market games with indivisible commodities and non-convex preferences

The Review of Economic Studies, Ltd.

Market Games with Indivisible Commodities and Non-Convex PreferencesAuthor(s): Claude HenrySource: The Review of Economic Studies, Vol. 39, No. 1 (Jan., 1972), pp. 73-76Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2296444 .

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Market Games with Indivisible

Commodities and Non-convex

Preferences 1 CLAUDE HENRY

Laboratoire d'Econome'trie de L'Ecole Polytechnique, Paris

INTRODUCTION

A few years ago Scarf [6] proved that the core of a balanced n-person game is never empty; as a corollary of this theorem he showed that a pure exchange economy with upper semicontinuous quasi-concave utility functions and compact convex consumption sets including the corresponding bundles of initial resources, always has a non-empty core. This result cannot in general be extended to economies with non-convex preferences, e.g. when indivisible commodities are available (for a counterexample see Henry [4]); for such economies it may therefore be interesting to find allocations which satisfy properties as close as possible to those defining the core; this is a game-theoretical approach to a problem whose competitive equilibrium aspects have been studied by Starr [7] and Dierker [2].

Consider a pure exchange economy E where every consumer has a continuous utility function and a compact consumption set including the bundle of his initial resources; Theorem 1 makes it possible to define a convexed economy - nearest to E. It then becomes possible to apply Scarf's result to L and, coming back to E, to prove (Theorem 2 and its Corollary) that, if the distances between the consumers' utility functions in E and their respective least quasi-concave majorants-which appear to be their utility functions in t- are bounded, then there exists at least one allocation in E such that

(1) it is blocked by no coalition of consumers;

(2) for any commodity, either divisible or indivisible, the difference between the total quantity allocated and the total resources can be made as small as required with respect to the number of consumers, as this number becomes sufficiently large.

Here are some cases where the condition " the distances between the consumers' utility functions in E and their respective least quasi-concave majorants are bounded " is satisfied: if all the consumption sets are included in a common compact subset of the commodity space; or if for every commodity available in E there exists a quantity satiating any consumer; or if the only non-convexities present in E are those resulting from the indivisibility of some commodities, as in Henry [4]. In all these cases, as soon as economy E is sufficiently large, one can always find an allocation which may be said to almost lie in the core even if this, strictly speaking, is empty.

LEAST QUASI-CONCAVE MAJORANT OF A UTILITY FUNCTION

Consider a consumer in an economy E with m _ 2 commodities, and let XcR' be his consumption set and u:X- R:x->u(x) his utility function.

:First version received July 1970: final version received March 1971 (Eds.). 73



74 REVIEW OF ECONOMIC STUDIES

Two assumptions on X and u will be employed: (1) X is compact; but X is not assumed to be convex, not even connected, as the

commodities involved may be either divisible or indivisible; (2) u is continuous on X, with respect to the topology induced on X by the Euclidean

topology on Rm. If x is any commodity bundle in X, 6(x) will be the set of all the commodity bundles in X which are at least as desired as x. Denoting by conv X and conv 6(x) the convex hulls respectively of X and 6(x), consider the subset of X

D(z) = {x E X I z E conv (x)},

where z is any point in conv X. D(z) will prove useful for the construction of a quasi- concave utility function on conv X; this will be done in

Theorem 1. Given a compact consumption set X and a continuous utility function u on X, the least quasi-concave function defined on conv X which is minorized by u on X is

U:conv X-4R:z-+ U(z) = max u(x) x E D(z);

moreover U is an upper semicontinuous univalued function on conv X.

Remark. That U is an upper semicontinuous univalued function on conv X does not imply that it is an upper semicontinuous correspondence in (conv X) x R; it simply means that, for every z in conv X and every strictly positive e, there exists a neighbourhood of z on which U is less than U(z) + s.

Proof of Theorem 1. If U is a quasi-concave function on conv X, it is clearly the least one which is minorized by u on X. Indeed let V be any quasi-concave function on conv X which is minorized by u on X, z any point in conv X and x any point in D(z); as z is a point in conv 6(x), it can be written as a convex combination of points xk in 6(x); for at least one of these points, say xl, V(z) _ V(xl) holds; hence V(z) ? V(xl) ? u(xl) ? u(x).

As a correspondence in (conv X) x X, D is upper semicontinuous. Indeed this is equivalent to the statement that the correspondence conv 6 in Xx (conv X) is upper semicontinuous, a result which is immediately derived from the upper semicontinuity of the correspondence 6 in Xx X, because the convex hull of a compact subset K of R' is compact and can be obtained by forming all convex combinations of at most m + 1 points in K at a time (these are consequences of Caratheodory's theorem; see for example Rockafellar [5], p. 155-158).

D being upper semicontinuous, U is defined everywhere on conv X, as u reaches its supremum on the compact set D(z), z E conv X. Moreover the upper semicontinuity of the correspondence D implies that U is an upper semicontinuous univalued function on conv X (see for example Berge [1], theorem 2, p. 122).

It remains to prove that U is quasi-concave on conv X; this is an immediate con- sequence of the logical equivalence:

U(z) > U(y) D(z) =DD(y), where y and z are any two points in conv X. From the very definition of U,

D(z) v D(y) => U(z) _ U(y) is trivial. Suppose D

U(z) ? U(y) -> D(z) D D(y)

is false; then there would exist w E X such that w E D(y), hence U(y) _ u(w), but w 0 D(z), hence z 0 conv 3(w). Consider any x in D(z); as z E conv 3(x) but z 0 conv 3(w), 3(w) ca (x),

i.e. u(w) > u(x). It can therefore be concluded that, for every x in D(z), U(y) > u(x), a conclusion which

contradicts U(z) _ U(y). Q.E.D.



