded as more fundamental than his conception, the process of recontract as described in these pages." (F.Y. Edgeworth, 1925: p.311)

The process of "recontract" that Edgeworth is referring to here is what we will call the process of "Edgeworthian exchange". He laid it out in his famous treatise, Mathematical Psychics (1881) and in several articles (e.g. Edgeworth, 1889, 1891a, 1891b).

Edgeworth's work led to a brief controversy with Alfred Marshall in the 1890s, but was basically dead soon afterwards. It remained dormant until it was resurrected by Martin Shubik (1959) as the theory of the "core". The literature on the Edgeworthian core that exploded in the 1960s and proceeded through the 1970s is vast. The Edgeworthian revival was conducted through a series of now classical pieces of work such as Gérard Debreu and Herbert Scarf (1963), Robert Aumann (1964), Karl Vind (1964, 1965), Werner Hildenbrand (1970, 1974), Truman Bewley (1973), David Schmeidler (1973), Binyamin Shitovitz (1973), Donald Brown and Abraham Robinson (1973), Robert Anderson (1978) and many more. We shall review the Edgeworthian revival later, but, now, let us begin at the beginning, and examine Edgeworth's original 1881 contribution directly.

(2) Barter Exchange

Edgeworth's interest in the process of barter exchange was sparked by a rather unsatisfactory discussion of the matter in William Stanley Jevons's Theory of Political Economy (1871). "Barter exchange" refers to the exchange of goods between parties without the mediation of the market -- i.e. without "prices" being used as guides. So, suppose there are two agents -- using Edgeworth's nomenclature, let us call one Robinson Crusoe and the other Friday. Let x represent the amount of a good that Robinson possesses and y represents the amount of labor Friday has. As Edgeworth assumed that Robinson owns all of x and Friday owns all of y, then this is a situation of bilateral monopoly.

Engaging in "barter exchange" means that Robinson will give some amount D x to Friday in return for some amount of labor D y. The rate at which good y is exchanged for good x is then D y/D x. Now, according to Jevons:

"The ratio of exchange of any two commodities will be the reciprocal of the ratio of the final degrees of utility of the quantities of commodity available for consumption after the exchange is completed." (Jevons, 1871: p.95).

where, by "final degrees of utility" Jevons is referring to marginal utilities. In other words, we will find that in equilibrium:

MUxR/MUyR = D y/D x = MUxF/MUyF

where MUih is the marginal utility of good i Î (x, y) to household h Î (Robinson, Friday). Notice that this formula implies that the indifference curves of Robinson and Friday are tangent to each other.

Now, Jevons went on to assert that in an barter exchange economy "there can only be one ratio of exchange of one uniform commodity at any moment" (Jevons, 1871: p.87), i.e. that the bilateral exchange ratio D y/D x was unique, indeeed, so unique that he reasoned that it could be simply rewritten as px/py, the price ratio obtained on the market. Then substituting px/py for D y/D x in our previous equation, we can rewrite it as:

MUxR/px = MUyR/py = MUxF/px = MUyF/py

which is well known as Gossen's Second Law.

Now, the market price ratio px/py requires a "market", i.e. price-mediated multi-lateral exchange while the ratio D y/D x obtained via bilateral barter exchange is not "price-mediated". Why should they be the same? Jevons's reasoning here is a bit garbled. As it turns out, Jevons believed that it was easier to obtain a unique ratio of exchange in bilateral barter exchange than it is to obtain a unique set of market prices in multi-lateral, price-mediated exchange. Thus, for Jevons, the central question is how does a multi-lateral situation yield a bilateral solution? Jevons proposed that one could divide both sides of the market into a pair of "trading bodies" which, with the assistance of arbitrageurs, would "act like" bilateral exchangers and thus yield the unique bilateral solution (Jevons, 1871: p.88-95). This is how px/py = D y/D x and Gossen's law would follow.

