Irreducibility, resource relatedness and survival with individual non-convexities

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E U I W O R K I N G P A P E R No. 8 9 / 4 2 7

Irreducibility, Resource Relatedness, and Survival

with Individual Non-Convexities

PETER J. HAMMOND © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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E U R O P E A N U N I V E R S I T Y I N S T I T U T E , F L O R E N C E D E P AR TM E N T OF ECO N O M IC S

E U I W O R K I N G P A P E R No. 8 9 / 4 2 7

Irreducibility, Resource Relatedness, and Survival

with Individual Non-Convexities

PETER J. HAMMOND © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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N o part o f this paper may be reproduced in any form without permission o f the author.

European University Institute Badia Fiesolana - 50016 San Domenico (FI) -

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IR R E D U C IB IL IT Y , RESO UR CE R ELATED NESS A N D SURVIVAL

W IT H IN D IV ID U A L N O N -C O N V E X IT IE S

Pe t e r J . Ha m m o n d

Department of Economics, European University Institute, Badia Fiesolana, 50016 S. Domenico (FI), Italy and Stanford University, CA 94305-6072, U.S.A.

June 1989; revised July and December 1989

A revised version o f the second part o f a paper presented to the C onference on “ General Equilib rium and Growth: The Legacy o f Lionel M cK enzie” at the University o f Rochester, M ay 5th-7th, 1989. The first part o f that paper, establishing other properties o f com pensated equilibria, remains incomplete.

Abstract

Standard results in general equilibrium theory, such as existence and the second efficiency and core equivalence theorems, are most easily proved for compensated equilibria. A new condition establishes that, even with individual non-convexities, in compensated equilibrium any agent with a cheaper feasible net trade is also in uncompensated equilibrium. Obvious generalizations of McKenzie’s irreducibility assumption then imply that (almost) no agent is at a cheapest point, so the easier and more general results for compensated equilibria become true for uncompen sated equilibria. Survival of all consumers in uncompensated equilibrium depends on satisfying an additional assumption similar to irreducibility.

ACKNOWLEDGEMENTS

This work extends som e o f the results presented in joint papers with Jeffrey C oles — see C oles and Hammond (1986, 1989a, 1989b). That work was in turn largely inspired by Lionel M cK en zie’s presidential address to the Econometric Society in Vienna, 1977 — see M cK enzie (1981). 1 am grateful to Jeff for encouraging me to extend these results, and to Kenneth Arrow, David Gale, and the discussant Sidney Afriat for helpful suggestions. They are not responsible for m y errors.

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IR R E D U C IB L E E C O N O M IE S

1. Introduction

1.1. General Equilibrium Theory in the 1950’s

The heyday of mathematical economics in general, and of general equilibrium theory in particular, was surely the decade of the 1950’s. For the first time general logically consistent models of a competitive market economy were developed and existence of a competitive equilibrium was proved. First Arrow (1951) proved the two efficiency theorems of welfare economics. The first of these theorems shows that any competitive equilibrium is Pareto efficient. The second shows that, under assumptions such as convexity and continuity which have since become standard, almost any Pareto efficient allocation can be achieved through competitive mar kets in general equilibrium, provided that lump-sum redistribution is used to ensure that each individual has the appropriate amount of puchasing power.

After these efficiency theorems, the next step was to give a proper proof of the existence of equilibrium for a given exogenous distribution of purchasing power. McKenzie (1954a) gave such a proof for a world economy with continuous aggregate demand functions for each nation. This was done using newly available fixed point theorems in mathematics such as Kakutani’s (1941).1 McKenzie’s proof used arguments which were obviously capable of significant generalization. Also, since he was using Graham’s (1948) model of international trade in which whole nations were being considered as economic units, McKenzie (1954a, b) naturally saw no need to assume that each nation has the preferences of a representative national consumer. Indeed, Gorman’s (1953) work the year before had already shown how restrictive such an assumption would be, even if it were posible to classify all individuals as nationals of their own nations in which they live permanently.

1 Despite considering demand correspondences rather than (single-valued) demand functions, in his later general existence p roof M cK enzie (1959) was able to use just B rouw er’s fixed point theorem for continuous functions, rather than Kakutani’s theorem for correspondences. See Khan (1989) for further details.

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In the very next issue of Econometrica, Arrow and Debreu (1954) gave the first published general proof of existence for an economy in which preferences and endowments, rather than demand functions, are exogenous. Of course, one could well argue that this model is somewhat more appropriate for general ex istence proofs than Graham’s, because one should perhaps consider a model of international trade in which each consumer in each nation is treated as a separate economic unit. Anyway, these results were then extended, in rapid succession, in such works as those of McKenzie (1955, 1959, 1961), Gale (1955), Nikaido (1956, 1957), and Debreu (1959, 1962).

Although no excuse for revisiting the 1950’s is required on this occasion, I shall nevertheless provide one. It should come as no surprise that the arguments of these path-breaking papers can be somewhat streamlined thirty or more years later. Nor that the assumptions can be considerably relaxed. In the process of doing so, however, it has become very useful to adopt an approach first hinted at in Arrow and Debreu (1954), and then put into practice in Arrow and Hahn (1971).2 An important part of this approach to extending the results of general equilibrium theory involves concepts closely related to McKenzie’s (1959, 1961, 1981, 1987) irreducibility assumption. The key role played by this assumption (or Arrow and Hahn’s notion of “resource relatedness” ) has often been neglected.3

1.2. From Compensated to Uncompensated Equilibrium

Indeed, following the approach of Arrow (1951), it is almost trivial to prove the first efficiency theorem. In its weak form, this states that, without any as sumptions at all, any competitive equilibrium allocation is weakly Pareto efficient, in the sense that there is no other physically feasible allocation which moves all agents simultaneously to a preferred allocation. In its stronger and more familiar form, the first efficiency theorem states that if agents have locally non-satiated preferences, then any competitive equilibrium allocation is Pareto efficient in the

2 See also M oore (1975).

3 For example, it is not discussed at all in such surveys as Duffie and Sonnenschein (1989), or such advanced texts as Hildenbrand and Kirman (1988).

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sense that there is no other physically feasible allocation which moves some agent or agents to a preferred allocation without simultaneously moving others to a dis- preferred allocation. Equally trivial is the theorem which says that any competitive equilibrium allocation is in the core.

