chich, arbitrage, choice and the core

38 Pages·1997·0.63 MB·English

SocChoiceWelfare(1997)14:161—198
Market arbitrage, social choice and the core
Graciela Chichilnisky*
Programon InformationandResources,ColumbiaUniversity,
405LowMemorialLibrary,NewYork,NY10027,USA
Received:30December1994/Accepted:12August1996
Abstract. This paper establishes a clear connection between equilibrium
theory, game theory and social choice theory by showing that, for a well
definedsocialchoiceproblem,aconditionwhichisnecessaryandsufficientto
solvethisproblem—limitedarbitrage—isthesameastheconditionwhichis
necessary and sufficient to establish the existence of an equilibrium and the
core.Theconnectionisstrengthenedbyestablishingthatamarketallocation,
which is in the core, can always be realized as a social allocation, i.e. an
allocationwhichisoptimalaccordingtoanorderingchosenbyasocialchoice
rule. Limited arbitrage characterizes those economies without Condorcet
triples, and those for which Arrow’s paradox can be resolved on choices of
large utility values.
1. Introduction
Marketsprovideawidelyusedsolutiontotheproblemofallocatingresources
among the members of the economy. A market equilibrium is individually
optimaland clears the markets. The efficiencyof competitivemarket alloca-
tions is what makes them desirable.
*UNESCO Chair in Mathematics and Economics and Director, Program on In-
formation and Resources, Columbia University. email: gc9(cid:64)columbia.edu. Support
from Stanford Institute for Theoretical Economics, NSF Grant SES92-16028, the
Sloan Foundation, the Salinbemi Chair at the University of Siena and the Leif
JohansenAwardattheUniversityofOslocontributedtotheresearchreportedhere.
The main results of this paper were presented at ‘‘Columbia Celebrates Arrow’s
Contributions’’atColumbiaUniversity,October1991,andatinvitedaddressestothe
International Congress of Mathematicians Paris, July 4, 1994, a Colloquium at the
UniversityofCaliforniaatBerkeley,March1993,attheAmericanEconomicAssoci-
ationAnnualMeetingsinBoston,January3—51994,andasakeynoteaddresstothe
WorldCongressonNonLinearAnalysisinAthens,July16,1996.
(cid:49)(cid:54)(cid:50) G.Chichilnisky
A different solution to the resource allocation problem is provided by
social choice rules. These rules derive a social preference as a function of
individual preferences in a fair fashion, for example through voting. An
allocationwhichisoptimalaccordingtoasocialpreference,iscalledasocial
allocation.
A third solution to the problem of resource allocations is the core: these
are allocations from which no coalition would want to deviate.
This paper establishes a clear connection between equilibrium theory,
socialchoicetheoryandgametheorybyshowingthat,forawelldefinedsocial
choice problem, a condition which is necessary and sufficient to solve the
socialchoiceproblem—limitedarbitrage—isthesameastheconditionwhich
isnecessaryandsufficienttoestablishtheexistenceofanequilibriumandthe
core.(cid:49) The connection is strengthened by establishing that a market alloca-
tion, which is in the core, can always be realized as a social allocation.
How does limited arbitrage work? Limited arbitrage is a condition on
traders’endowmentsandpreferences.Introducedin[18]itboundspotential
gains from trade, and the scope for mutually advantageous reallocation in
theeconomy.Inmathematicalterms,limitedarbitrageisthenon-emptyinter-
section of a family of cones. Somewhat surprisingly, this nonempty inter-
sectionis thesameasatopologicalcondition:thecontractibilityof spacesof
preferences.Thisconnection arises from a new result: a family of convex sets
intersects if and only if every subfamily has a contractible union, proved in
Chichilnisky[15,19].Ithasbeenshownthatcontractibilityisarestrictionon
thediversityofpreferences[44].Throughitsconnectionwithcontractibility,
therefore,limitedarbitrageimpliesarestrictiononthetradersdiversity[22],
[23,26,28].Indeed,thecentralroleoflimitedarbitrageinresourceallocation
is explained by its connection with social diversity [22], [23], [26], [28].
Social diversity is crucial for resource allocation.