Jevons's reasoning irked Francis Ysidro Edgworth (1881). Edgeworth believed that Jevons had it exactly the wrong way around. He did not dispute Gossen's law in market exchange, and recognized that for this to be true, then indeed px/py = D y/D x. But it is market prices, Edgeworth argued, that are unique and easy to determine, while it is the bilateral exchange ratio that is quite difficult to determine. For Edgeworth, the central question is quite the opposite: why (or when) should a bilateral situation yield the multi-lateral solution? Edgeworth asserted that only in situations of "perfect competition" will we find that D y/D x = px/py; otherwise, D y/D x could be quite different.

Edgeworth's principal endeavor was to demonstrate that bilateral barter exchange was not determinate, i.e. that there are many ratios through which mutually beneficial exchange could be conducted between a pair of trading parties which fulfills Jevons' conditions. Edgeworth illustrated his argument with the famous net trade diagram shown in Figure 1a. This is adapted from Edgeworth (1881: p.28) which is, incidentally, the first depiction of indifference curves in economics. For those more comfortable with the "Edgeworth-Bowley box", we reproduce effectively the same information in Figure 1b. Note that in Figure 1a, the quantities measured along the axes are net trades, D x and D y, whereas in Figure 1b we are measuring allocations, i.e. net trades plus endowments, x = ex + D x and y = ey + D y, where ex and ey are the endowments of goods x and y respectively. Notice as the endowment vector e is in the bottom right corner of the box in Figure 1b, then this implies that Robinson owns all of x and Friday owns all of y, i.e. we have a bilateral monopoly.

[Note: Figure 1b has been baptized "Edgeworth-Bowley" box because of its resemblance to Edgeworth's net trade diagram and because Arthur Bowley (1924: p.5) drew a "box" around Edgeworth's figure. However, the actual box diagram in Figure 1b was first drawn by Vilfredo Pareto (1906: p.138). Thus, some modern authors, e.g. Maurice Allais (1989: p.67, 620) insist on calling Figure 1b a "Pareto Box".]

Engaging in "barter exchange" means that Robinson will give some amount D x to Friday in return for some amount of labor D y. Robinson is best off if Friday gives him all his labor while he pays nothing (represented by point OF in Figures 1a and 1b). Friday is best off if he gets all of Robinson's endowment of x while providing no labor (represented by point OR in Figures 1a and 1b). Thus, in the net trade diagram in Figure 1a, Robinson's utility levels rise as his indifference curves ascend to the northwest (thus URD > URC > URe) while while Friday's utility rises as his indifference curves approaches the southeast (thus UFC > UFD > UFe). In the box Figure 1b, Robinson's utility rises to the northeast and Friday's rises to the southwest.

Fig. 1a - Indeterminacy of Exchange in a Net Trade Diagram

Fig. 1b - Indeterminacy of Exchange in an Edgeworth-Bowley Box

Now, recall that Jevons's barter exchange formula, MUxR/MUyR = D y/D x = MUxF/MUyF, implies that the ratio of exchange will be obtained by the tangencies of the Robinson's and Friday's indifference curves. But what Edgeworth demonstrated (and Jevons did not realize) was that there were many such ratios where this is fulfilled. As we see in Figures 1a and 1b, we have tangent indifference curves at many different trades. For example, both trades C and D fulfill Jevons's conditions. Yet it is clear that D y/D x ¹ D y¢ /D x¢ , i.e. the exchange ratio obtained with exchange C is different from the ratio obtained with exchange D and there is no reason to assume that C will be chosen rather than D (or vice-versa) since both are mutually beneficial to both trading parties. As Edgeworth demonstrated, there is a whole range of trades like this, thus the actual trade (and thus resulting exchange ratio) in bilateral barter exchange will be quite indeterminate.

This can be understood by examining the net trade diagram in Figure 2a or the box diagram in Figure 2b. We have the same Robinson-Friday economy where UR and UF represent the autarky (no trade) indifference curves. Edgeworth (1881: p.19) differentiated between contracts, settlements and final settlements. A contract is any point in the net trade space in Figure 2a or 2b. In Figures 2a, 2b, the exchanges represented by A, B, C, D and F are all "contracts". In Figures 2a, 2b, the entire shaded region, the area of the "lens" formed by the autarky indifference curves, represent contracts which are mutually beneficial -- such as C, D and F. In contrast, other contracts outside this lens are not mutually beneficial because one of the agents will have less utility than if he had not traded at all. So, contract A, for instance, is detrimental to Friday, while contract B is detrimental to Robinson.