The departures from standard assumptions which interest us include non convexity of feasible sets or of preferences, discontinuous preferences, the possible non-survival of some individuals, boundary problems, and exceptional cases such as that discussed by Arrow (1951). Such departures from standard assumptions can do nothing to perturb these very simple and robust results. They do, however, create serious difficulties for the existence of the competitive equilibrium which is predicated in the first efficiency theorem. They may also make the core empty, or lead to violations of the second efficiency and core equivalence theorems. One of the principal aims of our work, therefore, is to generalize the conditions under which these important results in general equilibrium theory retain their validity, especially in continuum economies for which individual non-convexities may be come smoothed out.

To this end, an approach which appears to be very helpful is to break the problem up into two parts. It turns out to be relatively easy to prove weak versions of the existence, second efficiency, and core equivalence theorems, using a weaker notion of compensated competitive equilibrium. This terminology is apparently due to Arrow and Hahn (1971), though the concept is surely due to Arrow and Debreu (1954, Section 5.3.3). In compensated equilibrium every agent is minimizing expenditure at equilibrium prices, subject to not falling below an upper contour set associated with the preference relation (cf. McKenzie, 1957). Whereas uncompensated equilibrium is the more familiar and natural kind of equilibrium in which every agent is maximizing preferences subject to the budget constraint that net expenditure at equilibrium prices cannot be positive. This distinction is brought out in Section 2. The weak version of the existence theorem is simply that a compensated equilibrium exists. The weak version of the second efficiency theorem is that (almost) any Pareto efficient allocation can be achieved

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as a compensated equilibrium, provided that appropriate lump sum redistribution of wealth has occurred. The weak version of the core equivalence theorem is that any allocation in the core of a continuum economy must be a compensated equilibrium. These three weak theorems have been stated and proved in various ways by numerous different authors. The assumptions under which they are valid do not necessarily require individuals’ feasible sets to be convex, nor do they require that all individuals survive in compensated equilibrium. They even allow some discontinuities in preferences.

Of course, these weak theorems concerning compensated equilibrium still leave us with the problem of proving the existence of an uncompensated competitive equilibrium, as well as the usual versions of the second efficiency and core equiva lence theorems which refer to uncompensated equilibria. One way to prove these results, obviously, is to pass as directly as possible from a compensated equilibrium to an uncompensated equilibrium. This is precisely where irreducibility becomes important. Section 3 discusses some of the precursors to the idea of irreducibility, especially the extremely implausible interiority assumption used originally by Ar row (1951) and by Arrow and Debreu (1954). As is well known, the main function of this assumption is to ensure directly that all agents have cheaper points in their feasible sets. If individuals have convex feasible sets and continuous preferences, this ensures that any compensated equilibrium is also an uncompensated equi librium. But since our work is intended to allow both non-convex feasible sets and discontinuous preferences, a different assumption concerning particular- cones generated by the feasible set and by the set of preferred net trade vectors is used to establish this standard conclusion. This is also taken up in Section 3, along with an aggregate interiority condition which rules out Arrow’s (1951) exceptional case. This aggregate interiority condition is sufficient for at least one individual to have a cheaper feasible net trade vector.

Thereafter, Section 4 presents a “non-oligarchy” condition which proves ad equate to show that a particular- compensated equilibrium allocation is an un compensated equilibrium. For the special case of “linear exchange” economies,

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in which each consumer’s indifference map consists of parallel line segments in the non-negative orthant, this condition is virtually equivalent to irreducibility, according to Gale’s (1957) and Eaves’ (1976) definition, or to Gale’s (1976) defi nition of “self-sufficiency.” The results of this section confirm that an important part of a more appealing sufficient condition for compensated equilibria to become uncompensated equilibria involves considering the extent to which different agents in the economy are able to benefit from one another’s resources. Moreover, a re sult due to Spivak (1978) is extended to show that a “convexified non-oligarchy” condition is necessary as well as sufficient for every agent to have a cheaper point in compensated equilibrium with lump-sum redistribution.

Irreducible economies finally make their entrance in Section 5. McKenzie’s fundamental contribution was incompletely anticipated in the latter part of Arrow and Debreu (1954). As mentioned above, an earlier paper by Gale (1957) contained a somewhat related idea for linear exchange economies. Irreducibility was duly recognized by Debreu (1962). Arrow and Hahn’s (1971) later concept of resource relatedness turns out to be a generalization. Section 5 presents a condition which encompasses virtually all previous conditions of this kind,4 and is also suitable for economies in which individuals may have non-convex feasible sets.

In addition, irreducibility was extended by Moore (1975) and McKenzie (19S1, 1987) to finite economies in which consumers may have to rely on others to sur vive. The last part of Section 5 generalizes their sufficient conditions which ensure that any compensated equilibrium in the economy is actually an uncompensated equilibrium in which all consumers are able to survive. The “convexified inter dependence” condition presented here is necessary as well as sufficient for every agent to have a cheaper point in compensated equilibrium.

Finally, following Hildenbrand’s (1972) modification of irreducibility, Section 6 shows how to extend the analysis to continuum economies. There is no need to consider convexified versions of the earlier non-oligarchy and interdependence’

4 The sole exception o f which I am aware is G ale’s (1976) special condition for linear exchange econom ies. © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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conditions because the continuum of agents itself ensures convexity. Section 7 contains a few concluding remarks.

2. Preliminaries

2.1. Basic Assumptions

2.1.1. A Finite Set o f Agents

Assume that there is a finite set A of economic agents.

2.1.2. The Commodity Space

Assume that there is a fixed finite number t of physical commodities, so that the commodity space is the Euclidean space 9ft*.

2.1.3. Consumers’ Feasible Sets

Next, assume that every agent a £ A has a fixed feasible set X a C 3ft* of net trade vectors x a satisfying 0 £ X a. If agent a £ A happens to have a fixed endowment u a £ 3ft*, then each x a £ X a is equal to the difference ca —coa between a feasible consumption vector ca and this endowment vector. By following Rader (1964) and others, however, and considering only feasible net trade vectors, the formulation here allows domestic production activities such as storage, as well as the kind of economy with small farmers considered in Coles and Hammond (1989a). Note the assumption that autarky is feasible, but note too that this does

not imply that autarky enables the consumer to survive. 2.1.4. Feasible Allocations

An allocation x := ( xa)a^A € IIa€4 is a profile of net trade vectors which are individually feasible for all agents a £ A. A feasible allocation x also has the property that the aggregate net trade vector YlaeA Xa = 0* N°te that only an exchange economy is being considered, and that free disposal is assumed only to the extent that some individual agents can dispose of goods freely. Let T denote the set of all such feasible allocations.