For example, social diversity in the form of contractibilityof preferences
is necessary and sufficient for the existence of continuous anonymous social
choice rules respecting unanimity, Chichilnisky and Heal [33,18,12]. Since
limited arbitrage is a form of contractibility, this explains the connection
betweenlimited arbitrage and social choice. I show below that limited arbi-
trage is also closely connected with Arrow’s impossibility theorem: it is
necessaryandsufficienttoeliminateCondorcettriples—orcyclicalbehavior—
on choices over feasible allocations of utility value which approaches the
supremum of utilities, and for solving Arrow’s theorem on such choices.(cid:50)
Diversity has a fundamental role in other forms of resource allocation.
Indeed, markets exist because people are different. Only when traders have
differentpreferencesandendowmentsdotheyhaveareasontotrade.Further-
moreaconditionwhichlimitsdiversity,namelylimitedarbitrage,isnecessary
(cid:49)These results have been announced and presented in a number of papers starting
in 1991: Chichilnisky [18,16,17,22,21,23,24,20,26,31,27] Chichilnisky and Heal
[32];thesepapersandtheliteraturearediscussedinacommentbyMonteiroPageand
Wooders[50]andits responseChichilnisky[25].
(cid:50)This is an ordinal result, which is independent from the utility representation of
preferences.Seealso[20,21,22,23].
Marketarbitrage,socialchoiceandthecore (cid:49)(cid:54)(cid:51)
and sufficient for the existence of a competitive equilibrium in general
economies [16,32,23,24,26]. It is also necessary and sufficient for the
uniqueness of equilibrium in strictly regular economies [27]. Elsewhere
I showed also that limited arbitrage is necessary and sufficient for the
existence of Pareto efficient allocations [28,29,30,16,24]. Thus a failure of
limited arbitrage prevents optimal coalition formation: it leads to an
emptycore[21,28].Diversityisthereforeequallyimportantforgametheoret-
ical allocations.
Theseresultshighlightnovelfeaturesofresourceallocation.Forexample,
market allocations are generally considered to be practical, more so than
social allocations. One reason for this is that markets are viewed as
having equilibria generally, while social choice theory has always stressed
paradoxes and non-existence results. Our results, however, show that this
is not accurate. The same restriction on diversity is necessary for the
existence of market equilibrium and for the existence of appropriate social
choice rules.(cid:51)
Acarefulexaminationoftheliteratureillustratestheimportanceofdiver-
sity in resource allocation. The social choice literature has focused on inter-
personal diversity as a reason for the non-existence of social choice rules.
Several ways of resolving social choice problems have been proposed by
limiting the diversity of the individual preferences, a ‘domain restriction’ on
preferences,Black,[8,9],PattanaikandSen[55],andChichilniskyandHeal
[33]. These works propose ways of limiting the diversity of individuals. The
issueofexistenceofuniversalsocialchoicerules—aproblemwhichinitsmore
generalform hasnosolution—is turnedintothequestion:forwhat societies
can the social choice problem be resolved? Or: how much diversity can
a society function with?(cid:52)
The issue of diversity has also been studied in the context of market
equilibrium. There are widely used conditions which ensure the existence of
a competitive equilibrium. These are also restrictions in social diversity:
theyrequirethattheendowmentsofanyhouseholdaredesired,indirectlyor
directly, by others, so that their incomes cannot fall to zero: examples are
McKenzie’sirreducibility condition [48,49,50], and the resource relatedness
conditioninArrowandHahn[3].Theseconditionsaredifferentfromlimited
arbitrage,butneverthelessrestrictthediversityofindividuals’preferencesand
endowments.Limitedarbitrageprovidesanotherlimitationinsocialdiversity:
it does so parsimoniously, because it is a minimal condition, in the sense of
(cid:51)Competitiveequilibriaarenotalwaysavailable:whenthetradersendowmentsand
preferencesareverydifferenttheeconomycanfailtohaveacompetitiveequilibrium.
ArrowandHahn([3],Chapter4,1)providedexamplesofstandardmarketeconomies
with no competitive equilibrium. Their example highlights the role of interpersonal
differences between the traders, differences in their endowments and in their prefer-
ences,inpreventingtheexistenceofacompetitiveequilibrium.Thefailureofexistence
arises when some traders have zero income, while others wish and can afford un-
boundedly large allocations which a bounded economy cannot accommodate ([3],
Chapter4, p.80).
