American Economic Association
Economic Theory in the Mathematical Mode
Author(s): Gerard Debreu
Source: The American Economic Review, Vol. 74, No. 3 (Jun., 1984), pp. 267-278
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/1804007
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Economic Theory in the MathematicalMode
By GERARD DEBREU*
I
If a symbolicdate were to be chosen
for the birthof mathematical
economics,
our profession,
in rare unanimousagreement,wouldselect1838,theyearin which
AugustinCournotpublishedhis Recherches
In thatyear,JohnvonNeumannand Oskar
Morgenstem
publishedthe firsteditionof
the Theoryof Gamesand EconomicBehavior,
an eventthatannounceda profoundand
extensive
transformation
ofeconomic
theory.
In the following
decade,powerful
intellectual stimulialso came frommany other
sur les PrincipesMathematiques
de la Theorie directions.
In additionto vonNeumannand
desRichesses.
Students
ofthehistory
ofeco- Morgenstern'sbook, Wassily Leontief's
nomicanalysiscouldpointoutcontributions input-outputanalysis, Paul Samuelson's
madeto mathematical
economics
as earlyas
Foundationsof Economic Analysis,Tjalling
thebeginning
oftheeighteenth
century.
They Koopmans'activity
analysisof production,
could also pointout JohannHeinrichvon andGeorgeDantzig'ssimplex
were
algorithm
topicsof discussion,
notablyat the
Thunen'sDer IsolierteStaat,1826,a proto- frequent
whenI joinediton June
typicalexampleof theuse of mathematical CowlesCommission
reasoningin economictheorywith little 1, 1950.To becomeassociatedat thattime
interactive
groupwhichpromathematical
formalism.
ButCournotstands witha strongly
forthetype
outas thefirst
greatbuilderofmathematical videdtheoptimalenvironment
models explainingeconomicphenomena. of researchthat I wantedto do was an
privilege.
in thenineteenth
cen- exceptional
Amonghissuccessors
forthatresearch
One leadingmotivation
the
turyand the earlytwentieth
century,
of generalecowillbe givenin thislec- was thestudyof thetheory
highest
prominence
Its goals wereto make
turetoLeonWalras(1834-1910),thefounder nomicequilibrium.
to generalize
rigorous,
it,to simof the mathematical
theoryof generaleco- thetheory
nomicequilibrium,
to FrancisY. Edgeworth plifyit,and to extendit in newdirections.
(1845-1926),and to Vilfredo
Pareto(1848The executionof such a programrequired
in thetheory
ofseveralproblems
1923).All threelivedlongenoughintothe thesolution
twentieth
and demand.It led to
to haveincreased,
forall
ofpreferences,
century
utility,
intoeconomic
ofnew
NobelLaureates,
thevalueoftheeconomics theintroduction
theory
borrowed
fromdiverse
prize,had it,liketheotherprizes,beenini- analytical
techniques
it made
fieldsof mathematics.
Occasionally
tiatedin 1901.
it necessaryto find answersto purely
If 1838 is the symbolicbirthdateof
questions.The numberof remathematical
economics,1944 is the sym- mathematical
small
involvedwas, at first,
bolicbeginning
of its contemporary
period. searchworkers
butin theearly1960's
and slowlyincreasing,
it beganto growmorerapidly.
The mostprimitive
of theconceptsof the
*University
of California,
CA 94720.This
Berkeley,
in Stockarticleis thelectureGerardDebreudelivered
I willsurvey
anddiscussis thatofthe
theory
holm,Sweden,December8, 1983,whenhe received
the
commodity
space.One makesa listofall the
Nobel Prize in EconomicSciences,The articleis
in theeconomy.
Let / be their
commodities
?)
copyrighttheNobelFoundation.
It is published
here
finite
number.
Havingchosena unitofmeawiththepermission
of theNobel Foundation,
and is
in thevolumeLes Prix Nobel 1983.
included
surement
foreach one of them,and a sign
I thankRobertAnderson,Frank Hahn, Werner convention
to distinguish
inputsfromoutHildenbrand,
HerbertScarf,StephenSmale,and espeare positive,
a
consumer
puts
(for
inputs
comciallyGeorgeand HelenBreak,formanyhelpful
outputsnegative;fora producerinputsare
ments.
267
268
THE AMHERICANECONOMIC REVIEW
JUNE 1984
negative,
outputspositive),
one can describe thatled me to our collaboration
was sometheactionof an economicagentby a vector whatdifferent.
Afterhavingbeeninfluenced
in thecommodity
spaceR'. The factthatthe at theEcoleNormaleSuperieure
in theearly
commodity
spacehas thestructure
of a real 1940's by the axiomaticapproachof N.
vectorspaceis a basicreasonforthesuccess Bourbakito mathematics,
I becameinterof themathematization
of economictheory. estedin economics
towardtheendofWorld
In particular
convexity
properties
of setsin War II. The traditionof the School of
themein thetheory
ofgeneral Lausanne had been kept alive in France,
RI,a recurring
economicequilibrium,
can be fullyexploited. notablyby Fran9oisDivisiaand by Maurice
If,in addition,
onechoosesa unitofaccount, Allais,and it was in Allais' formulation
in
and ifone specifies
thepriceof eachone of A la Recherched'une DisciplineEconomique
the1 commodities,
onedefines
a price-vector (1943) thatI firstmet,and was captivated
in R', a conceptdual to thatof a com- by, the theoryof generaleconomicequimodity-vector.