A settlement is a "class of contracts to the variation of which the consent of both parties cannot be obtained" (Edgeworth, 1881, p.28). In modern terms, a "settlement" is a Pareto-optimal contract. In Figures 2a and 2b, contracts A, B, C and D are settlements because if we start at any of these contracts, at least one of the parties will not consent to a move away from it. In contrast, contract F is not a settlement: starting at F, both Robinson and Friday will agree to move to another contract (such as C or D). Thus contracts C and D are "Pareto-superior" to contract F. However, note that from point F, one cannot obtain agreement to move to either A or B. Thus, neither A nor B are "Pareto-comparable" to contract F.

Fig. 2a - Settlements in the Net Trade Diagram

Fig. 2b - Settlements in the Edgeworth-Bowley Box

Now, that for any settlement (A, B, C, D), the indifference curves of Robinson and Friday are tangent to each other, whereas at non-settlements, such as F, they are not. This property was noted by Edgeworth explicitly:

"If we enquire in what directions [Robinson and Friday] will consent to move together, the answer is, in any direction between their respective lines of indifference, in a direction positive as it may be called for both. At what point then will they refuse to move at all? When their lines of indifference are coincident." (Edgeworth, 1881: p.22)

and thereafter gave the formula for the tangency of indifference curves. The set of settlements in the economy corresponds to the set of contracts where indifference curves are tangent to each other. This is shown by the line connected OF to OR in Figures 2a and 2b -- what Edgeworth (1881: p.29) called the contract curve.

As we know, the tangency of indifference curves is Jevons's formula. Thus, Edgeworth claimed, Jevons's condition is fulfilled at any point along the contract curve. Now, in bilateral barter exchange, only mutually beneficial settlements will be considered -- and thus only the portion of the contract curve within the lens (i.e. the thick line in Figures 2a, 2b) is relevant. This represents what Edgeworth called the region of final settlements. Any final settlement, i.e. any of the points on this segment, will be serve as a solution to bilateral barter exchange. Thus, Edgeworth concludes, Jevons's assertion that the solution is unique is wrong. Bilateral barter exchange is quite indeterminate. Note that this does not mean that a ratio of exchange would not be attained. Rather, Edgeworth argued that it was impossible to predict, from the outset, which ratio of exchange would be finally attained in barter exchange. Alternatively stated, we know the parties will be able to agree on a specific final settlement -- and will trade accordingly -- but there are many trades which are plausible candidates, and there is no reason to presume that one rather than another will be chosen.

[Note: The nomenclature surrounding this can be somewhat confusing. Peter Newman (1965) calls the portion of the contract curve within the lens formed by UF and UR as the "contract curve", and refers to the rest of it as the "efficiency locus". The modern custom is to call the entire locus the "contract curve" (as Edgeworth initially did, although he then wavered on this) and refer to the part within the lens merely as "the core" (as we shall see later). This is what we shall adhere to for much of the rest of this survey. Whatever the case, this is an unfortunate choice of terms. For consistency, Edgeworth ought to have called the entire locus the settlement curve and the segment within the lens the final settlement curve.]

(3) Recontracting and the Core

Edgeworth challenged Jevons's assertion of determinacy of barter exchange by showing that "Jevons's conditions" that (1) indifference curves are tangent at the solution trade and (2) that the solution trade is mutually beneficial, will yield a set of solutions represented by all the points on the region of final settlements. But this is not the end of the story. Jevons basic intuition might still be correct if a reason can be found to choose one of these points over all the others. And why not? Further higgling or bargaining between Robinson and Friday could make some solutions more likely than others. Determinacy could still result. Edgeworth has not yet proved that indeterminacy will result necessarily.

Edgeworth ruled out determinacy by modelling the higgling process as a "recontracting" process and strictly forbidding any other bargaining process. It is Edgeworth's particular "recontracting" vision of barter exchange that yields indeterminacy. It is not self-evident (as we will discuss later) that other forms of higgling will not yield a determinate solution.

Formally, Edgeworth defined a final settlement as "a settlement which cannot be varied by recontract within the field of competition" (1881: p.19). As both the terms "recontract" and "competition" as used by Edgeworth can be a bit slippery, they require a little more exploration.