(cid:52)Black’s singlepeakedness condition restricts diversity and solves the problem pro-
posedbyCondorcet’s[34]paradoxofmajorityvoting.
(cid:49)(cid:54)(cid:52) G.Chichilnisky
being necessary as well as sufficient for existence. In addition, it applies
simultaneously to resource allocation by market equilibrium, by social
choice(cid:53)andthecore.Indeedlimitedarbitragemeasurespreciselytheamount
ofdiversityacceptableforthefunctioningofmarkets,cooperativegamesand
social choice.
Thereis a differentperspective ondiversity, onewhich views diversityas
a positive factor in a group’s adaptation to its environment. This view
contrasts with the role of diversity on resource allocation, in which the
functioningofeconomicinstitutionsimposeslimitsondiversity.Thecontrast
suggestsanopen,andperhapsunsettling,question:Aretheformsofdiversity
implied in our economic institutions consistent with humans’ successful ad-
aptation to change? Are our economic institutions sustainable?
2. Definitions and examples
An Arrow Debreu market (cid:69)"(cid:77)X,(cid:41) ,u ,h"1,2,H(cid:78) has H52 traders,
(cid:104) (cid:104)
indexedh"1,2,H,N52commoditiesandconsumptionortradingspace(cid:54)
X"R(cid:78) orX"R(cid:78);elsewhereIhaveconsideredthecasewhereinsteadofR(cid:78)
(cid:96)
one has a Hilbert space of infinite dimension.(cid:55) The vector (cid:41) 3R(cid:78) denotes
(cid:104) (cid:96)
traderh’spropertyrightsorinitialendowmentand(cid:41)"((cid:43)(cid:72)(cid:104)(cid:47)(cid:49)(cid:41)(cid:104))isthetotal
endowment of the economy.(cid:56) Traders may have zero endowments of some
goods.Eachtraderhhasacontinuousandconvexpreferencerepresentedby
u :XPR.Thispapertreatsinaunifiedwaygeneralconvexpreferenceswhere
(cid:104)
simultaneously on every indifference surface for a given trader, the map
g(x)"Du(x)/(cid:68)(cid:68)Du(x)(cid:68)(cid:68)assigningtoeachvectorthenormalizedutilitygradient
iseitheropenorclosed.Thereforeeither(i) allindifferencesurfacescontainno
halflines(themapgisopen)or(ii) thenormalizedgradientstoanyclosedset
of indifferent vectors define a closed set (the map g is closed). Some traders
may have preferences of one type, and some of the other. Case (i) includes
strictly convex preferences, and case (ii) linear preferences. All the
assumptionsandtheresultsinthispaperareordinal,(cid:57)thereforewithoutloss
of generality one normalizes utilities so that for all h,u (0)"0 and
(cid:104)
sup u (x)"R. Preferences are increasing, i.e. x’y N u (x)5u (y).
(cid:77)x:x3X(cid:78) (cid:104) (cid:104) (cid:104)
WhenX"R(cid:78) eitherindifferencesurfacesof positiveutilityarecontainedin
(cid:96)
the interior of X,R(cid:78) , such as Cobb—Douglasutilities, or if an indifference
(cid:96)(cid:96)
(cid:53)ChichilniskyandHeal[33]establisheddomainrestrictionswhicharenecessaryand
sufficientfortheexistenceofsocialchoiceruleswhichsatisfytheaxiomsof[12,13]:
contractibilityofthespaceofpreferencesisnecessaryandsufficientfortheexistenceof
socialchoicerules.Althoughtheseworksdealwithsomewhatdifferentaxioms,theyall
findthesametypeofsolution:arestrictionof individuals’diversity.
(cid:54)R(cid:78)"(cid:77)(x ,2,x )3R(cid:78):∀i,x50(cid:78).
(cid:96) (cid:49) (cid:78) (cid:105)
(cid:55)Chichilnisky and Heal [31,32] proved that limited arbitrage is necessary and
sufficient for the existence of a competitive equilibrium in economies with infinitely
manymarkets.
(cid:56)If x,y3R(cid:78), x5y8∀ix5y, x’y8x5y and for some i,x’y, and x<y
(cid:105) (cid:105) (cid:105) (cid:105)
8∀i,x’y.
(cid:105) (cid:105)
(cid:57)Namelyindependentofthe utilityrepresentations.