The valueof thecommodity- librium.
in theuncomTo somebody
trained
vectorz relative
to theprice-vector
p is then promising
rigorof Bourbaki,
counting
equatheinnerproductp z.
in theWalrasiansystem
tionsand unknowns
One of theaimsof themathematical
the- could not be satisfactory,
and the nagging
orythatWalrasfoundedin 1874-77 is to questionof existence
was posed.But in the
explaintheprice-vector
and the actionsof late 1940'sseveralessentialelements
of the
thevariousagentsobservedin an economy answerwerestillnotreadilyavailable.
in termsofan equilibrium
resulting
fromthe
In the meantime,
an easierproblemwas
interaction
of thoseagentsthrough
markets solved,and its solutioncontributed
signififorcommodities.
In suchan equilibrium,
ev- cantlyto thatof theexistence
problem.At
eryproducer
maximizes
hisprofit
relative
to theturnof thecentury,
Paretohad givena
theprice-vector
in hisproduction
set; every characterization
of an optimalstateof an
consumer
in hiscon- economyin termsof a pricesystem,
satisfies
hispreferences
using
sumptionset underthe budgetconstraint the differential
calculus.A long phase of
defined
bythevalueofhisendowment-vectordevelopment
of Pareto'sideas in the same
and hisshareoftheprofits
oftheproducers; mathematical
framework
came to a resting
and for everycommodity,
total demand pointwiththeindependent
contributions
of
equals totalsupply.Walrasand his succes- OscarLange(1942) and of Allais(1943).In
sorsforsixdecadesperceived
thathistheory the summerof 1950,Arrow,at theSecond
wouldbe vacuouswithoutan argument
in BerkeleySymposium
on Mathematical
Staofat leastone equi- tisticsand Probability,
supportoftheexistence
and I, at a meeting
of
librium,and noted thatin his model the the Econometric
Societyat Harvard,sepnumberof equationsequals thenumberof aratelytreatedthesameproblemby means
an argument
unknowns,
that cannotcon- of thetheoryof convexsets.Two theorems
vincea mathematician.
One must,however, are at the centerof that area of welfare
immediately
add thatthemathematical
tools economics.The firstassertsthatif all the
thatlatermadethesolutionof theexistence agentsof an economyare in equilibrium
problempossibledid notexistwhenWalras relativeto a givenprice-vector,
thestateof
wroteone of thegreatest
classics,if notthe theeconomyis Paretooptimal.Its proofis
greatest,of our science.It was Abraham one ofthesimplest
in mathematical
economfromGustavCassel's(1918) ics. The secondprovidesa deepereconomic
Wald,starting
formulation
of the Walrasianmodel,who insightand restson a property
of convex
in Viennain 1935-36 provided sets.It assertsthatassociatedwitha Pareto
eventually
the firstsolutionin a seriesof papersthat optimalstates of an economy,thereis a
attracted
so littleattention
thattheproblem price-vector
towhichall theagents
p relative
wasnotattackedagainuntiltheearly1950's. are in equilibrium
(underconditionsthat,
KennethArrowhas toldin hisNobellec- hereas elsewhere,
I cannotfullyspecify).
Its
ture(1974) aboutthepaththathe followed proofis basedon theobservation
thatin the
to thepointwhereitjoinedmine.The route commodity
space R', thea priorigivenen-
VOL. 74 NO. 3
269
DEBREU: ECONOMIC THEORY IN THE MA THEMA TICAL MODE
w2 (al()
t---
A1
2
2~~~~~~~~~~~~~~~
eX
I1
al
(a2
FIGuRE2
H
by JohnNash in his one-pagenoteof 1950
Games"
Pointsin N-Person
on "Equilibrium
and by MortonSlaterin his unpublished
e of the economyis a
dowment-vector
paper,also of1950,on Lagrangemultipliers.
pointofthesetE ofall theendow- Again therewas an ideal tool, this time
boundary
withwhichit is possibleto
ment-vectors
fortheproofthatI gave
Kakutani'stheorem,
at
of all consumers
in 1952 of the existenceof a social equisatisfythe preferences
Nash'sresult.Sincethe
leastas wellas in thestates. Undercondi- libriumgeneralizing
thattheset E is convex,there transposition
fromthecase of twoagentsto
tionsinsuring
we shall
H forE through thecase of n agentsis immediate,
hyperplane
is a supporting
whichlendsitselfto
to thehyperplane consider
e. A vectorp orthogonal
onlytheformer
Let thefirst
towardsE has all therequired a diagrammatic
representation.
H, pointing
of agentchoosean action a, in the a priori
(See Figure1.) The treatment
properties.
theproblemthusgivenbymeansofconvex- givensetA1,and thesecondagentchoosean
moregeneraland actiona2 in thea priorigivensetA2. Knowity theorywas rigorous,
by meansof the ing a2, the firstagenthas a set tt(a2) of
simplerthanthetreatment
knowingal,
calculusthathadbeentraditional equivalentreactions.Similarly,
differential
the- thesecondagenthas a sett2(aj) of equivahyperplane
sincePareto.Thesupporting
theo- lent reactions.(See Figure2.) tt(a2) and
theHahn-Banach
orem(moregenerally
rem,Debreu,1954a) seemedto fittheeco- 1L2(al) maybe one-element
sets,but in the
case of an economywithsome
Especiallyrelevant important
nomicproblemperfectly.
underconstantreturns
is thefactthattherestate- producersoperating
to mynarrative
in set-theoretical to scale, theywill not be. The state a=
economics
mentofwelfare
ifand onlyif a1 E
of severalof (a,, a2) is an equilibrium
termsforceda reexamination
of gen- A1(a2)and a2 E t2(aj), thatis,ifandonlyif
theprimitive
conceptsof thetheory
Thiswas ofgreat a E ,u(a) = Al(a2)X A2(aj).
eraleconomicequilibrium.
stateif
probIn otherwords,a is an equilibrium
valueforthesolutionof theexistence
and onlyif it is a fixedpointof thecorrelem.