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2.1.5. Consumers’ Preferences

It is also assumed that every agent a G A has a (complete and transitive) weak preference ordering fca on the set X a, and an associated strict preference relation

y a which is globally non-satiated — i.e., the preferred set Pa( x a) := { x'a G X a \ x'a >■« x a } >s non-empty for every x a G X a.

Note especially that there is no assumption of monotonicity, free disposal, or even of local (as opposed to global) non-satiation. Nor any assumption of convexity or continuity. This allows consideration of indivisible goods and also of deterministic incentive compatible state contingent commodity contracts (as in Hammond, 1989). It also allows the kind of preference discontinuities which arose naturally in Coles and Hammond (1989a, b).

2.1.6. The Classical Hypotheses

Some later results, however, will require standard convexity and continuity hypotheses. Indeed, say that agents’ preferences satisfy the classical hypotheses provided that, for all a G A:

(i) the set X a of feasible net trade vectors is convex;

(ii) whenever x'a fcax a, x " fc„x a, and also 0 < A < 1, then (1 — A)xa + Ax" fcax„, and if also x " x a, then (1 — A)xa + Ax" y a x a;

(iii) both the feasible set X a and, for every x a G X a, the “lower contour” set consisting of all net trade vectors x'a that satisfy x a fcax a, are closed sets; (iv) for every x„ G X a, the net trade vector x a is on the boundary of the (non

empty) preferred set P „(x a) := { x'a G X a | x'a X a x„ }.

The above classical hypotheses are automatically satisfied when the set of all feasible allocations is convex, and when preferences axe convex, continuous, and locally non-satiated. Of coruse, these latter are standard assumptions in general equilibrium theory for finite economies; indeed, (iii) above is usually supplemented by the requirement that each upper contour set Ua( x a) = { x'a \ x'a &0x „ } is also

closed. © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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2.1.7. The Price Domain

The set of all allowable price vectors will b e l p G S f ^ l p ^ O } . Note that negative prices are allowed because there has been no assumption of free disposal.

2.2. Equilibrium

2.2.1. The Budget, Demand, and Compensated Demand Correspondences

For each agent a £ A, wealth level wa, and price vector p ^ 0, define the

budget set

Ba(p, wa) := { x £ X a \px < w a }

of feasible net trade vectors satisfying the budget constraint. Note that B a(p, wa) is never empty when wa > 0 because of the assumption that 0 £ X a.

Next define, for every a £ A and p ^ 0, the uncompensated demand set

Ca (Piwa) ■ = { * £ B a ( p , W a ) I X 1 y a X => p x ' > W a } = arg max{ | x £ B a(p, wa) }

X

and also the compensated demand set

( a (p ,w a) := { x £ B a(p,w a) | x' y a X ^ p x ' > w a }.

Note how C,a(p,wa) C Ca(P>wa) as an immediate implication of the above defini tions.

2.2.2. Compensated and Uncompensated Equilibria

An uncompensated (resp. compensated) equilibrium is a pair (x ,p ) consisting of a feasible allocation x and a price vector p such that, for all a £ A, both p x u = 0 and x a £ C^(P,0) (resp. C f(P,0)).

An uncompensated (resp. compensated) equilibrium with lump-sum redistribu

tion is a pair (x ,p ) consisting of a feasible allocation x and a price vector p such

that x a £ (V ( p ,p x a) (resp. £a { p ,p x a)) for all a £ A.

Note how budget exhaustion is built into these definitions. It would be automatically satisfied, of course, if each agent had preferences which were locally

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non-satiated at every point of the relevant budget set B a(p ,w a) or B a(p, 0), but no assumption of local non-satiation is made here.

3. Cheaper Points for Individuals

3.1. Arrow’s Exceptional Case

Recall that the second efficiency theorem states conditions under which a Pareto efficient allocation is an uncompensated equilibrium with lump-sum redis tribution — or, in a weaker version, under which such an allocation is a compen sated equilibrium with lump-sum redistribution. Both Arrow (1951) and Debreu (1951) announced versions of this theorem in the same year. It seems clear that Arrow was the first to realize the nature of the problem of how to prove that a compensated equilibrium with expenditure minimization by agents is also an un compensated equilibrium with preference maximization by agents. Indeed, in that paper he gave a famous example of an “exceptional case” in which expenditure minimization is insufficient to ensure preference maximization. On the other hand Debreu (1951) simply passes from compensated to uncompensated equilibrium without any attempt at a proof. He was apparently unaware of the possibility of any exceptional case, even though he acknowledges having seen an early version of Arrow’s paper. Of course this logical flaw was later set right in his succeeding works such as Debreu (1954, 1959, 1962).

3.2. Inferiority

3.2.1. Inferiority o f a Particular Allocation

In fact Arrow’s paper was presented at the same symposium as Kuhn and Tucker’s (1951) classic work on non-linear programming. They also gave an ex ample in which a constrained maximum could not be supported by shadow prices — something which is impossible for linear programs in finite dimensional spaces, because for those the well known duality theorem applies. The simplest instance of a constraint qualification is Slater’s condition, requiring that a concave program have a point in the interior of its feasible set.

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Arrow gave two alternative conditions which rule out exceptional cases. The first of these is a direct parallel of the Slater constraint qualification. This is the “interiority assumption” which simply assumes that the Pareto efficient allocation being considered gives each agent a net trade vector in the interior of his feasible set. For the case of convex feasible sets and continuous preferences, this is certainly enough to convert a compensated equilibrium with lump-sum redistribution into an uncompensated equilibrium with identical lump-sum redistribution, and to make the second efficiency theorem true.