Edgeworth defined the term "recontract" quite carefully as the process of breaking a proposed trade by organizing another trade. The choice of words here is not incidental. A contract is merely a suggestion by one of the parties, it is an offer of a trade. This is why one of the parties to the contract can recontract, i.e. break off from the original suggested contract "without the consent of others" (Edgeworth, 1881, p.17) and contract with someone else. This, as noted, is supposed to capture the process of "higgling" in the market: a seller may make a tentative offer to a buyer and the buyer may agree -- this is a contract for a trade; but, before the trade is consummated, upon hearing that there is a better deal somewhere else, the buyer can easily break off the original contract and go to the other seller -- this is recontracing. As he writes in a famous passage:

"'`Is it peace or war?' asks the lover of `Maud', of economic competition, and answers hastily: It is both, pax or pact between contractors during contract, war, when some of the contractors without the consent of others recontract. Thus an auctioneer having been in contract with the last bider (to sell at such a price if no higher bid) recontracts with a higher bidder. So a landlord on expiry of lease recontracts, it may be, with a new tenant." (Edgeworth, 1881: p.17)

The suggestive, hypothetical nature of a "contract" in Edgeworth's definition contrasts sharply with the common usage of the term. When we think of "contract" it is precisly supposed to be something that is "binding" to the parties involved. However, in Edgeworth, a contract it is not binding but always subject to recontract.

However, an important caveat is in order: recontracting is not negotiation or bargaining in the common sense of those terms; one cannot break a contract simply because one wishes to do better and hopes for a better deal by making a counter-offer. Rejection has to be credible in some manner. In order for a rejection not to be mere capriciousness, the person who rejects must have an alternative. Breaking a contract is only possible if one can do better and has a better deal elsewhere to go to. Thus, a proposed contract is binding if there are no alternatives to which one of the parties can escape to. This is precisely why Edgeworth used the term "recontracting" rather than "bargaining". Friday can only "reject" a proposition by "recontracting" with someone else, otherwise he is committed to the proposed trade.

We can then see how Edgeworth's "recontracting" logic will lead to a "mapping of the competitive field" in the manner shown in Figures 2a and 2b. At the simplest level Robinson can offer Friday a contract. Friday has little choice but to accept the offer or reject it -- in which case he gets nothing at all. "Rejecting an offer" is equivalent to recontracting with one's self. In Figures 2a/2b, "nothing", for Friday, is represented by the autarky indifference curve UF. Thus, Friday will reject any offer by Robinson Crusoe (like contract A), which gives him less utility than not trading at all. Similarly, Robinson will reject any contract offered by Friday that will give him a utility that is lower than UR. Thus, only the shaded area in the "lens" formed by the autarky indifference curves, UR and UF cannot be "blocked" by Robinson or Friday individually recontracting with their own selves.

Now, within this lens, both Robinson and Friday will recontract with each other to exclude all points that do not lie on the contract curve. For example, if Robinson offers Friday contract F, then Friday will reject it because he can make a better contract with Robinson -- namely contract C. As C is better than F for both, then Robinson and Friday will recontract from F to C. However, as noted earlier, it is not certain that C will actually be chosen. Contract D, for instance, is also acceptable to Robinson and Friday. Notice that it is one of the fundamental rules of Edgeworth's recontracting process that Friday cannot reject trade D in the hope that Robinson would offer him trade C instead (which is personally better for him). If Friday capriciously says "no" to proposed trade D, that "no" is not credible and Robinson would have no incentive to change it, for he knows that Friday has no alternative. At D, as Friday has no one to recontract with, he must accept it (or rather, he cannot reject it).

There is no reason, then, to suppose from the outset (i.e. from the endowment/origin) that Robinson and Friday will settle on trade C as opposed to trade D (or vice-versa) via the recontracting logic. There is, in fact, a deadlock between these trades. This is what Edgeworth meant by the "indeterminacy of barter", i.e. the idea that there is more than one "final settlement" which Robinson and Friday can agree upon in barter exchange via recontracting. The "region of final settlement" and what has been termed by later economists as "the core" of the economy. All that is known, Edgeworth asserts, is that the actual final settlement that will be undertaken lies somewhere on this curve, but where exactly cannot be ascertained beforehand.