In theyearI joinedtheCowlesCommis- spondencea |-+ ,u(a) fromA = A1 x A2 to A
sion, I learnedabout the Lemmain von itself.Conditionsinsuringthat Kakutani's
theory theoremappliesto A and y guaranteethe
Neumann'sarticleof 1937on growth
state. In our
in 1941 existenceof an equilibrium
thatShizuoKakutanireformulated
I alsolearnedabout articleof 1954,Arrowand I casta competias a fixedpointtheorem.
made tiveeconomyin theformof a socialsystem
theapplications
ofKakutani'stheorem
FIGURE1
270
THE AMERICANECONOMIC REVIEW
of the precedingtype.The agentsare the
consumers,
the producers,
and a fictitious
pricesetter.
An appropriate
definition
ofthe
set of reactionsof the price setterto an
excessdemandvectormakestheconceptof
equilibrium
forthatsocialsystem
equivalent
to theconceptofcompetitive
equilibrium
for
theoriginal
In thismanner
a proof
economy.
on Kakutani's
ofexistence,
resting
ultimately
of
theorem,
was obtainedforan equilibrium
an economymade up of interacting
consumersand producers.
In the early1950's,
comeforsolutions
thetimehad undoubtedly
of theexistence
problem.In additionto the
workofArrowandme,begunindependently
and completed
jointly,LionelMcKenzieat
Duke University
provedtheexistence
of an
in Graham'sModel of World
"Equilibrium
Trade and Other CompetitiveSystems"
(1954), also using Kakutani'stheorem.A
differentapproach taken independently
by David Gale (1955) in Copenhagen,
HukukaneNikaido (1956) in Tokyo,and
Debreu(1956)in Chicagopermitted
thesubstantialsimplification
givenin myTheory
of
Value(1959)of thecomplexproofofArrow
and Debreu.
Followingthat approachwe considera
p different
from0 in RQ , the
price-vector
closedpositiveorthant
of R'. The reactions
oftheconsumers
in the
and oftheproducers
to p yieldan excessdemandvector
economy
z in RI,whosecoordinates
represent
foreach
commodity
the (positive,zero,or negative)
excess of demandover supply.Since the
vectorz maynot be uniquelydetermined,
one is led to introduce
theset Z(p) of the
excessdemandvectorsassociatedwithp, a
if p is multiplied
setwhichis invariant
bya
strictly
positivereal number.If everycommodityin the economycan be freelydisif
posedof,p* is an equilibrium
price-vector
and onlyif thereis in Z(p*) a vectorall of
whosecoordinates
are negative
or zero,that
is, if and onlyif Z(p*) intersects
R', the
closednegativeorthant
of R'. The factthat
everyconsumersatisfieshis budgetconstraint
impliesthatall thepointsofZ( p) are
in or belowthehyperplane
through
theorigin of R' orthogonal
to p. (See Figure3.)
Additionalconditionson Z suggestedby
Kakutani'stheorem
establish
theexistence
of
an equilibrium
p*.
JUNE1984
p
/
CX
~~~~~~~~R+
0
-R_
FIGURE
3
a
is nowconsidered
A proofof existence
necessaryadjunctof a modelproposinga
and in a
conceptof economicequilibrium,
recentsurvey(Debreu,1982)morethan350
existenceproofsof
containing
publications
thattypewerelisted.One of themostcomplex amongthese,becauseof thegenerality
at whichit aimed,wasmyarticle(1962).
During the past threedecades,several
ofexistence
to theproblem
otherapproaches
a
havebeen developed.Withoutattempting
suchas thosepreparedfor
survey
systematic
(1981-84)byStephen
Arrowand Intriligator
Smale (ch. 8), by Debreu (ch. 15), by E.
Dierker(ch. 17), and by HerbertScarf(ch.
twoofthem
mention
21),onemustexplicitly
here.
Givenan arbitrary
positivepricestrictly
vectorp, we nowconsiderthecase in which
and of the
the reactionsof the consumers
a unique
determine
intheeconomy
producers
excessdemandvectorF( p). We also assume
ofeveryconsumer
thatthebudgetconstraint
Thenone has
is exactlysatisfied.