HENRY MARKET GAMES 75

Remarks:

(1) In general U is not continuous, as shown by the following counterexample:

X = {a, b}, with u(a) < u(b);

conv X = [a, b], conv b(a) = [a, b], conv 3(b) = {b};

D(z) = {a} if z E [a, b[ and D(b) = {a, b};

U(z) = u(a) if z E [a, b[ and U(b) = u(b).'

(2) From the equivalence

U(z) ? U(y):D(z) v D(y), it results that

U(z) > U(y) if every conv 6(x) which contains y also contain z.

This necessary and sufficient condition is precisely that used by Starr [6] to convex a continuous preordering defined on Rm as consumption set; indeed he writes X,kY if, for all w in Rm+, x E conv A(w) whenever y E conv A(w), where A(w) = {v e Rm j v > w}. In the case considered in Theorem 1 Starr's convexed preordering need not in general be continuous and Debreu's theorem cannot be invoked to claim the existence of a utility function for that preordering; this is precisely the reason why I chose another way to convex the initial utility function u.

UNBLOCKED ALLOCATIONS WITH LEAST EXCESS DEMAND

Suppose E is a pure exchange economy with n > m consumers. Let i denote any of these consumers and xi any commodity bundle in i's consumption set Xi; i(x') being the set of all the commodity bundles in Xi which i desires at least as much as xi, consider any z in conv 6'(xL) and a point x in 6'(x') nearest to z; then denote by di (xi, z) the coordinates of the vector x-z and by

d' = max max d (x'; z) xi e xi z e conv yi(xi)

the coordinates of a vector d' which can be used to measure the distance between i's utility function ui and its least quasi-concave majorant Ui.

Theorem 2. If every consumer in a pure exchange economy has a continuous utility function and a compact consumption set including the bundle of his initial resources, then there exists at least one unblocked allocation {xi I i1, ..., n} such that, for every commodity h-l,...~m,

n n

Zx < E o4+m. max d, i=h1 i=1 i= .n

where co is the bundle of its initial resources.

Remark. Of course co'>0; but we do not make the quite unrealistic assumption coi >> 0.

Proof of Theorem 2. As a corollary of his theorem on the core of a balanced game, Scarf [6] has shown that a pure exchange economy, with upper semicontinuous quasi- concave utility functions and compact convex consumption sets including the corresponding bundles of initial resources, always has a non-empty core; one can thus choose an allocation {yiI i = 1, ..., n} in the core of economy P = {co', conv X', U' I i = 1, ..., n}.

1 This counterexample is due to the referee, whom I want to thank for all the suggestions he made to improve the initial version of this paper.



76 REVIEW OF ECONOMIC STUDIES

For every consumer i select x' in D'(y') = {x E Xi I y' E conv 3'(x')} in such a way that n n n

Ui(yi) = u'(x'). AS E yi = coi, oi is thus a point in the convex hull of the sum of i=l i =] i=l

the compact sets 3i (xi) i = 1, ..., n. We can then apply a theorem proved by Folkman n n n

and Shapley [3] to write point E coi as a sum coi = E zi where, for every consumer

i, zi E conv 3' (xi), and for at least n-rm among the n consumers i, zi E 3' (xi); indeed this theorem reads: " let K', i = 1, ..., n, be compact sets in Rm, n>rm, and let x be any point

n in the convex hull of the sum E K'; then in the convex hull of every K', i=1,..., n,

i = 1 n

one can find a point y' in such a way that x = E yi and that, with at most m exceptions,

yi E K'. Without any restriction on the generality of the proof, one may suppose that zi E 3' (xi)

for every i = m rn+, ..., n, which implies ui (zi) > u'(x') = U'(y'); let then i = zi. When i = 1, ..., m, it may happen that u <(z') u'(x') = U'(y'); let then xi be a point

in 3' (x') such that, for every h = 1, ..., m, m -zg ? d4; of course

ui(xi) > ui(xi) = ui(yi).

An allocation {x' I i = 1, .. ., n} has thus been defined in economy E in such a way that it is unblocked-as {yi i = 1, ..., n} is in the convexed economy L-and satisfies the following conditions, h = 1, ..., m:

n n m n

Z5E < + E h d? < E c+ m. max d'. Q.E.D. i=1 i=1 i= 1 i=1 i= .

Corollary to Theorem 2. Making the same assumptions as in Theorem 2, if for every commodity h there exists dh such that dh > dh for every consumer i, i.e. if the di are bounded, then

ni n

nli=1 nli= 1 n

for every commodity h the difference between the total quantity allocated and the total resources can thus be made as little as wanted with respect to the number of consumers.

REFERENCES

[1] Berge, C. Espaces topologiques et fonctions multivoques, Dunod (Paris 1966) 2nd edition.

[2] Dierker, E. "Equilibrium analysis of exchange economies with indivisible commodities ", Econometrica (forthcoming).

[3] Folkman, J. H. and Shapley, L. S. "Starr's problem ", Unpublished, reported by Starr [7].

[4] Henry, C. "Indivisibilites dans une economie d'echanges ", Econometrica 38, 542-558, 1970.

[5] Rockafellar, R. T. Convex analysis, Princeton University Press (1970).

[6] Scarf, H. E. " The core of an n-person game ", Econometrica 35, 50-69, 1967.

[7] Starr, R. M. " Quasi-equilibria in markets with non-convex preferences ". Econo- metrica 37, 25-38, 1969.



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