So far we have been discussing everything in terms of two-person or "bilateral" barter exchange -- e.g. exchange between Robinson and Friday -- but Edgeworth demonstrated that his story is still relevant when there are many people engaged in barter exchange. It is important to note that Edgeworth did not break down barter exchange with many people into a series of bilateral barter exchanges (e.g. taking various pairs of people in sequence). Rather, he was concerned with multi-lateral barter exchange: everyone exchanges with everyone else at once. In multi-lateral exchange, any "contract" or proposition to trade must be fully specified: it must state what everyone in the economy is to receive and surrender.

As anybody can propose a multi-lateral trade, then anybody in the economy can reject it. However, as we know, rejection has to be credible in the sense that he must have an "alternative" for recontracting. Now, with multi-lateral exchange, the set of alternatives is more interesting than in the bilateral case. The simplest alternative -- not trading at all (recontracting with one's self) -- remains available to every trader individually. Thus a proposed multi-lateral trade which gives someone in the economy less than he would get by not participating, will necessarily be blocked by that person. But with multi-lateral exchange, a new type of alternative emerges: namely, recontracting with some, but not all, of the participants, forming what has been called a "blocking coalition".

To see this, suppose there are now three agents, Robinson, Friday and Quixote. Imagine that Robinson proposes a trade that Friday and Quixote cannot individually reject. But, together, Friday and Quixote can perhaps do something about it. Specifically, Friday can propose to Quixote to withdraw from Robinson's proposition and to make a bilateral trade among themselves -- leaving Robinson out of it. If this private Quixote-Friday trade is better for both Quixote and Friday than the original was, then certainly they will undertake it as an alternative to Robinson's proposition. Thus, Quixote and Friday are said to have formed a "coalition" to block Robinson's proposed trade. This Quixote-Friday alternative sub-trade, is a "credible" threat to Robinson's proposition. It is a proper "recontract".

[Note: But is it really credible? What if the sub-trade that Friday proposes to Quixote can itself be bettered, e.g. if Robinson can make his own offer to Quixote for a sub-trade to get out of his coalition with Friday? In this case, Friday's threat about forming a coalition with Quixote to block Robinson's original proposition is actually not very credible -- because the Friday-Quixote coalition itself is not very stable. Edgeworth's theory of the core does not require such a high degree of credibility. Nonetheless, game theorists have explored this possibility and come up with concepts analogous to the "core" but which require that every blocking coalition is more credible in the sense that it itself cannot be subsequently blocked by another coalition (and that the subsequent blocking coalition is not itself blockable by another coalition, etc.). The stable set, the bargaining set, the kernel, etc. represent various iterations of this idea.]

In sum, the Edgeworthian "core" represents the set of trades/allocations which cannot be "blocked" by any coalition via a recontracting process. The recontracting process works as follows: someone proposes a trade that reallocates endowment to everybody else, and either someone disputes it (and that allocation is eliminated from consideration) or no one does (in which case it becomes part of the core). In order to "dispute" a proposed allocation, a person must prove it is a credible dispute: i.e. that they either lose by it (i.e. it is better for him not trade at all) or that he can get a better deal by forming a coalition with some other people in the economy who, trading among themselves, can all do better (or at least no worse) than the originally proposed allocation.

What about competition? This is a bit of a hornet's nest. Edgeworth viewed competition merely as the number of parties available for exchange. This may not seem self-evident: why should sheer numbers of people imply "more" competition? Many economists have objected to this definition; competition, they argue, is about "strategic" behavior or the degree of ability to influence one's own prices, not about mere "numbers".

Edgeworth explicitly acknowledged Augustin Cournot (1838) for the connection. Cournot had clearly expounded the idea that "monopoly power" -- or the ability to influence one's own price -- is reduced as the number of competing sellers increases. For Cournot, a large number of firms would be necessary to swamp the market power of any individual firm. "Going beyond Cournot, not without trembling" (1881: p.47), Edgeworth inserted this notion into his barter exchange economy. Of course, in this context, there are no "prices" to be influenced or uninfluenced. So for Edgeworth, "competition" was simply defined as "large numbers of traders" without any obvious justification.