Walras'Law
p *F(p) = 0.
thattheprice-vector
Thisequalitysuggests
ittothestrictly
byrestricting
p be normalized
positivepartS of theunitspherein R', for
to p,
thenthevectorF( p), beingorthogonal
can be represented
as beingtangentto the
VOL. 74 NO. 3
DEBREU: ECONOMIC THEOR Y IN THE MA THEMA TICA L MODE
271
newaspectto thetheory
addedan important
ofgeneraleconomicequilibrium.
givenbya
ofequilibrium
The explanation
if
wouldbe complete
modelof theeconomy
wereunique,and thesearch
theequilibrium
for satisfactoryconditionsguaranteeing
uniquenesshas been activelypursued(an
F(p)
is foundinArrowandHahn,
survey
excellent
of the
1971,ch. 9). However,the strength
p
thatwereproposedmadeit clear
conditions
was
bythelate1960'sthatglobaluniqueness
and thatone
a requirement
too demanding
withlocaluniquewouldhaveto be satisfied
of an economy
thatproperty
ness.Actually,
evenunderstrong
could not be guaranteed
of the
aboutthecharacteristics
assumptions
economicagents.Butonecan prove,as I did
in
in 1970,that,undersuitableconditions,
economies
of
set
the
all
economies,
of
set
the
FIGURE 4
that do not have a set of locallyunique
of
The exactmeaning
is negligible.
equilibria
the termsI have just used and the basic
resulton whichthe proofof
sphereS at p. (See Figure4.) In mathemati- mathematical
restscan be foundin
assertion
thepreceding
cal terms,the excess demandfunctionF
StephenSmaleinto
which
representatheorem
Sard's
definesa vectorfieldon S. This
in thesummer
in
conversations
me
general
the
troduced
to
the
key
be
to
out
tionturned
partsof thesolution
of excessdemandfunctions of 1968.The different
characterization
Soundon theSouth
thatI willdiscusslater.It also providesan fellintoplaceat Milford
of
On
theafternoon
Zealand.
New
boundary
of
a
Island
existenceproofin the case of
and
Francoise
wife
my
terms
when
1969,
9,
in
July
economic
meaning
on
F,
condition
rain and overcast
thatexcessdemandbecomeslargewhensome I arrived,intermittent
me to
pricestend to zero, and in mathematical weatherthatdulledtheviewtempted
long
a
become
had
what
on
inward
more
once
work
termsthattheexcessdemandpoints
ideas
this
time,
and,
continuous
problem,
a
For
tantalizing
S.
of
boundary
near the
The next morninga
impliesthatthere quicklycrystallized.
vectorfield,thisproperty
is at least one point p* of S for which cloudlessskyrevealedtheSoundin itsmidF( p*) =O. This equalityof demandand wintersplendor.
to whichI alThe "suitableconditions"
expressesthat
supplyforeverycommodity
which,
conditions
luded are differentiability
price-vector.
p* is an equilibrium
unare essentially
The secondapproachconcernsthedevel- in thepresentsituation,
it
forthecom- avoidable.As forthe term"negligible,"
algorithms
opmentof efficient
set
a
finite-dimensional
of
case
area
in
the
an
means,
equilibria,
putationof approximate
"containedin a closedsetof
in whichScarf(1973)playedthe of economies,
of research
of Lebesguemeasurezero."The mainideas of
leadingrole. The searchforalgorithms
in the
intuitively
of theproofcan be conveyed
thatclassis a naturalpartoftheprogram
m
with
economy
an
exchange
Yet
of
case
simple
equilibrium.
studyof generaleconomic
The demandfunction
from consumers.
fi of the
cameunexpectedly
thedecisivestimulus
the solutionof a problemin gametheory, ith consumerassociateswith everypair
p
positiveprice-vector
whenC. E. Lemkeand J.T. Howson(1964) (p, wi) of a strictly
the
income)
wealth
(or
of
a
solution
and
positive
wi
the
for
providedan algorithm
games.The com- demand fi(p, wi) in the closed positive
non-zero-sum
two-person
space.The ith
putationof equilibriahas foundits wayin- orthantR ofthecommodity
by his demand
and has consumeris characterized
to a largenumberof applications
272
THE AMERICANECONOMIC REVIEW
JUNE1984
function
fi and by his endowment-vector
ei
in thestrictly
positiveorthantP of R'. The
functions
fiarekeptfixedandassumedtobe
continuously
differentiable.
Therefore,
the
economyis describedby the list e =
in
(el,..., em) of the m endowment-vectors
Pm. The price-vector
p beingrestricted
to
sCI
belongto S, thestrictly
positivepartof the
unitsphere,
theexcessdemandvectorassociated witha pair (p,e) in S x Pm is
T
m
F( p, e)=
E [fi( p, p-*ei)-ei] -
i=1
M (Smale,1974;
Theequilibrium
manifold
Balasko,1975) is the subsetof S x Pm definedby F(p, e) = O, an equalitywhich,because of Walras' Law, imposesonly I-I
constraints.
Under the assumptions
made,
M is a differentiable
manifoldand its
dimensionis dimM = dimPm + dimS -
e
FIGURE
pm
5
fromchronological
Departing
order,I now
returnto the late 1950's and to the early
1960's,and to thebeginning
ofthetheory
of
the coreof an economy.Edgeworth
(1881)
had givena persuasive
argument
in support
ofthecommonimprecise
beliefthatmarkets
(i-i) = im = dimPtm.Now let T be theproas thenumberof
jectionfromM intoPm,anddefinea critical becomemorecompetitive
in sucha waythateach
e as an economy
suchthatit is the theiragentsincreases
economy
of a point(e, p) of M wherethe one of themtendsto becomenegligible.