3.2.2. Interiority of Initial Endowments

A somewhat different interiority assumption was also used in the first exis tence theorem of Arrow and Debreu (1954), and in immediately succeeding works such as Gale (1955), Nikaido (1956), and Debreu (1959). For the second efficiency theorem, the interiority assumption could refer to the particular Pareto efficient allocation whose competitive properties are being demonstrated. For an existence theorem, however, it makes no sense to refer to the equilibrium allocation until we know whether one exists. Nor should we assume that a compensated equilibrium allocation is an interior allocation, since there are many examples where it will not be — for instance, if some individuals are unconcerned about the consumption of some goods, compensated equilibrium at strictly positive prices implies that these individuals are at the lower boundaries of their consumption sets. For this rea son, then, it seemed more natural to assume that all individuals had endowment vectors in the interior of their consumption sets — or, somewhat more generally, that the zero net trade vector was in the interior of each individual’s set of feasible net trades. Since feasibility was nearly always assumed to imply survival, this was actually a kind of “strict survival” assumption. But in fact, as remarked above, there is actually no need to assume that feasibility implies survival. Armed with this survival assumption, as well as the usual assumptions that individuals’ feasible sets are convex and their preferences are continuous, it becomes quite straightfor ward to show that any compensated equilibrium is actually an uncompensated

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equilibrium.5

The absurdity of this interiority assumption was immediately apparent. Early writers would typically apologize for it and would sometimes seek to weaken it.6 After all, in a model of competitive equilibrium international trade in the world economy, non-traded goods such as local construction and other labour services have to be differentiated by the country in which they are supplied. Then the inte riority assumption requires each individual to be able to supply every non-traded good in every country of the world simultaneously. Not even internationally mobile professors of economics have this capability! Indeed, once commodities become distinguished by time and geographical location, as they should be, interiority requires all agents to be both omnipresent and immortal. This kind of consid eration surely motivated Nikaido’s (1957) ingenious relaxation of the interiority assumption. But he had to assume instead that all agents have the non-negative orthant as their consumption set, that there is a positive total endowment of every commodity, and also that preferences are strictly monotone — in particular, that every good is strictly desirable.7 The last assumption is hardly more acceptable than interiority when one considers goods differentiated by location. Chinese tea is wonderful almost anywhere at almost any time, but all the tea in China only benefits those who are actually there.

5 O f course Debreu (1959) and many successors prefer to use the assumption that en dow ments are interior in order to demonstrate upper hemi-continuity o f the uncompensated rather than o f the compensated or o f the quasi demand correspondence, and then to prove existence o f an uncompensated equilibrium by applying a fixed point theorem directly to the aggregate excess uncompensated demand correspondence. This, however, tends to obscure the role o f the objectionable interiority assumption in the p roof o f existence.

6 Kenneth Arrow tells me that at first both he and Debreu, in independent unpublished work, had overlooked the need for some additional assumption such as interiority. Later they held up the publication o f their joint paper Arrow and Debreu (1954) while they formulated a less unsatisfactory alternative to interiority. This alternative is discussed in Section 5.1 below.

7 Actually, despite N ikaido’s contribution, many subsequent papers in general equilibrium theory seem to com bine non-negative orthant consumption sets, strictly m onotone preferences, and the interiority assumption which he showed to be unnecessary in the presence o f the other two assumptions. © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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3.3. Cheaper Feasible Points

3.3.1. Introduction

Despite its evident absurdity, the interiority assumption remains the one which is most frequently encountered in the general equilibrium literature. Yet al ternatives have been proposed, starting with a second assumption in Arrow (1951) which was weaker than the interiority condition he had used first. Whereas interi ority requires the agent to be able to supply a little of every good, Arrow’s second assumption is that every agent can supply a little of at least one good having a positive price in the compensated equilibium price system. Unfortunately, how ever, this assumption has the obvious defect that it can only be checked after the compensated equilibium price system is known. Or at least it is necessary to know which goods have zero prices and which goods have positive prices in this system.

3.3.2. Cheaper Points for Non-Convex Feasible Sets

The usual demonstration that the existence of a cheaper feasible point for an individual converts a compensated into an uncompensated equilibrium for that individual depends upon the feasible set being convex and the strict preference set being open relative to the feasible set. To generalize this to non-convex feasible sets and preferences that may not be continuous requires additional assumptions. When some goods are indivisible, it has been common to assume that divisible goods are “overridingly desirable,” meaning that any commodity bundle with in divisible goods can be improved by moving to another with only divisible goods. Assumption 1 below seems at first to be a significant generalization of this strong assumption.

For every a & A and x a 6 X a, let:

K a(*a) ■— co{ x'a e I 3xa € X a & 3A > 0 : x'a = x a + A (x a - x a) };

K a ( x °) : = c o { x 'a £ | 3z<i 6 Pa(x a) & 3A > 0 : x'a = x a + A (x a - x a) }; denote the smallest convex cones with vertex x a that are generated by the set X a of feasible net trade vectors, and by the set Pa(x a) := { x'a 6 X a \ x'a y a x a } of

net trade vectors which are strictly preferred to x a, respectively.

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ASSUMPTION 1. For every a G A and x a g X a, the convex cone K '£ (xa) is open relative to the convex cone K a(x a).

Note that often K a(x „ ) will be the whole space 5ft*, but this still makes As sumption 1 meaningful. The convex cone K £ ( x a) may also be the whole space

if preferences are non-convex. Of course, when the feasible set X a is convex, Assumption 1 is automatically satisfied if preferences are continuous, because then

Pa( x a) is open relative to X a, which makes K ^ ( x a) open relative to K a( x a).

In Coles and Hammond (1989b), Assumption 1 unfortunately does not apply to points on the boundary of the survival set at which there are preference discon tinuities. In that paper, therefore, it was necessary to use a different assumption, ensuring that the set of individuals on the boundaries of their survival set has measure zero. Nor perhaps is Assumption 1 as easy to check as it should be. Nevertheless the assumption gives what is needed, by ruling out such troublesome examples as those discussed in Broome (1972), Mas-Colell (1977), and Khan and Yamazaki (1981).

3.3.3. Overriding Desirability of Divisible Goods

When some goods are indivisible, however, Assumption 1 often turns out to be no weaker than the assumption of overriding desirability. To see this, consider the case with just one divisible and one indivisible good. Indeed, suppose that A’„ = 3f+ x Z+ where Z+ denotes the subset of whose members have (non-negative) integer co-ordinates, representing quantities of the indivisible good. Suppose too that preferences are strictly monotone in the sense that, if x a = (ya, z a) and x’a = (y'a,z'a) are both members of X a with y'a > ya and z'a > za, then x'a ~ ax a, with strict preference unless both y'a = ya and z'a = za.