Marketarbitrage,socialchoiceandthecore (cid:49)(cid:54)(cid:53)
surface of positive utility intersects a boundary ray,(cid:49)(cid:48) it does so trans-
versally.(cid:49)(cid:49)
Definition 1. A preference is uniformly non-satiated when it is represented
by a utility u with a bounded rate of increase, e.g. for smooth preferences.(cid:49)(cid:50)
(cid:104)
(cid:38)(cid:101), K’0:∀x3X, K’(cid:68)(cid:68)Du (x)(cid:68)(cid:68)’(cid:101).
(cid:104)
Uniformlynon-satiatedpreferencesarerathercommon:forexample,prefer-
ences represented by linear utilities are uniformly non-satiated.(cid:49)(cid:51)
Proposition2. If a utility function u :R(cid:78)PR is uniformly non-satiated its
(cid:104)
indifferencesurfaces are within a uniform distance of each other, i.e. ∀r,s3R,
(cid:38)N(r,s)3R such that x3u(cid:126)(cid:49)(r)N(cid:38)y3u(cid:126)(cid:49)(s) with (cid:68)(cid:68)x!y(cid:68)(cid:68)4N(r,s).
(cid:104) (cid:104)
Proof. This is immediate from the definition. (cid:106)
Assumption1. WhenX"R(cid:78),thepreferencesin theeconomy(cid:69)areuniformly
non-satiated.
The space of feasible allocations is (cid:174)"(cid:77)(x(cid:49),2,x(cid:72))3X(cid:72):(cid:43)(cid:72)(cid:104)(cid:47)(cid:49)x(cid:104)"(cid:41)(cid:78).
Theset ofsupportstoindividuallyrationalaffordableefficientresourcealloca-
tions is
S((cid:69))"(cid:77)v3R(cid:78): if (x ,2,x )3(cid:174) with u (x )5u ((cid:41) )∀h"1,2,H,
(cid:49) (cid:72) (cid:104) (cid:104) (cid:104) (cid:104)
(cid:83)v,x !(cid:41) (cid:84)"0, and ∀h u (z )5u (x ) then (cid:83)v,z !x (cid:84)50(cid:78).
(cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104)
(1)
The set of prices orthogonal to the endowments is(cid:49)(cid:52)
(cid:78)"(cid:77)v3R(cid:78)!(cid:77)0(cid:78): (cid:38)h with (cid:83)v,(cid:41) (cid:84)"0(cid:78). (2)
(cid:96) (cid:104)
(cid:49)(cid:48)AboundaryrayrinR(cid:78)isasetwhichconsistsofallthepositivemultiplesofavector
(cid:96)
(cid:108)3(cid:76)R(cid:78) i.e.r"(cid:77)w3R(cid:78):(cid:38)(cid:106)’0s.t.w"(cid:106)v(cid:78).
(cid:96) (cid:96)
(cid:49)(cid:49)Thismeansthatifx3(cid:76)R(cid:78)andu(x)’0,thenthegradientDu(x)isnotorthogonalto
(cid:96)
(cid:76)R(cid:78)at(cid:106)x,∀(cid:106)50.Thisconditionincludesstrictlyconvexpreferences.Cobb—Douglas
(cid:96)
and CES preferences, many Leontieff preferences u(x,y)"min(ax,by), preferences
whichareindifferenttooneormorecommodities,suchasu(x,y,z)"(cid:74)x#y,prefer-
enceswithindifferencesurfaceswhichcontainraysof(cid:76)R(cid:78) suchasu(x,y,z)"x,and
(cid:96)
preferencesdefinedonaneighborhoodofthepositiveorthantorthewholespace,and
whichareincreasingalongtheboundaries,e.g.u(x,y,z)"x#y#z.
(cid:49)(cid:50)Smoothness is used to simplify notation only: uniform nonsatiation requires no
smoothness.Whenpreferencesadmitnosmoothutilityrepresentation,thenonerequires
(cid:38)(cid:101),K’0:∀x,y,K(cid:68)(cid:68)x!y(cid:68)(cid:68)’(cid:68)u(y)!u(x)(cid:68)andsup (cid:68)u(y#x)!u(x)(cid:68)5(cid:101)(cid:100).