He
projection
shownthathis "contractJacobianof T is singular,geometrically had specifically
wherethe tangentlinearmanifoldof di- curve"tendsto thesetof competitive
equimensionIm does notprojectontoPm. (See
libria in a two-commodity
economywith
thesetofcriti- equal numbers
of consumers
Figure5.) By Sard'stheorem
of each one of
cal economiesis closed and of Lebesgue two types.His brilliant
stimucontribution
measurezero. A regulareconomy,outside lated no further
workuntilMartinShubik
the negligiblecriticalset, not only has a
(1959) linkedEdgeworth'scontractcurve
discrete
setofequilibria;it also has a neigh- withthegametheoretical
conceptofthecore
varies (D. B. Gillies,1953).The firstextension
borhoodin whichthesetofequilibria
of
as a function
oftheparameters Edgeworth's
resultwas obtainedby Scarf
continuously
to
theeconomy.The studyof regular (1962), and the completegeneralization
defining
number
ofcommodia basisfortheanalysis thecaseofan arbitrary
economies
thusforms
was givenby
ofthedeterminateness
ofequilibrium
and of tiesand of typesof consumers
thestability
ofeconomicsystems.
Moreover, Debreu and Scarf(1963). Associatedwith
our joint paper is one of my mostvivid
the continuity
of the set of equilibriain a
of theinstantwhena problemis
of a regulareconomyinsures memories
neighborhood
had metme
thattheexplanation
ofequilibrium
provided solved.Scarf,thenat Stanford,
by themodelis robustwithrespectto un- at the San Franciscoairportin December
to Palo Altoon
of the 1961,and as he was driving
avoidableerrorsin themeasurement
one ofus,in one sentence,
Once again,a mathematical
re- thefreeway,
proparameters.
was foundto fitexactly videda keyto thesolution;theother,also in
sult,Sard'stheorem,
theother
immediately
provided
theneedsof economictheory.
The studyof onesentence,
regulareconomieshas been an activere- key;and thelockclickedopen.Once again,
resultwas the supsearcharea in the last decade,and Smale, the basic mathematical
theorem
forconvexsets.
hyperplane
Balasko,and AndreuMas-Colell(1984) are porting
The theorem
thatwe had provedremained
amongitsmaincontributors.
VOL. 74 NO. 3
DEBREU: ECONOMIC THEOR Y IN THE MA THEMA TICA L MODE
273
special,becauseit appliedonlyto economies
Intimately
linkedwiththe contemporary
witha givennumberof typesof consumers development
of the theoryof generalecowas thatof thetheory
and an equal, increasingnumberof con- nomicequilibrium
of
were preferences,
utility,
anddemand.Newresults
sumersof each type.Generalizations
in
soon forthcoming.
RobertAumann(1964) in thelatterwerein somecasesrequired,
introduced
theconceptof an atomlessmea- othersmotivated
by theformer.
The primiofpreferences
of
sure space of economicagents,a natural tiveconceptsin thetheory
of theconceptof a consumerare his consumption
set X, a
mathematical
formulation
relation<,
an economywitha largenumberof agents, subsetof R', and hispreference
preorder
on X. We shallsaythat
all of themnegligible.
Undernotablyweak a complete
Aumannprovedthatforsuchan
a real-valued
functionu on X is a utility
conditions,
if it represents
thepreference
relaeconomythecorecoincideswiththeset of function
KarlVind(1964)then tion < in thesensethat
competitive
equilibria.
pointedout thatthe propermathematical
_
resultwas
toolfortheproofof thatstriking
Ix < y] -r*[u(x)
u(y)].
on theconvexity,
Lyapunov's
(1940)theorem
andcompactness,
oftherangeofan atomless
A necessaryand sufficient
conditionfor
vectormeasure.Out of theexistence
finite-dimensional
function
ofa continuous
utility
litera- is thatthe set G={(x,y)eXXXlxxy}
thesecontributions
grewan extensive
ture that includedamongits highpoints be closedrelativeto X x X (Debreu,1954b;
YakarKannai's(1970)andTrumanBewley's 1964). Althoughmore abstractthan the
(1973) articles,and that culminatedin familiar
conceptof an infinite
familyof inWerner
Hildenbrand's
book(1974).Thiswas difference
setsin R', theconceptof a single
in Arrowand Intriligator set G in R' x R' is farsimpleras twomore
surveyed
recently
instances
illustrate.
(1982)byHildenbrand
(ch. 18).
In a different
direction,
a formalization
of
simiTo saythatan agenthas preferences
an economywitha largenumberof negligi- lar to thatof anothermeansfora matheble agentswas proposedby Donald Brown maticaleconomist
thata topologyhas been
and AbrahamRobinson(1972), who in- introduced
This
on the set of preferences.
troducedthe sophisticatedtechniquesof was done by Kannai (1970), in an article
Nonstandard
Analysisin economictheory. whose publicationwas long delayed.The
this approacheventually
Remarkably,
led prospectof comparing
twopreference
relato the elementary
inequalitiesof Robert tions < and <' on the two consumption
Anderson(1978) on theextentof competi- setsX and X' (nowassumedto be closed)is
tiveness
in thecorein an econ- dauntingif one thinksof each preference
ofallocations
number
ofagents.
omywitha finite
relationas an infinite
familyof indifference
In themid-1970's,
thetheoryof thecore setsin R'. It becomesappealingifonethinks
and the theoryof regulareconomieswere ofeachpreference
relation
as a closedsubset
joinedin thestudyoftherateofconvergence of R' x R' (Debreu,1969).The topologyon
of thecore to the set of competitive
was at thebasisof the
equi- thesetofpreferences
libria. Lloyd Shapley (1975) had shown theory
of thecorein Hildenbrand
(1974).It
fortheworkthatKannai
thatconvergence
slow. was indispensable
could be arbitrarily
Debreu(1975) thenprovedthatin thecase (1974) and Mas-Colell(1974) did on the
of increasingequal numbersof agentsof approximation
of a convexpreference
relarelationsrepreeach of a finite
numberof types,therateof tion by convexpreference
to theset of competitive
functions.