Next, say that divisible goods are overridingly desirable if, whenever x a = (y0, z«) and x'a = (y'a,z'a) are both members of X a, then there exists ya £ R+ such that (ya, z 'a) X a (ya,z a).8

8 This assumption appears to have originated with B room e (1972). It has since been used by, amongst others, M as-Colell (1977), Khan and Yamazaki (1981), and Hammond, Kaneko

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Thus, even though z'a may involve much less of the indivisible good than za does, the move from za to z'a can always be outweighed by a move from ya to ya in the divisible good.

Le m m a. Under the hypotheses o f the first paragraph o f this Section, Assumption

1 implies that divisible goods are overridingly desirable.

P R O O F : Suppose that (ya, za) is any member of X a with z „ > 0 . Then the hy potheses imply that the convex cone K a(Va,Za) with vertex (ya,z a) which is gen erated by the feasible set X a = 3?_|_ X Z+ is either the whole of 5R2, in case ya > 0 , or else is just { (y, z) £ 5H2 | y > 0 }, in case ya = 0.

Suppose it were true that (ya, za) ^ a(ya, za — 1) whenever ya > 0. Then the convex cone I\ £(ya, za) with vertex (ya, z a) which is generated by the preference set Pa(ya,z a) must be a subset of { (y, z) £ 5R2 | z > za }. Also, strict monotonicity of preferences implies that the set { ( y , z) £ 3?2 | y > ya,z — za } is part of the boundary of K jf(y a, za). This contradicts Assumption 1, which implies in particular that the boundary of K jf(y a, za) must be a subset of the boundary of

K a(ya, z a).

Therefore, for every (y „,z a) £ X a with z„ > 0, there exists ya £ 3? for which (ya, za — 1) (ya, z a). Becasue preferences are transitive, an easy argument

by induction on the integer z„ then establishes that the one divisible good is overridingly desirable.

Although this lemma has been proved only for the case on one divisible and one indivisible good, it is clear' that the argument can be greatly generalized to allow many divisible and indivisible goods. The issue of whether there is an ad equate weaker assumption than overriding desirability of divisible goods will be taken up in Section 3.3.5 below.

and W ooders (1989). Henry (1970) earlier formulated a condition that the divisible good be necessary for survival. Unfortunately, however, the condition is stated in a way which appears to be inconsistent with his other assumptions. For, in the notation used here, the assumption requires that, whenever xa = (ya,za) and x'a = (y'a,z'a) are both members o f X a, then there exists ya£ 51+ such that (ya,za) y a (ya, z'a). This can never hold, o f course, if (ya, z„) = (0 ,0 ), the worst point in X a. I f instead the assumption is restricted to those xa = (ya, za) £ SK+ x Z + for which ya > 0, then there is n o inconsistency, but the assumption becom es quite different from overriding desirability. © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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3.3.4. The Cheaper Point Theorem

T H E O R E M . Suppose that x „ £ CC( p ,p x a) JS a n y compensated equilibrium for individual a at the price vector p ^ 0. Then, if Assumption 1 is satisfied, and if there is also a “cheaper feasible point” x„ £ X a with p x a < p x „ , it follows that

x„ £ Cu (p ,p x a) — i.e., that x„ is an uncompensated equilibrium.

P R O O F : Suppose that x„ is a compensated equilibrium which is not also a n uncompensated equilibrium. Then there exists x'a £ X a such that x'a y a x a and yet, because x „ is a compensated equilibrium and x„ is a cheaper point,

px'a = p x a > p x a. Let L := [x„,x'a] denote the line segment joining the two

points x a and x'a. It is wholly contained in the convex cone K a[xa )■ But the convex cone I ( ^ ( x a) is open relative to A'a(.r„), by Assumption 1. So L Cl (x „ ) is open relative to L. Since x'a £ this implies that there must be some small e > 0 for which e < 1 and also

x\ := x'a + c ( x a - x'a) £ L n K £ ( x a).

Because x ' £ K ^ ( x a), there must exist x " and A > 0 such that x " = x „+ A (x a — x a) and x " y a x a. This implies, however, that

p x " = p x a + Ap(x' - x a) = (1 - A)px„ + Apx'„ + A ep(xa - x ' a) = (1 — A e)p x a + A e p .tj < p x a

because px'n = p x a > p x „. Yet this contradicts the hypothesis that x„ £ C C(P,P*a)- I

3.3.5. The Necessity o f Assumption 1

Though Assumption 1 has some undesirable implications, as discussed in Sec tion 3.3.3, the following result shows that there is a sense in which it is in e v it a b le .

Le m m a. Suppose there exists x„ £ X a for which the convex cone (x a) is

not open relative to I ia( x a). Then there exists a price vector p ^ 0 such that

*« e ( a ( p ,p x « ) and yet x„ g tf(p,pxa), even though p i » < p x „ for some Xa £ X a — i.e., there is a compensated equilibrium net trade vector that is not an uncompensated equilibrium, even though there is a cheaper point.

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PROOF: Because K £ ( x a) is not open relative to K a( x a), the convex cone K £ ( x a) is a proper subset of and there must also exist some x'a which is a boundary point of K £ ( x a) and yet an interior point of K a( x a) — i.e., there must exist

x'a

£

(£„)

n bd7t'^(xa)n

i n t K a(xa)-9

Because x'a is a boundary point of K ^ ( x a), which is a cone with vertex x a, there must be a price vector p ^ 0 and an associated hyperplane p x a = a = p x a =

px'a through the two points x a and x'a which supports the convex cone K ^ ( x a)

at x'a — i.e., p x a > a whenever x a £ K £ ( x a)- Since Pa( x a) C K „ ( x a ) , it follows that x a y a x a implies p x a > p x a. This shows that x a £ C~ (PtP io

)-Because x'a £ K ^ ( x a), there exist a finite collection of net trade vectors

XÌ 6 Pa(£a) and associate positive scalars \] ( j = 1 to J ) for which x'a = x a + YLj-i (x Ja — %a)- But it has already been proved that x a )~a x a implies p x a > p x a, and so p x 3a > p x a for j = 1 to J. Since p was constructed to satisfy p x a = px'a, it follows that 0 = p(x'a — x „) = ^3 P ÌXÌ — x a) > 0 and so that J2j=i X P (x a ~ i o ) = 0. Since AJ > 0 and p x ]a > p x a ( j = 1 to J), this implies

that p x ]a = p x a for j ~ 1 to J. Because x]a x a (j = 1 to J), it follows that x a $ Ca (P ,P x

a)-Finally, because x'a £ in t if a(x a) and p ^ 0, there must exist x a £ K a( x a) such that p x a < p x a. Then there must exist a finite collection of net trade vectors x\ £ X a and associate positive scalars p.1 (j = 1 to J) for which x a — io + Y/j= l pH H ~ io )- Then 0 > p ( x a - x a) = J2j=i P3 P (x a ~ i o j which

implies that there must exist some j (j = 1 to J ) for which p x ]a < p x a. So, defining x „ := x j then gives an i , £ X a for which p x a < p x a.