(cid:68)(cid:68)x(cid:68)(cid:68)((cid:100)
(cid:49)(cid:51)Thisincludeslinear preferences,piecewise linearpreferences,Leontiefpreferences,
preferences with indifference surfaces which intersect the boundary of the positive
orthant(ArrowandHahn1971)andsmoothutilitieswhicharetransversaltoit.
(cid:49)(cid:52)Nisemptywhen∀h,(cid:41) <0.
(cid:104)
(cid:49)(cid:54)(cid:54) G.Chichilnisky
Fig.1. This preference is not uniformly non-satiated because two indifferencesurfaces spread
apartforever
The utility possibility set of the economy (cid:69) is the set of feasible and individ-
ually rational utility allocations:
”((cid:69))"(cid:77)(» ,2,» ): ∀h,»"u (x )5u ((cid:41) )50,
(cid:49) (cid:72) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104)
for some (x ,2,x )3(cid:174)(cid:78).
(cid:49) (cid:72)
TheParetofrontieroftheeconomy(cid:69)isthesetoffeasible,individuallyrational
and efficient utility allocations:
P((cid:69))"(cid:77)”3”((cid:69)):&(cid:38)…3”((cid:69)):…’”(cid:78).
A competitive equilibrium of (cid:69) consists of a price vector p*3R(cid:78) and a fea-
(cid:96)
sibleallocation(x*(cid:49),2,x*(cid:72))3(cid:174)suchthatx*(cid:104) optimizesu(cid:104)overthebudgetset
B (p*)"(cid:77)x3X:(cid:83)x,p*(cid:84)"(cid:83)(cid:41) ,p*(cid:84)(cid:78).
(cid:104) (cid:104)
(cid:50)(cid:46)(cid:49)(cid:46) (cid:71)(cid:108)(cid:111)(cid:98)(cid:97)(cid:108) (cid:97)(cid:110)(cid:100) (cid:109)(cid:97)(cid:114)(cid:107)(cid:101)(cid:116) (cid:99)(cid:111)(cid:110)(cid:101)(cid:115)
Two cases, X"R(cid:78) and X"R(cid:78), are considered separately.
(cid:96)
(cid:102) Consider first X"R(cid:78).
Definition3. Fortraderhdefinethe(cid:103)(cid:108)(cid:111)(cid:98)(cid:97)(cid:108)(cid:99)(cid:111)(cid:110)(cid:101)ofdirectionsalongwhichutility
increaseswithout bound:
A ((cid:41) )"(cid:77)x3X: ∀y3X,(cid:38)(cid:106)’0: u ((cid:41) #(cid:106)x)’u (y)(cid:78)
(cid:104) (cid:104) (cid:104) (cid:104) (cid:104)
Marketarbitrage,socialchoiceandthecore (cid:49)(cid:54)(cid:55)
Thisconewasintroducedin[18,23,24];itcontainsglobalinformationonthe
economyandisnewintheliterature.(cid:49)(cid:53)Inordinalterms,theraysofthiscone
intersectallindifferencesurfacescorrespondingtobundlespreferredbyu to
(cid:104)
(cid:41) .Thisconeandthepartofitsboundaryalongwhichutilityneverceasesto
(cid:104)
increase define:
G ((cid:41) )"(cid:77)x3X:&(cid:38)Max u ((cid:41) #(cid:106)x)(cid:78)
(cid:104) (cid:104) (cid:106)50 (cid:104) (cid:104)
Thiscone is identicalto theglobal cone definedinChichilnisky(1995,p. 85,
4);(cid:49)(cid:54) it treats all convex preferences in a unified way and under Assumption
1ithasasimplestructure:whenpreferenceshavehalflinesintheirindifferen-
cesG ((cid:41) )equalsA ((cid:41) );whenindifferencescontainnohalflines,thenG ((cid:41) )
(cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104)
is its closure.
Definition 4. „he (cid:109)(cid:97)(cid:114)(cid:107)(cid:101)(cid:116) (cid:99)(cid:111)(cid:110)(cid:101) of trader h is
D ((cid:41) )"(cid:77)z3X: ∀y3G ((cid:41) ), (cid:83)z,y(cid:84)’0(cid:78) (3)
(cid:104) (cid:104) (cid:104) (cid:104)
D istheconeofpricesassigningstrictlypositivevaluetoalldirectionsof
(cid:104)
net trades leading to eventually increasing utility. This is a convex cone.