convergence
equi- sentablebyconcaveutility
libriaof a regulareconomyis of the same
The otherinstancepertainsto preference
utilorderas the reciprocalof the numberof relations
representable
by differentiable
The traditional
agents.The extensionof this resultfrom ityfunctions.
approach,by
replicatedeconomiesto more generalse- focusingon the consumption
set X in R,
wasprovided
questions
(extenquencesofeconomies
byBirgit raiseddelicateintegrability
Grodal(1975).
sivelysurveyed
byLeonidHurwicz,
1971,ch.
274
THE AMERICAN ECONOMIC REVIEW
9). In contrast,a differentiable
preference
relationS can simplybe definedby the
conditionthatthe boundaryof the associated set G is a differentiable
manifoldin
R' X R' (Debreu,1972).
In all thesedevelopments,
the theoryof
preferences
was stimulated
and helpedby
questionsaskedabouttheutility
function
u
suchas "Whenis u continuous?,"
"Whenis
u concave?,"
"Whenis u differentiable?"
Yet
anotherinstanceis providedby thestudyof
a preference
relation< defined
on theproduct X of n setsX1,...., Xn.Thequestionnow
is whether
thepreference
relation< can be
represented
by a utility
function
oftheform
JUNE 1984
Under weak standardassumptions,
the
function
F (1) is continuous
and (2) satisfies
Walras' Law. Hugo Sonnenschein
(1972,
1973) asked whetherthesetwo properties
characterize
F. Specifically,
givenF satisfying (1) and (2), can one findm consumers
withdemandfunctions
fi and endowmentvectorsei satisfying
(a)? Sonnenschein
conjecturedthattheanswerwas affirmative
and
made thefirstattackon thisproblem.Rolf
Mantel(1974)provedSonnenschein's
conjecturein thecaseofcontinuously
differentiable
demandfunctions,
and Debreu(1974)in the
generalcase.Theproofappearing
in thislast
articlewas inspiredby, and restson, the
representation
oftheexcessdemandfunction
n
F as a vector-field
on the strictly
positive
u(X)=
partof theunitsphere.The characterization
Ui(xi)
i =1
of aggregate
excessdemandfunctions
so obtainedhas severalapplications.
It showsthat
wherex is the n-list(xI,..., xJ) and for thehypothesis
of preference
satisfaction
(or
everyi, xi E Xi. This problemwas studied equivalently
ofutility
maximization)
putsesno restriction
on F, thata theorem
by Leontief(1947a,b) and by Samuelson sentially
(1947, ch. 7), by meansof the differential on theexistence
of a generaleconomicequicalculus.It can be studiedby topological librium
is equivalent
toa fixedpointtheorem
methods(Debreu, 1960) whichbringout (via an observationof HirofumiUzawa,
more clearly the essentialindependence 1962),and thatanydynamic
behavior
canbe
property
on whichthesolutionis based.
observedforan economyoperating
undera
process(as the examplesof
ofpref- t'atonnement
The lastexamplefromthetheory
of Scarf,1960,presaged).
anddemandwillbe theprob- globalinstability
erences,
utility,
of theexcessde- One impactofthatcharacterization
hasbeen
lemof thecharacterization
of researchon aggregate
of an economy.
We consider theredirection
mandfunction
detowarda specification
an exchangeeconomy4 withm consumers. mandfunctions
ofthe
As before,
thedemandfunction
of thecharacteristics
of theecofiof theith distribution
nomicagents.The first
consumer
associateswitha pair(p, wi) of a
theoretical
resultexplaining the "Law of Demand" (Hildenprice-vector
p in thestrictly
positivepartS
of theunitspherein R' and of a wealth(or brand,1983)wasa product
ofthatredirected
real research.
income)wiin thesetR+ ofnonnegative
vectorf1(p,w) in
a consumption
numbers,
R' of R'. If the
theclosedpositiveorthant
II
relation5 i on
ithconsumer
hasa preference
that
in somedetail,as tradiHavingsurveyed
R+, thenfi(p,
wi) is a commodity-vector
tionrequires,theworkcitedby the Royal
satisfies< underthebudgetconstraint
p z
< wi.Theeconomy
6'is defined
byspecifying SwedishAcademyof Sciences,I turnto isin economictheory.
forthe ith consumer(i=1,...,m) the de- suesofmethodology
in thetheory
mandfunction
f1and theendowment-vector Contemporary
developments
in RI. The aggregate
excessdemandfunc- ofgeneral
economic
tookWalras'
equilibrium
F defined workas theirpointofdeparture,
is thefunction
tionoftheeconomy
butsomeof
Walras' ideas had a long lineagethatinby
cluded Adam Smith's(1776) profoundinm
sight.Smith'sidea thatthemanyagentsof
(a)
F( p) = E? [ ( p,pei)-ei
an economy,
makingindependent
decisions,
i =1
VOL. 74 NO. 3
DEBREU: ECONOMIC THEORYIN THE MATHEMATICALMODE
275
theory
of an effective
do notcreateutterchaos but actuallycon- othermajorattributes
Again,their
and generality.