When considered in the context of Edgeworth's process of recontracting, the idea of competition as "sheer numbers" seems to make at least some sense, although the logic is not watertight. Intuitively, the more people there are in a marketplace, the easier it becomes to recontract, i.e. to break a contract and take up another. From a seller's viewpoint, it is more difficult to insist on an outlandishly self-serving trade if the buyer has many other sellers he can go to. Conversely, from a buyer's point of view, it is difficult to convince the seller to give him a good bargain if there are many other buyers mulling around the place.

Be that as it may, Edgeworth did not spend much ink on this part. But he was very interested in the connection between the degree of competition (or rather, the number of traders) and the degree of indeterminacy of contract. Specifically, he argued that as the number of people increases, the degree of indeterminacy (i.e. the size of the core) is reduced; in the limiting case of perfect competition (i.e. an infinite number of people), contract will in fact be absolutely determinate, i.e. the only contract will be the unique market equilibrium contract, i.e. D y/D x = px/py. This proposition has become known as "Edgeworth's conjecture", which we will turn to in more detail later.

(3) The Core: mathematical restatement

It is now possible to make a mathematical restatement of the concept of the Edgeworthian "core" in a general, multi-lateral barter context. Let us denote the set of all agents in the economy by H. Let the preferences of the hth agent be denoted by the binary relation ³ h and let his endowment be a vector eh. Thus, the hth agent is completely described by his preference-endowment pair, (³ h, eh). An allocation is denoted x = {xh}hÎ H, where xh is the amount received by the hth agent.

Let us define a "coalition" S as a group of agents, thus S Í H, who trade among themselves. Now we define "blocking":

Blocking: A proposed allocation x is "blocked" by the coalition S if there is some other alternative allocation y such that:

(i) yh ³h xh for all h Î S
(ii) there is a t Î S such that yt >t xt
(iii) å hÎ S yh £ å hÎ S eh

So, (i) says that no one in the coalition is worse off at the alternative allocation; (ii) says that at least one person (the t th) in the coalition is strictly better off while (iii) says that the alternative allocation must be feasible for the coalition members (i.e. that the "subtrade" is possible).

[Note: sometimes (i) is strengthened by requiring yh >h xh for all h Î S, so that all members of the coalition must be strictly better off at the alternative. Also, some writers prefer to say "coalition S improves upon allocation x" rather than "coalition S blocks x" because, as Werner Hildenbrand explains, "The core expresses what coalitions can or cannot do for themselves, not what they can or cannot do to their opponents" (Hildenbrand, 1974: p. 128). We respectfully, but adamantly, disagree. We believe that the term "improving upon" gives the impression that the coalition trades are actually being undertaken, and not merely used as credible threats in a multi-lateral negotiating process (which is really what the "mapping of the competitive field" is all about). Consequently, we shall adhere to the "blocking" language.]

Let us proceed with more definitions:

Unblockable: an allocation x = {xh}hÎ H is "unblockable" if there is no coalition S Í H with alternative allocation {yh}hÎ H that can "block" x.

and, finally, the definition of the Edgeworthian core:

Core: the "core" of an economy, denoted C(E), is the set of all allocations x = {xh}hÎ H in the economy which are unblockable.

Simple enough. Now, let us turn to proving the simplest proposition, namely that the core lies somewhere on the contract curve (as we saw in Figures 2a and 2b). As the contract curve is the set of all Pareto-optimal points, this amounts to merely proving that all allocations in the core are Pareto-optimal.

Theorem: Every allocation in the core, x Î C(E), is Pareto-optimal.

Proof: Let x Î C(E). Suppose it is not Pareto-optimal. Then, there is an allocation y ={yh}hÎ H such that yh ³ h xh for all h Î H (and at least strictly better for some t Î H). If this is true for all h Î H, then, for any S Í H, it is true for all h Î S. Thus, we can choose a coalition S such that yh ³ h xh for all h Î S and strictly for some t Î H. Thus, x ={xh}hÎ H can be blocked. Thus x Ï C(E). A contradiction.§

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