The following proposition establishes the structure of the global cones,
andisusedinprovingtheconnectionbetweenlimitedarbitrage,equilibrium
and the core:
Proposition5. If the function u :R(cid:78)PR is uniformly nonsatiated: (i) „he
(cid:104)
interior of the global cone is
G(cid:111)((cid:41) )"A(cid:111)((cid:41) )
(cid:104) (cid:104) (cid:104) (cid:104)
"(cid:77)z3G ((cid:41) ): lim u ((cid:41) #(cid:106)z)"R(cid:78)O(cid:48).
(cid:104) (cid:104) (cid:106)PR (cid:104) (cid:104)
(ii) „he boundary of the cone G ((cid:41) ), (cid:76)G ((cid:41) ), contains (a) those directions
(cid:104) (cid:104) (cid:104) (cid:104)
along which utility increases towards a bounded value that is never reached:
B ((cid:41) )"(cid:77)z3(cid:76)A ((cid:41) ): ∀(cid:106)’0, u ((cid:41) #(cid:106)z)Olim u ((cid:41) #(cid:106)z)(R(cid:78)
(cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104)
(cid:106)PR
and (b) those directions along which the utility eventually achieves a constant
value:
C ((cid:41) )"(cid:77)z3(cid:76)A ((cid:41) ): (cid:38)N:(cid:106),(cid:107)’NNu ((cid:41) #(cid:106)z)"u ((cid:41) #(cid:107)z)(cid:78),
(cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104)
(cid:49)(cid:53)TheglobalconeA ((cid:111) ,(cid:41) )haspointsincommonwithDebreu’s[34]‘‘asymptotic
(cid:104) (cid:104) (cid:104)
cone’’ corresponding to the preferred set of u at the initial endowment (cid:41) , in that
(cid:104) (cid:104)
alonganyofitsraysutilityincreases.UnderAssumption1,itsclosureAM (cid:104)((cid:41)(cid:104)),equals
the‘‘recession’’coneusedbyRockafellerin1970,butnotgenerally:alongtheraysin
A ((cid:41) )utilityincreasesbeyondtheutilitylevelofanyothervectorinthespace.This
(cid:104) (cid:104)
conditionneednotbesatisfiedbyDebreu’sasymptoticcones[35]orby‘‘recession’’
cones. For example, for Leontief type preferences the recession cone through the
endowmentistheclosureoftheuppercontour,whichincludestheindifferencecurve
itself.Bycontrast,theglobalconeA ((cid:41) )istheinterioroftheuppercontourset.The
(cid:104) (cid:104)
globalconeofaconstantfunctionisemptywhiletherecessionconeisthewholespace.
RelatedconceptsappearedinChichilnisky[10,11];otherwisethereisnoprecedentin
theliteratureforglobalcones.
(cid:49)(cid:54)The coneG ((cid:41) ) is that in Chichilnisky[18] underAssumption1, and it appears
(cid:104) (cid:104)
alsoinMonteiroPageandWooders[51]whichisacommentonChichilnisky[18].
(cid:49)(cid:54)(cid:56) G.Chichilnisky
Fig.2. This preference has a ‘fan’ of directions along which the utility values reach a bound.
Assumption1isnotsatisfied.Allthedirectionsinthefanareintherecessionconebutnoneinthe
globalconeG(cid:104)orintheconeA(cid:104).
(iii) theinterioroftheglobalcone,itsboundaryanditsclosureandthecones
G and D are uniform across all vectors in the space, i.e. ∀(cid:41),(cid:34)3X:
(cid:104) (cid:104)
G(cid:111)((cid:41))"G(cid:111)((cid:34))"A(cid:111),
(cid:104) (cid:104) (cid:104)
B ((cid:41))(cid:88)C ((cid:41))L(cid:76)G ((cid:41))"(cid:76)G ((cid:34))"(cid:76)G ,
(cid:104) (cid:104) (cid:104) (cid:104) (cid:104)
G ((cid:41))"G ((cid:34))"G ,D ((cid:41))"D ((cid:34))"D ,
(cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104)
and in particular
G ((cid:41))"G ((cid:34))"G .
