Value Cores for Finite Agents

Value Cores for Finite Agents* PAUL VAN MOESEKE

University of Otago, Dunedin, New Zealand

The standard definition o f the competitive equilibrium, at given prices, presents the difficulty that finite agents (i.e. agents with finite budget and production sets), and a fortiori coalitions of such agents, need not be price takers and will upset the equilibrium. Since 1963 an extensive literature has sprung up, which has dealt with the problem, essentially by considering infinite numbers of infinitesimal agents.

The present article outlines two alternative approaches to deal with this problem, both of which, realistically, admit (a finite number o f ) finite agents and redefine the core in terms o f the value structure of the economy, viz. prices in the presence of controls, and cost and revenue functions in their absence. It is shown that the notion o f value core, here introduced, coincides with that of competitive equilibrium, which is formulated as a characteristic-function game.

I Introduction The present paper introduces the cone@ of

value cores to give a realistic characterization of the stability of the competitive equilibrium among finite agents, where retrading on the open market (rather than just barter among a coalition’s members) is a permissible strategy open to any coalition. The value core in fact redefines the core in terms of the value structure of the economy, viz. prices in the presence of controls (Section 11); cosat and revenue functions in their absence (Section 111).

* Cores relative to price controls were introduced for measure spaces of agents in a paper (van Moeseke, 1977a) read before the Economic Society of Australia und New Zealand at the Sixth Conference of Economists in Hobart, May 1977. Section 11 of the present apticle originated in response to a number of queries received there, regarding the finite version of cores defined relatively to price controls.

I am indebted to a referee for a number of penetrating comments, especially regarding the specification of the generalized Competitive equilibrium in Section 111. This research was sponsored by the National Science Foundation in Brussels: I am grateful to Secretary-General J. Traest for help and encouragement and to Dr F. Vande Ginste for research assistance.

The latter approach in fact generalizes the competitive market to an asymmetric CO- operative game.

A basic difficulty with the standard treat- ment (Arrow and Debreu, 1954, Debreu, 1954, Koopmans and Bausch, 1959) of the competitive equilibrium is that ( a finite number of) individual agents may have any finite size (in terms of budget sets, production sets, and endowments). Indeed, finite agents, let alone coalitions of them, are not necessarily price takers so that the price system for which an allocation is competitive will not normally remain constant.

There are at least three ways of dealing with this difficulty, only the first of which has, to my knowledge, been pursued in extenso in the literature.

(a) One may consider a countable number (Debreu and Scarf, 1963), or even a continuum (Aumann, 1964), or more gcncrally a measure space (Debreu, 1967) of infinitesimal agents.

(b) Prices may be regarded as price controls, which also implies an incomes policy since input prices are controlled as well. This

76

1979 VALUE CORES FOR FINITE AGENTS 77

is the approach of Section I1 (and of van Moeseke (1977b) where the social optimality of competitive allocations is investigated in the prescnce of controls and capitation) -

(c) One may relax controls and broaden the definition of the competitive equilibrium to allow for nonlinear valuation so that neither cost ( to consumers )nor revenue (of producers) need, in principle, be linear functions of commodity quantities transacted.

This model (Section 111) delineates the competitive market as a cooperative game of the following institutional type: agents con- sume as individuals and can combine in cartlels for production. Every consumer has an expenditure function (cost of his consumption bundle) and every potential (production) coalition an objective function. The approach can be ‘compared, and contrasted, with the oligopoly model in Gabsnewicz and Vial (1972) where agents are price takers on the consumption side and firms behave as the players of a noncooperative game.1

Whichever tack is taken the competitive equilibrium by definition is so formulated as to express individual rationality in the sense that no agent can, within his own means, improve upon his share of the allocation. One naturally wants to know whether, and to what extent, this concept also covers group rationality in the sense that no coalition of agents can improve upon (or block) the Competitive allocation: if so it is in the core. (A weaker property, Pareto optimality, expresses collec- tive rationality, which merely rules out blocking by the total coalition.)