tribute
toproducing
a socialoptimum,
raises are simplicity
to makethemdeindeeda scientific
questionofcentralimpor- aestheticappeal suffices
of
forthedesigner
to answerit havestimulated sirableendsin themselves
tance.Attempts
the studyof severalof the problemsthat a theory.But theirvalue to the scientific
Simgoesfarbeyondaesthetics.
community
everyeconomicsystemmustsolve,suchas
theefficiency
of resourceallocation,
thede- plicitymakes a theoryusable by a great
of numberof researchworkers.Generality
centralization
of decisions,theincentives
decisionmakers,the treatment
of informa- makesit applicableto a broadclassofproblems.
tion.
theaxiomatization
manner,
In thepastfewdecades,thatwiderangeof
In yetanother
problems
hasbeenthesubjectofan axiomatic of economictheoryhas helped its pracby makingavailableto themthe
analysisin whichprimitive
conceptsarecho- titioners
languageof mathematics.
themare for- superblyefficient
sen, assumptions
concerning
with
themto communicate
are derivedfrom It has permitted
mulated,and conclusions
witha greateconthoseassumptions
bymeansofmathematical each other,and to think,
fromany intended omyof means.At the same time,the diareasoningdisconnected
interpretation
of theprimitive
concepts.
The logue betweeneconomistsand mathematiThe example
benefits
of the axiomatization
of economic cianshasbecomemoreintense.
of the firstmagnitude
of a mathematician
havebeennumerous.
theory
Makingtheasof a theoryentirely
sumptions
explicitper- like Johnvon Neumanndevotinga signifimitsa sounder
judgment
abouttheextentto cant fractionof his researchto economic
whichit applies to a particularsituation. problemshas not been unique. SimultaAxiomatization
mayalso givereadyanswers neously,economictheoryhas begunto inAmongtheclearestinto newquestions
whena novelinterpretation fluence
mathematics.
of
thetheory
of primitive
As an
conceptsis discovered.
stancesareKakutani'stheorem,
(Hildenbrand,
ofcorrespondences
illustration,
considertheconceptof a com- integration
ofapforthecomputation
modity,
whichhadmeanttraditionally
a good 1974),algorithms
or a servicewhosephysicalproperties
and
proximatefixedpoints(Scarf'sch. 21 in
1981-84),and ofapwhose deliverydate and location are
Arrowand Intriligator,
of equations
solutionsof systems
In the case of an uncertainen- proximate
specified.
vironment,
Arrow(1953) added to those (Smale's ch. 8 in Arrowand Intriligator,
1981).
characteristics
of a commodity
theeventin
whichdelivery
willtakeplace.In thismanner
III
one obtains,without
anychangein theform
of the model,a theoryof uncertainty
in
scientists
try
of theircareers,
In narratives
whichall the resultsof the theoryof certo which
themaininfluences
to acknowledge
taintyare available(Debreu,1959,ch. 7).
on mathemati- theyresponded,and the supporttheyreAxiomatization,
by insisting
and fromdifferto a
ceivedfromotherscientists
has repeatedly
led economists
cal rigor,
eventhoughsuchattempts
of theproblemsthey ent institutions,
deeperunderstanding
To all
successful.
to be entirely
werestudying,
and to theuse of mathemati- are unlikely
I havenamed,
cal techniques
thatfitted
thoseproblems
bet- thepersonsand organizations
education
ter.It hasestablished
which I want to add the outstanding
securebasesfrom
It
couldstartin newdirections.
systemI have knownin France,and the
exploration
Scientifique
CentreNationalde la Recherche
has freedresearchers
fromthenecessity
of
whichmade my conversionfrommathein
theworkoftheirpredecessors
questioning
possible.Aftermymove
maticstoeconomics
fulfills
an
everydetail.Rigorundoubtedly
to theUnitedStatesin 1950,I wasassociated
intellectual
needofmanycontemporary
eco(Chicago,Yale,
withthreegreatuniversities
nomictheorists,
whotherefore
seekit forits
researchis a
wherescientific
and Berkeley)
own sake,but it is also an attribute
of a
naturalwayof life;and duringthelast two
thatis an effective
tool.Two
theory
thinking
276
THE AMERICANECONOMIC REVIEW
JUNE1984
ofSciencesof theU.S.A.,
tionalAcademy
1972,69, 1258-60.
Sozialokonomie,
Cassel,K. G., Theoretische
1918.
Leipzig:C. F. Winter,
Cournot,
A., Recherchessur les Principes
de la Theorie
desRichesses,
Mathematiques
1838.
Paris:L. Hachette,
Dantzig,
G. B., "Maximizationof a Linear
REFERENCES
Functionof VariablesSubjectto Linear
in T. C. Koopmans,ed., Acd'uneDiscipline
Inequalities,"
Allais,M., A la Recherche
Nationale,
Paris:Impnmerie
andAllocation,
Economique,
Analysis
ofProduction
tivity
1943.
New York:Wiley& Sons,1951,339-47.
CoreEquiv- Debreu,G., "The Coefficient
of Resource
R. M.,"An Elementary
Anderson,
NovemJuly1951,19,
Econometrica,
Utilization,"
alenceTheorem,"Econometrica,
273-92.
ber1978,46, 1483-87.