M (cid:104) M (cid:104) M (cid:104)
(iv) For general non-satiated preferences G ((cid:41) ) and D ((cid:41) ) may not be
(cid:104) (cid:104) (cid:104) (cid:104)
uniform.
Proof. See [23] and [26], Proposition 3, Appendix.
(cid:102) Consider next the case: X"R(cid:78).
(cid:96)
Definition 6. „he market cone of trader h is
D(cid:96)((cid:41) )"D ((cid:41) )(cid:87)S((cid:69)) if S((cid:69))L(cid:78),
(cid:104) (cid:104) (cid:104) (cid:104)
"D ((cid:41) ) otherwise. (4)
(cid:104) (cid:104)
where S((cid:69)) and (cid:78) are defined in (1) and (2).(cid:49)(cid:55)
(cid:49)(cid:55)ThemarketconeD(cid:96)isthewholeconsumptionsetX"R(cid:78)(cid:96)whenS(E)hasavector
(cid:104)
assigningstrictly positive income to all individuals. If some trader has zero income,
thenthistradermusthaveaboundaryendowment.
Marketarbitrage,socialchoiceandthecore (cid:49)(cid:54)(cid:57)
ThereisnoanalogtoProposition5whenX"R(cid:78);indeed,whenX"R(cid:78)
(cid:96) (cid:96)
the market cones D(cid:96)((cid:41) ) typically vary with the initial endowments.
(cid:104) (cid:104)
Proposition 7. …hen X"R(cid:78) and an indifference surface of u corresponding
(cid:96) (cid:104)
toapositiveconsumptionbundlex’0intersectsaboundaryrayrL(cid:76)X,then
r3G (0).
(cid:104)
Proof. Recallthatweassumedu (0)"0,andthatthepreference’sindifference
(cid:104)
surfacesofpositiveutilityareeither(a)containedintheinteriorofR(cid:78),R(cid:78) ,
(cid:96) (cid:96)(cid:96)
or(b)theyintersectaboundaryrayrofR(cid:78) anddosotransversally.Incase(a)
(cid:96)
thepropositionissatisfiedtrivially,becausenoindifferencesurfaceofstrictly
positivevalueeverintersectstheboundaryofR(cid:78).Incase(b)theproposition
(cid:96)
follows immediately from the definition of transversality. Observe that it is
possible that sup (u (x))(R. (cid:106)
x3r (cid:104)
Definition 8. „he core of the economy E is the set of allocations which no
coalition can improve upon within its own endowments:
C((cid:69))"(cid:77)(x ,2,x )3R(cid:78)(cid:93)(cid:72):(cid:43)(x !(cid:41) )"0 and &JL(cid:77)1,2,H(cid:78):
(cid:49) (cid:72) (cid:104) (cid:104)
(cid:104)
(cid:93)(cid:43) (y!(cid:41) )"0,∀3J,u (y )5u (x ),
(cid:106)(cid:124)(cid:74) (cid:106) (cid:106) (cid:106) (cid:106) (cid:106) (cid:106) (cid:106)
and (cid:38) 3J:u (y )’u (x )(cid:78).
(cid:106) (cid:106) (cid:106) (cid:106) (cid:106)
3. Limited arbitrage: definition and examples
This section provides the definition of limited arbitrage. It gives an intuitive
interpretationforlimitedarbitrageintermsofgainsfromtrade,andcontrasts
limited arbitrage with the arbitrage concept used in financial markets. It
provides examples of economies with and without limited arbitrage.
Definition 9. …hen X"R(cid:78), (cid:69) satisfies (cid:108)(cid:105)(cid:109)(cid:105)(cid:116)(cid:101)(cid:100) (cid:97)(cid:114)(cid:98)(cid:105)(cid:116)(cid:114)(cid:97)(cid:103)(cid:101) if
(cid:72)
(‚A) (cid:89) D O(cid:48).