Interest consequently focuses on alternative assumptions assuring the equivalence of competitive and core aIlocations, an idea dating back to Edgeworth’s conjecture ( 1881 ) that what we now call the core shrinks to the Wal- rasian equilibrium if the exchange economy becomes ‘large’. The idea has been tak,en up again since 1963, when the corresponding equivalence theorem was established for economies with a denumerable number, and then with a continuum, of agents in the papers mentioned

Approaches (a) and (b) were combined by the present author in van Moeseke (1977), where the equivalence of core and competitive allocations under price controls was established for measure spaces of agents, a result depending cru- cially on the Measurable Choice theorem proved in Aumann (1969).

under (a ) above. The latter approach has given rise to a very extensive literature (all of it under our heading ( a ) ) which it would be otiose to recapitulate here. (See Hildenbrand, 1974, Hildenbrand and Kirrnan, 1976 for references up to 1976.) Suffice it to indicate later extensions to production economies in Hilden- brand (1968), to the case of coalition pre- ferences in Vind (1964), and to nonstandard exchange economies in Brown and Robinson (1975) and Brown (1976). The logical rela- tionship between competitive or core allocations on the one hand, and alternative game- theoretical equilibrium concspts such as Shap- ley value and bargaining set on the other, have recently been investigated in, for example, Au- mann (1975) and Maschler ( 1976).

IZ Value Cores with Price Controls For simplicity the notions below arc defined

for an economy where every agent is both consumer and producer (or resource holder : e.g. a proletarian owns his labour aq sole resource),

By Y C Rm denote the (additive) commodity space, by Z = (1, . . ., n } the set of agents, and by E = ( { Y J , {R) ; all i E I ) the

economy where, for all i, Y , and R are subsets

of Y , respectively Y X Y , and designate agent i’s production set (or resource endowment) and preference preorder, respectively. Let further S C I denote a coalition of agents.

An allocation (x) is a sequence { x ~ ; all i E I > such that

I

1

Ex, E ZY%. Throughout the paper summation, unless

For typographical simplicity we write x ( R xi (instead of R) and sup pyi instead

of sup p y i on Yi. Here p denotes the mtuple of price controls. Further, the sequence {xi; all i E Slis abbreviated ( x ) ~ but we write (x) instead of ( x ) ~ . Without risk of confusion we use the same symbol for agent i and the coalition consisting of agent i.

An allocation is a Walrus (competitive) allocation relative to p (the controlled prices) if, for all i , the consumption choice x,* is maximal on i‘s budget set:

indexed, runs over I ( i s . C is short for X) . I

i

78 THE ECONOMIC RECORD MARCH

PXi* G sup PY, (W1) px, < sup p y , implies x,* R xt. (W2)

A coalition S of agents blocks (the term im- proves upon is used in Hildenbrand, 1974 and Hildenbrand and Kirman, 1976) allocation (x*) at p if for some (x),

C P X , G C SUP PYa

x, P x,* for some j E S x, R xi* for all i E S.

(B1)

(B2) (B3)

An allocation is in the value core relative to p if no coalition blocks it at p . This notion is a special case of the value core (section 111): it is a value core in the presence of price controls. Note that ( B l ) replaces

s S

Ex, E CY, (B1') s s

in the definition of the standard (or commodity) core,2 where (Bl') indicates that barter among the members of S, within the limits of its aggregate production set CY,, is

the only strategy available to them to improve upon a given allocation. However, agents cannot he prevented from availing themselves of all strategies open to them: under price controls these include trading at those prices, as expressed by ( B l ) . Clearly (B l ) includes the strategies available under (Bl') since any se quence (x), satisfying (Bl') satisfies ( B l ) .

For the equivalence result below we need the following axiom and lemmata, ( L ) Local Nonsaturation: Every neighbour- hood of any x, owns an x: such that x,' P x+

S

Lemma 2.1: If (L), (W2) hold then x, R xt* implies px, 2 sup py,. (2.1 )

Proof: Observe that (W2) is equivalent to x, P x4* implies px, > sup py+ (2.2)

It remains to exclude (xi -xi*) A (px' < pyJ which, by ( L ) , would imply that there is an x,' such that

(2.3) As x i P x,* (2.3) conflicts with (2.2). QED.

Lemma 2.2: Any allocation (x*) such that (2.4)

xi' P xi and p x i < pyc.

ZPX,* < C SUP PYC S S

is blocked by S. Proof: By (2.4) pxd* < sup pyL for at least

one i E S. By ( L ) choose xi P xc* such that px; < sup pya and define ( x ) ~ such that xi = x;; xi = x,*, all j E S such that j + i. Then

2 Compare, for example, the necessity part of theorem 2.1 below with theorem 1 in Arrow and Hahn (1971), ch. 8.2.

by (B 1,2,3) S blocks (x*) at p . QED. The necessity part of theorem 2.1 below

parallels the proof for the standard notion of core in Arrow and Hahn (1971), chapter 8.2.