3.4. Aggregate Inferiority

3.4.1. The Interiority Assumption

In a finite economy, in order for some individual to have a cheaper feasible net trade, it is necessary that society as a whole should have some cheaper point. So assume:

ASSUMPTION 2. 0 belongs to the interior o f Ka, the convex hull o f the set Xa '■=

Y l a e A X a

-9 The boundary o f any set K will be denoted by b d / f , and the interior by int K.

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This assumption can be motivated as follows. Earlier it was assumed that 0 E X a for all a E A. So if Assumption 2 is false, it can only be because 0 is a boundary point of the convex set Ka- But then there would be a hyperplane p x = 0 through 0 which supports I\a in the sense that p x > 0 whenever x E I\a-

Note how X a C Xa C Ka for all a E A, because 0 E X a> for all a' E A \ {a '}. So, for all a E A, this would imply that p x > 0 whenever x E X a. Thus almost no agent in the economy can trade in the open half-space H of directions x satisfying

p x < 0. If almost nobody can trade in any such direction, net demands for vectors

in the opposite half-space —H are completely ineffective. They are like asking for the moon — or at least, for a return trip there this year (1989). It therefore makes sense to restrict attention to the proper linear subspace of net trade vectors satisfying p x = 0. Repeating this argument as many times as necessary results in a commodity space which is the linear space spanned by the relative interior of the convex set Ka- For this commodity space, of course, Assumption 2 is satisfied

by construction.

3.4.2. Cheaper Points for Some Individuals

LEMMA. Under Assumption 2, for every price vector p ^ 0, the set

{ a E A | 3xa E X a such that p x a < 0 }

o f individuals with feasible net trade vectors having negative value is non-empty.

PROOF: Assumption 2 implies that for every price vector p ^ 0 there exists x E Xa such that p x < 0. The rest is obvious.

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4. Non-Oligarchic Allocations

4.1. Definitions

Suppose that x is a feasible allocation. Let S be any proper subset of A. Say that “A \ S may strongly improve the condition of 5 in x ” if there exists an alternative feasible allocation x ' G J- such that x'a y a x a for all a G 5. And, as in Spivak (1978), say that “A \ S may improve the condition of 5 in x ” if there exists an alternative feasible allocation x ' G T such that x'a teax a for all a G S', and x'a, >-a‘ x a• for some a* G S.

Next, S is said to be a weak oligarchy at x if A \ S cannot strongly improve the condition of S in x. Equivalently, S is a weak oligarchy at x if there is no other feasible allocation x ' such that x'a >~a x a for all a G S — i.e., if

^ ] C „ 6 s Pa^Xa^ + € A \ S X a.

And S is said to be a strong oligarchy at x if A \ S cannot improve the condition of 5 in x. Equivalently, 5 is a strong oligarchy at x if there is no other feasible allocation x' such that x'a fcax a for all a G 5, and x'a. >-a• x a* for some a* £ S — i.e., if

1 ^ p a• (x a* ) -f

es\U

for all a* G S.

The last definitions receive their name because an omnipotent oligarchy would presumably choose a feasible allocation with the (strong oligarchy) property that no alternative feasible allocation could make some of its members better off without simultaneously making some other members worse off. Or it would at least choose a feasible allocation with the (weak oligarchy) property that no alternative feasible allocation could make all of its members better off simultaneously.

Conversely, the feasible allocation x is said to be strongly (resp. weakly) non-

oligarchic if there is no weak (resp. strong) oligarchy at x. Thus, x is strongly

non-oligarchic if and only if, for every partition A\ UA.2 of A into two disjoint non empty sets Ai,A.2, there exists another feasible allocation x ' such that x'a y a x a

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for all a € A\ — i.e., if

0 6 Pa( x “ > + E a6„ 2 X “ ■

And x is weakly non-oligarchic if and only if, for every partition A\ U -42 of A into two disjoint non-empty sets A \, A 2, there exists another feasible allocation x ' such that x'a ~ ax a for all a 6 and x'a. x am for some a* G i i — i.e., if there exists a* G i i such that

0 e jPa*(xa* ) + y ^

a€.4i\{a* }

<'•('•>+E.6„,*

Of course, any strongly non-ohgarchic allocation is weakly non-oligarchic.

In a non-ohgarchic allocation, by contrast to an oligarchic allocation, every coalition which is a proper subset of A, the set of all agents, does have an improve ment available, provided it can suitably exploit the resources of the complementary coalition. The force of the non-oligarchy assumption is that the allocation leaves every coalition with some resources which the complementary coalition would like to exploit if it could. Provided that some agents are in uncompensated equilibrium, this will ensure that the other agents do have cheaper points.

4.2. G ale’s Linear Exchange Model

Gale (1957, 1976) considered a linear exchange model in which each agent a has a feasible set X a = { x a £ | x a = :ra } of net trades, and a preference ordering

represented by the linear utility function Va(x a) := va x a = va(J x a(J for

some semi-positive vector va of coefficients. Typically x a is taken equal to —u:a. where u:a is agent a’s fixed endowment vector, but there is no need here for this specific assumption.