(cid:104)
(cid:104)(cid:47)(cid:49)
Definition 10. …hen X"R(cid:78), (cid:69) satisfies (cid:108)(cid:105)(cid:109)(cid:105)(cid:116)(cid:101)(cid:100) (cid:97)(cid:114)(cid:98)(cid:105)(cid:116)(cid:114)(cid:97)(cid:103)(cid:101) if
(cid:96)
(cid:72)
(‚A(cid:96)) (cid:89) D(cid:96)((cid:41) )O(cid:48). (5)
(cid:104) (cid:104)
(cid:104)(cid:47)(cid:49)
(cid:51)(cid:46)(cid:49)(cid:46) (cid:73)(cid:110)(cid:116)(cid:101)(cid:114)(cid:112)(cid:114)(cid:101)(cid:116)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110) (cid:111)(cid:102) (cid:108)(cid:105)(cid:109)(cid:105)(cid:116)(cid:101)(cid:100) (cid:97)(cid:114)(cid:98)(cid:105)(cid:116)(cid:114)(cid:97)(cid:103)(cid:101) (cid:97)(cid:115) (cid:98)(cid:111)(cid:117)(cid:110)(cid:100)(cid:101)(cid:100) (cid:103)(cid:97)(cid:105)(cid:110)(cid:115) (cid:102)(cid:114)(cid:111)(cid:109)
(cid:116)(cid:114)(cid:97)(cid:100)(cid:101) (cid:119)(cid:104)(cid:101)(cid:110) (cid:88)"(cid:82)(cid:78)
Gains from trade are defined by
(cid:71) (cid:72) (cid:72)
(cid:71)((cid:69))"sup (cid:43) u (x )!u ((cid:41) ) , where
(cid:104) (cid:104) (cid:104) (cid:104)
(cid:104)(cid:47)(cid:49)
(cid:49)(cid:55)(cid:48) G.Chichilnisky
Fig.3. Limitedarbitrageis
satisfied:feasibleallocationslead
toboundedutilityincreases
(cid:72)
(cid:43) (x !(cid:41) )"0, and ∀h, u (x )5u ((cid:41) )50.
(cid:104) (cid:104) (cid:104) (cid:104) (cid:104) (cid:104)
(cid:104)(cid:47)(cid:49)
The proposition below applies to preferences where the normals to
a closed set of indifferent vectors defines a closed set, case (ii); its corollary
following it applies both to case (i) and (ii):
Proposition11. Incase(ii),theeconomy(cid:69)satisfieslimitedarbitrageifandonly
if gains from trade are bounded i.e. if and only if(cid:49)(cid:56)
(cid:71)((cid:69))(R.
Foraproofseee.g.[23]andTheorem1provedintheAppendixof[26];its
sufficiencypartisvalidforallpreferencessatisfyingAssumption1,case(i)or
case (ii),
so that:
Corollary12. Foralleconomieswithuniformlynon-satiatedpreferences,limited
arbitrage implies bounded gains from trade.(cid:49)(cid:57)
(cid:49)(cid:56)TheexpressionG((cid:69))(Rholdswhen∀h,sup u (x)"Rasconsideredhere;
fiotrmsuosmtbeeproespitliavceedk,bwyhGen((cid:69)∀)(hssuupp(cid:77)x:x3X(cid:72):u(cid:43)x(x(cid:104)")((cid:41),uR(cid:104)(cid:77)(xx(cid:104):.)x53uX(cid:104)(cid:78)((cid:41)(cid:104))(cid:104)(cid:78)((cid:43)(cid:72)(cid:104)(cid:47)(cid:49)u(cid:104)(x(cid:104))!u(cid:104)((cid:41)(cid:104)))!k,
(cid:77)x:x3X(cid:78) (cid:104)
(cid:49)(cid:57)Thesimplestillustrationofthelinkbetweenlimitedarbitrageandno-arbitrageisan
economy(cid:69)wherethetraders’initialendowmentsarezero,(cid:41)"0forh"1,2,andthe
(cid:104)
set of gradients to indifference surfaces are closed. Here no-arbitrage at the initial
endowmentsmeansthattherearenotradeswhichcouldincreasethetraders’utilities
atzerocost:gainsfromtradein(cid:69)arezero.Bycontrast,(cid:69)haslimitedarbitragewhen
notradercanincreaseutilitybeyondagivenboundatzerocost:gainsfromtradeare
bounded.
Inbrief:no-aribtragerequiresthatthereshouldbenogainsfromtradeatzerocost
whilelimitedarbitragerequiresthatthereshouldbeonlyboundedutilityarbitrageor
limited gains from trade. In linear economies, limited arbitrage ‘‘collapses’’ into no-
arbitrage.