Theorem 2.1: If ( L ) holds, an allocation (x*) is competitive at p if, and only if, it is in the core relative to p.

Proof: Necessity. If S blocks the Competitive ( x * ) by (B2), (2.2)

pxj > sup pyj , some j E S and by (B3), (2.1)

px{. 2 suppy, all i C S such that i # j . Summation yields C p x , > Z sup pyi, contrary

S S to ( B l ) .

Sufficiency. If allocation ( x * ) is unblocked but not competitive then either ( W l ) or (W2), or both, are violated. If, contrary to (Wl) , pi* > sup pyr for some i then, as Zxi* E CY,, pxj* < s u p p y j for at least one j Z i. By lemma 2.2, j blocks, a contradiction.

If, contrary to (W2), x , P x t * for some x, such that p x i < suppy i then i blocks by definition, a contradiction. QED.

I l l Generalization: Value Cores without Price Controls

By Y again denote the commodity spawe (which, contrary to Section 11, need not be additive so that externalities are allowed for) and consider the economy E = ({Y,}, {R},

{g,), { f i } ; all i E I , all S C I ) , where Y, C Y is the production se t of coalition S and g, (defined on Y,) is, in the simplest interpreta- tion, its profit function, while fi (defined on Y ) is agent i's cost (or expenditure) function.

That is to say, the cost of agent i of commodity purchase x, is f c ( x c ) and coalition S earns g, ( y , ) by selling y s E Y,. Agents con- sume as individuals but may form coalitions in production. The numerical functions f , , g , replace the linear cast function px,, respectively the linear profit function Cpy i , at the controlled

prices p in Section 11. The notions of allocation, competitive allocation, and blocking will have to be generalized accordingly.

One should note that externalities in production are dealt with by the fact that Y need not, a priori, be additive. Further, gg can be interpreted a5 an objective function, rather t h a n just a profit function, having, beside profit,

I

s


other arguments as well, such as externalities in consumption, social indicators (cf. Fox and van Moeseke, 1973), and certain (favourable or adverse) aspects of monopoly power.

While g , is formulated without explicit re- ference to the f , ( i € S ) we shall see below (lemma 3.3) that Z sup gi = sup gs is a neces-

sary condition for the existence of a competitive equilibrium and furthermore that any competitive allocation ( x * ) satisfies (cf. proof lemma 3.3) fi (xi*) = supg,, all i , so that every consumer will prefer the consumption bundle x,* associated with the corresponding production plan-an observation that may be read parallel with the so-called Fisher separa- tion theorem (cf. theorem 1.1 in Milne, 1974).

This is perhaps the simplest characterization of competitive markets with finite agents and without price controls. While having the cost fi to consumer and the scale of production Y, depend only on decisions by agent i, respectively coalition S, may not be too un- realistic, the same cannot generally be claimed for treating the value gs of the objective as de- pendent solely on the coalition's own output. The approach is the same as that of cooperative-game theory, with supg, in the role of the characteristic function v( S ) , which is usually interpreted as the payoff the coalition can assure itself regardless of the lineup of all other players; while useful as a first approxi- mation to a generalized notion of competition, it is subject to the same limitations.

It is understood that in sup g,, supg, the sup is taken on the sets Y, and Ys, respectively. We further write supg instead of supg,. We also restrict g by the standard cooperative-game assumption that, for any disjoint S, T C I

In particular, for ail S,

S

SUP gs u T B SUP gs + SUP g,.

sup g , 2 C sup g,.

(3.1)

(3.2)

Aggregate consumption is denoted x . (As we also wish to allow for externalities in consumption x need not, even if Y is additive, satisfy x = Zx,.) An allocation is now an ( x ) such that x E Y,. By (3.1) sup g is the 1.u.b. to net revenue (say) from any allocation and we assume that expenditure cannot exceed it:

Ui (x,) < supg, all (x) such that x E Y,.

S

(3 .3 )

An allocation ( x ) is balanced if

k k U d (Xi) 2 c sup g, (3.4)

for any partition (SJ of I: i.e. aggregate revenue (or net national product) cannot exceed aggregate expenditure (or national income), regardless of coalition structure. By (3.4) one has, in particular, for balanced ( X I ,

Cfi (Xi> 2 sup g, + sup g,'

Zfi (Xi) 2 c sup g,.

fi (Xi*) < sup g ,

(all S C 1 ) (3.5)

(3.6) An allocation (x*) is competitive if it is

(C1) f i (xi) < sup gi implies xi* R x . ~ . (C2)

As always competitiveness expresses individual rationality (optimality of choice xi* within the agent's budget set).