In this linear exchange economy, the allocation x is said to be reducible (Eaves, 1976) if there exist partitions A = A\ U A 2 and G = G\ U G2 into two non-empty subsets of the set of agents A and of the set of goods G = { 1 , 2 , . . . , ^ } with the property that vag — 0 (« G A \; g E G'i); x ag —x ag — 0 (a E A 2] g E G> )■ © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Thus the only goods desired by the members of A\ are in the set G'2, and none of these can be supplied by the members of A 2. Consequently, a Pareto efficient allocation in this economy is reducible only if there exists an oligarchy (A i) which cannot be made better off even with the resources of the complementary coalition (which are goods in Gi). Notice that the allocation x is reducible only if the matrix M defined by

_

£

M aai := 1 Vag (Xa>9 ~ —o'ff) (a’ a = 2, . . .

is reducible in the sense of Gantmacher (1953, 1959) and Gale (1960), with M aa• = 0 for all a £ Aj and all a' £ A2. Of course, the element M aa.' of this matrix

represents the utility to a of agent a"s resources.

The converse statements are also true in this special economy. That is, if the proper subset S of A is an oligarchy at the allocation x, then it must be true that, for all a £ S, a' £ A \ S, and for g = 1 to i,

Vag > 0 =>• Xaig - x a,g = 0.

This condition defines what Gale (1976) calls “self-sufficiency,” although that ter minology would perhaps be better applied to feasible allocations rather than to ways of making a coalition better olf. The same condition holds, of course, if the matrix M is reducible, with an independent subset S. In either case, define

G2 ■= { s = 1 ,2 ,... , t | x ag - x ag = 0 (all a £ A \ S ) }.

Note that G2 is certainly non-empty because it contains any good y for which there exists some 0 6 S with vag > 0. This construction shows that the economy is reducible except in the uninteresting special case when x a = x a for all a £ A \ 5 , so that the coalition A \ S has no resources at all available for redistribution to the members of S.

Thus, for the special case of the linear exchange economy, the non-oligarchy condition is very closely related to irreducibility of the matrix M defined above.

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4.3. Convexified Non-Oligarchic Allocations: Definition

In fact only convex combinations of potential changes matter. Recall that the preference set is denoted by Pa(x a) = { x'a £ X a | x'a >-a x a }, and that the weak preference or upper contour set is denoted by Ua{ x a) = {x 'a £ X a \ x'a fc „x a }. Use c o f to denote the convex hull of any set V C lU.

Next, define the set

f C O[Pa(xa) U {x „} ] \ { x a} if X a £ CoPa( x a);

Pa( x ° ) := S

{ CO Pa(x a) if X a 6 C oP ,(x»).

Note that global non-satiation implies non-emptiness of Pa( x a) and so of P * (x u). Also, the set P * (x a) is convex because if x a ^ c o P a(x a) then x a must be an ex treme point of the set Pa(xa)U {x 0}, and so x a can be removed without destroying convexity. If preferences satisfy the classical hypthoses of Section 2.1.6, then x a will be a boundary point of the set Pa( x a) = co Pa( x a), and so P * (x a) = co Pa( x a). More generally, however, x a will be a boundary point of the set P * (x a) but not of

Pa(xa) nor even of c o P „(x a). Indeed, the construction of P * (x „) from P „(x a) is

a device intended to put x a on the boundary.

At the feasible allocation x the proper subset 5 of A is said to be a convexified

oligarchy if, for every non-empty subset A* of S, one has

° ^ E o€x. p° * ( * - ) +co yX ^ a e S \ A ‘ u a( x a) + y V ' A ~ i a £ A \ S x

The feasible allocation x is said to be convexified non-oligarchic if, for every partition Aj U A2 of A into two disjoint disjoint non-empty sets A j, A 2, neither subset is a convexified oligarchy.

Under the classical hypotheses of Section 2.1.6, any convexified non-oligarchic allocation x is obviously non-oligarchic. So “convexified” non-oligarchic allocations are only more general if the usual convexity hypotheses are violated.

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4.4. Non-Oligarchic Allocations and Uncompensated Equilibrium

Le m m a. Under Assumptions 1 and 2, any convexiGed non-oligarchic allocation x

which is a compensated equilibrium with lump-sum redistribution at some price

vector p ^ 0 must be an uncompensated equilibrium with lump-sum redistribution at the same price vector p.

PROOF: For the com pensated equilibrium (x ,p ) , let

A := { a £ A | x'a £ X a => px'a > p x a }

be the set of agents a £ A without any cheaper points x a £ X a satisfying p x a <

p x a. Because members of A \ A do have cheaper points, Assumption 1, together

with Theorem 3.3.4, implies that for them ( x ,p ) is an uncompensated equilibrium. Suppose that A is not empty. Then the aggregate interiority Assumption 2 implies that the complementary set A \ A must be non-empty. Since it cannot be a convexified oligarchy, there must exist a non-empty subset A ’ C A \ A for which

n6A* •P« ( * « ) + co

y

^-^aE(A\A)\Amu *(x ‘ ) + Z aeA

This implies the existence of net trade vectors x 3a £ X a (for a € A* and j — 1 to Ja; also for a £ A \ A* and j = 1 to J), nonnegative scalars p3a (for a £ A* and j = 0 to Ja) with < 1 and E /= o h i ~ 1 ( a^ a £ A*), and positive scalars \3

( j = 1 to J) with Z ^ = i ^ = 1) such that:

(i) 0 = Z a E A - [ h a x a + E /= ! h i X l\ + Z U V Z a E A \ A - X i ’

(ii) >~a x a for a £ A* and for j = 1 to Ja;

(iii) x { izax a for a £ (A \ A ) \ A* and for j = 1 to J.

Because (x ,p ) is a compensated equilibrium, (iii) implies that p x 3a > p x a for

all a £ (A \ A) \ A* = A \ (A U A*) and for all j = 1 to J. Because (x ,p ) is an uncompensated equilibrium for the members of A \ A, (ii) implies that p x 3a > p x a

for all a £ A* C A \ A and for j = 1 to Ja. Then, because A* ^ 0, and at least one p3a (j = 1 to J) is positive for each a £ A*, it follows that

E.6il. E -1,

**- *•) + E-=1 A**

**J Ea6(^A)\A. p(x“ - *•> >0**

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or that

P >P

T , aeA. E t . ^ x » + E t . v E a e A (^ . ) * i

Eag.4- Ej = l

X “

**Ej = l ^ Eag^l^^uji*) X°**

But then, because both E „ e / f [/i“ x a + E /= . /rj a:'] +Ey=i ajEae/t\4* x a and

E agj4 1» must equal zero, while E / = i = 1 and also, for all a £ .4*, both /.qj < 1 and E j= o Pa = 1; if follows that

P >P

-E

-E w /i°x“

* ■'a£A* y J » y ] y J a>v X d

or that E /= 1 E a g /i P(-Ta — ^o) < 0. This contradicts the assumption that no agent a f i has any cheaper point in the feasible set .Y0.

Therefore A must be empty after all, and (x ,p ) must be an uncompensated equilibrium for all the members of A.

Thus, in an economy satisfying both Assumptions 1 and 2, convexified non oligarchy is a sufficient condition for a particular compensated equilibrium with lump-sum redistribution to be an uncompensated equilibrium. This result does not, however, help prove existence of an uncompensated equilibrium. The reason is that consumers’ initial endowments may be such that the only compensated equi libria fail the convexified non-oligarchy condition. One way out of this problem is to assume “sagacious” lump-sum redistribution (Grandmont and McFadden, 1972), meaning that every agent always has a cheaper point at every price vector. The alternative is to apply ideas similar to those in this section to the autarkic allocation, as well as to a compensated equilibrium. This is precisely what McKen zie’s concept of irreducibility does, as well as Arrow and Hahn’s similar concept of resource relatedness. © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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4.5. Agents outside Convexified Oligarchies Are at Cheapest Points Following Spivak (1978)’s proof of a similar result for non-oligarchic alloca tions, one can show that convexified non-oligarchy is a necessary condition for every agent to have a cheaper point at every possible compensated equilibrium price vector. Indeed:

Lem m a. Let x be a feasible allocation at which the coalition S is a convexified

oligarchy. Then there exists a price vector p yf 0 such that (x, p) is a compensated equilibrium with lump-sum redistribution in which all agents outside S are at cheapest points — i.e., for all a G A \ S ,itm u st be true that, x a G X a =4* p x a > p x a.

PROOF: For every a' G S, define the convex set

Because S' is a convexified oligarchy at the feasible allocation x, it must be true that 0 ^ AVs(x) for every a' G S. Also K a's ( x ) is non-empty because of global non-satiation. So there exists a hyperplane p y = 0 through 0 with p ^ 0 such that

p y > 0 for all y G K a's (x ). In particular, because YlaeA Xa = ^ follows that S a sA p ( x a ~ •*») ~ 0 whenever x ai G P *,(xa'), £ Ua{ x a) (all a G S \ {a '} ) , and x a G X a (all a G A \ S).

Because x a’ is on the boundary of P *,(xai), because x a G Ua( x a) for a ll

a G S \ {a '} , and because also x a G X a for all a G A \ S, it then follows that: (i) p xa1 > pXa1 whenever x a> >~a' Xa'i (Ü) for all a G S \ {a 1}, one has p x a > p x a

whenever x a fcai a; (iii) for all a G A \ S, one has p x a > p x (l whenever x a G X a

and so also whenever i , These results together imply that (x ,p ) is indeed a compensated equilibrium with lump-sum redistribution in which all agents outside

S are at cheapest points.

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5. Interdependent Agents and Irreducible Economies

5.1. Arrow and Debreu’s Early Version of Irreducibility

It has already been pointed out that the first part of Arrow and Debreu (1954) relies on the implausible interiority assumption. But they do also present a different and more interesting second assumption. First, they assume that there is some set of goods V which all consumers desire. Also, that for any given feasible allocation, every consumer’s endowment includes some kind of labour service which can be converted into an increased aggregate net supply of some good in T>, without any offsetting reduction in the aggregate net supply of any other good. In a finite economy, any such labour service is bound to command a positive wage provided that any consumer at all is maximizing preferences subject to his budget constraint.

Now, it was evident that with convex feasible sets and continuous prefer ences, for any consumer having a net trade vector of negative value — i.e, able to sell something of positive value — any compensated equilibrium must be an uncompensated equilibrium, as in Theorem 3.3.4 above, which actually requires only the weaker Assumption 1 of Section 3.3.2. So, in a finite economy, the Arrow and Debreu assumption leaves just two possibilities. One possibility is that no body has a cheaper point at the compensated equilibrium price vector, and that nobody is in uncompensated equilibrium. This is what happens in Arrow’s excep tional case, but one could argue that the problem there is the attempt to price commodities such as return trips to the moon this year which cannot possibly be traded in any feasible allocation. The second possibility is more interesting. This is that every commodity can be traded, so every commodity can be supplied by somebody. Since at least one commodity should have a positive price, at least one consumer has a cheaper point, and so is in uncompensated equilibrium. But then, under the Arrow and Debreu assumption, all the desirable commodities in the set D must have positive prices, so everybody has a valuable labour service to supply, and therefore everybody has a cheaper feasible point. This shows that, in a finite economy, the compensated equilibrium is an uncompensated equilibrium for everybody, as required.

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Arrow and Debreu’s second assumption was unnamed. Yet it is actually a special case of the much more general irreducibility assumption introduced a few years later by McKenzie.

5.2. M cKenzie’s Irreducibility

As McKenzie (1959, p. 58) points out, he was “able to dispense with the categories of always desired goods and always productive goods used by Arrow and Debreu. An always desired good appears particularly implausible. It requires that every consumer be insatiable in this good within the supplies attainable by the whole market.” One can also note that Arrow and Debreu’s second assumption cannot apply in the usual kind of exchange economy in which each consumer has a fixed endowment vector of consumer goods, so that labour services are of no use to anyone.

In the introduction to McKenzie (1959, p. 55), he writes, “In loose terms, an economy is irreducible if it cannot be divided into two groups of consumers where one group is unable to supply any goods which the other group wants.” After stating the relevant assumption formally, this verbal description is expanded somewhat on pp. 58—9:

“ (The irreducibility] assumption says that how ever w e may partition the consumers into two groups i f the first group receives an aggregate trade w hich is an attainable output for the rest o f the market, the second group has within its feasible aggregate trades one which, if added to the good s already obtained by the first group, can be used to im prove the position o f som eone in that group, while damaging the position o f none there.”

Later, in McKenzie’s (1961) “ Corrections,” the formal assumption is restated in a way which “actually corresponds better with the verbal discussion,” and certainly seems easier to interpret. Indeed, the economy is said to be irreducible if and only if, for every feasible allocation x and every partition Ai U A2of A into two disjoint non-empty sets Ai and A 2, there exist a pair x' e