S blocks an allocation (x*) if for some (x), ( B 2 , 3 ) hold and

(b)

An allocation is in the value core if no coalition blocks it.

Assuming continuity of the fd the proof of the next lemma parallels that of lemma 2.1.

Lemma 3.1: If ( L ) , ((72) hold and fi is continuous then

(where S' is the compliment of S in I ) and

balanced and, for all i,

Zf* (4) G S U P g,. S

xi R xi* implies fi (xi) 3 sup g,. Lemma 3.2: Under ( L ) and continuity of the

f i any allocation ( x * ) such that

S is blocked by S.

Proof: Pick any j ES. Put xi = xi*, all i E S such that i f j , and select (by ( L ) and continuity of fj) an x j R x j * close enough to xj* such that C f i ( x ~ ) < supg,. Then ( B 2 , 3 )

and ( b ) are satisfied and S blocks. QED. The following lemma states a condition

necessary for the existence of a competitive allocation.

Lemma 3.3: If there exists a competitive allocation ( x * ) then, for any S,

If$ (Xi* 1 < sup g ,

S

I3 sup gi = sup g,. (3.7) S

Proof: As (x*) is balanced, by (3.6) and (CI 1

It4 ( X i * ) = c sup gp Again, by (C1) this implies fi (xi*) = sup g,,

80 THE ECONOMIC RECORD MARCH

all i, so that, by summing over S, Cfi (xg* ) = c sup g,. S S

(3A)

Now, if Cfi (xi*) > supg, then by (3.2) S

Zfi (xi*) > C supg,, contrary to (3.8). If S S

(xi*) < sup g, then by (3 .5) Cf, (x(*) S s’ > sup gs(, again contrary to (3.8). Hence

(3.9)

The lemma follows from (3 .8) , (3.9). QED. Theorem 3.4: Let there exist a competitive

allocation. Under local nonsaturation ( L ) and continuity of the f, an allocation ( x * ) is competitive if, and only if, it is in the core.

Proof: Necessity. If, contrary to assertion, ( x * ) is competitive and blocked by some S then by ( B 2 ) , (C2)

j j ( x j ) > snp gi, some j E S and by (B3) and lemma 3.1

fi (xi) 2 sup gt, all i E S, i # j . Summing yields Cfi (xi) > C supg, so that,

by lemma 3.3, Cf, (xt) > sup g,, contradict- S

ing ( b ) . Sufficiency. If, contrary to assertion, a core

allocation ( x * ) is not competitive then at least one of the following three cases hold:

Case (i): (x*) is not balanced. Then by (3.4) C f h (xi*) < supgB for at least one Sk

S h and S, blocks by lemma 3.2, a contradiction.

Cfi (Xi* ) = sup gfj. S

S S

h

Case (ii): ( x * ) violates (Cl) so that f h (xi*) > supg,, some i. (3.10)

AS ( x * ) is an allocation, by (3.3) Xf6 i(xI*) < sup g . As there exists a competitive allocation we may apply lemma 3.3 to Z so that sup g = C sup g,. Hence Cfd (xi*) < C sup g, and by (3.10) there is a j # i such that fj

( x i * ) < supgj and j blocks by lemma 3.2, a contradiction.

Case (iii): ( x * ) violates (C2) so that xt P x4* while fi (xi) < supg, and i blocks by definition, a contradiction. QED.

IV Concluding Remarks If supg, is interpreted as the characteristic

function of a cooperative game then lemma 3.3 indicates that the ‘existence of a competitive equilibrium requires that the game be inessential. Indeed, formula (3.7) holds if, and

only if, (3.1) holds with equality for all disjoint S, T C Z . Formula (3.7) further implies that

for all S C Z so that g defines a constant-sum game. It is well known (e.g. Luce and Raiffa, 1957, ch. 8) that, unless a constant-sum game is inessential, i.e. additive rather than super- additive, its core is empty.

This conclusion in turn hinges on definition (3.4) of a balanced allocation: from a macro- economic viewpoint it is interesting to note that the conclusion no longer holds if effective demand falls short of aggregate supply.

sup g = sup g, + sup ggl

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Value Cores for Finite Agents