"A Social EquilibriumExistence
,_
K. J., "An Extensionof the Basic
Arrow,
of the National
Theorem,"Proceedings
Theorems of Classical Welfare EcoofSciences,1952,38, 886-93.
of
Academy
in J.Neyman,
ed., Proceedings
nomics,"
and
the Second Berkeley Symposiumon
, (1954a)"ValuationEquilibrium
oftheNaProceedings
Statisticsand Probability, ParetoOptimum,"
Mathematical
tionalAcademyof Sciences,1954, 40,
of CaliforniaPress,
Berkeley:University
588-92.
1951,507-32.
ofa Prefer, (1954b)"Representation
"Le Role des Valeurs Boursieres
enceOrdering
Function,"
by a Numerical
pour la Repartitionla Meilleuredes
in R. M. Thrall et al., eds., Decision
Paris:CentreNaRisques,"Econometrie,
NewYork:Wiley& Sons,1954,
1953,
Scientifique,
Processes,
tionalde la Recherche
159-65.
41-48.
of theNadecadestheEconomicsProgram
tional Science Foundationhas givenme,
else, timeforthatremorethananything
a
haveprovided
search.All thoseinstitutions
forthetaskthathad to
superbenvironment
be performed.
_
__
, "General Economic Equilibrium:
Purpose,AnalyticTechniques,Collective
June
Review,
Economic
Choice,"American
1974,64, 253-72.
and Debreu,G., "Existence of an
Economy,"
fora Competitive
Equilibrium
Econometrica,
July1954,22, 265-90.
and Hahn,F. H., GeneralCompetitive
Analysis,San Francisco:Holden Day,
1971.
M. D., Handbookof
and Intriligator,
Vols. I, II, III,
Economics,
Mathematical
1981-84.
North-Holland,
Amsterdam:
of
R.J.,"Marketswitha Continuum
Aumann,
January-April,
Traders," Econometrica,
1964,32, 39-50.
Balasko,Y., "On the Graphof the Walras
SeptemEconometrica,
Correspondence,"
ber-November
1975,43,907-12.
Bewley,T. F., "Edgeworth's Conjecture,"
1973,
September-November
Econometrica,
41, 425-54.
D. J. andRobinson,
A.,"A LimitTheoBrown,
remon theCoresof LargeStandardExoftheNaProceedings
changeEconomies,"
,___.'"Market
Equilibrium,"Proceedings
ofSciences,1956,
oftheNationalAcademy
42, 876-78.
Theoryof Value, An Axiomatic
New
Analysisof EconomicEquilibrium,
York:Wiley& Sons,1959.
, "TopologicalMethodsin Cardinal
UtilityTheory,"in: K. J. Arrowet al.,
Methodsin theSocial
eds., Mathematical
Sciences,1959, Stanford:StanfordUniPress,1960,16-26.
versity
_
, "New Conceptsand Techniquesfor
EcoAnalysis,"International
Equilibrium
nomicReview,
1962,3, 257-73.
September
of Paretian
,"ContinuityProperties
EconomicReview,
Utility,"International
1964,5, 285-93.
September
in
EconomicAgents,"
, "Neighboring
du
La Decision,ColloquesInternationaux
CentreNationalde la RechercheScienNo. 171,Paris,1969,85-90.
tifique
,_
"Economies with a Finite Set of
May 1970,38,
Equilibria,"Econometrica,
387-92.
___
"Smooth Preferences," Econo-
VOL. 74 NO. 3
DEBREU: ECONOMIC THEOR Y IN THE MA THEMA TICA L MODE
metrica,
July1972,40, 603-15; "Smooth
Preferences:A Corrigendum,"
Econometrica,
July1976,44, 831-32.
,__
.
"Excess
Demand Functions,"Jour-
nal of Mathematical
Economics,
1974, 1,
15-21.
__
"The Rate of Convergenceof the
Core of an Economy,"Journal
ofMathematicalEconomics,
1975,2, 1-7.
, "RegularDifferentiable
Economies,"
AmericanEconomicReviewProceedings,
May 1976,66, 280-87.
____
"Existence of CompetitiveEqui-
in K. J.Arrowand M. D. Intrililibrium,"
gator,eds., Handbookof Mathematical
Economics,Vol. II, Amsterdam:
NorthHolland,1982,ch. 15.
and Scarf,H., "A Limit Theoremon
the Core of an Economy,"International
EconomicReview,September1963, 4,
235-46.
and Koopmans,T. C., "Additively
Decomposed Quasiconvex Functions,"
MathematicalProgramming,
1982, 24,
1-38.
Dierker,
E., "RegularEconomies,"in K. J.
Arrow and M. D. Intriligator,
eds.,
Handbook
ofMathematical
Economics,
Vol.
II, Amsterdam:
North-Holland,
1982,ch.
17.
Paris:
Divisia,F., EconomiqueRationnelle,
Doin,1928.
F. Y., MathematicalPsychics,
Edgeworth,
London:KeganPaul,1881.
Gale,D.,"The Law ofSupplyand Demand,"
Mathematica
Scandinavica,
1955, 3, 15569.
D. B., "Some Theoremson n-Person
Giflies,
Games," unpublisheddoctoraldisserta1953.
tion,Princeton
University,
of the
Grodal,
B.,"The Rate of Convergence
Corefora PurelyCompetitive
Sequenceof
EcoJournal
Economies,"
ofMathematical
nomics,
1975,2, 171-86.
Hildenbrand,
W.,CoreandEquilibria
ofa Large
Princeton:
Princeton
Economy,
University
Press,1974.
9 "Core of an Economy," in K. J.
Arrow and M. D. Intriligator,
eds.,
Vol.
Handbook
ofMathematical
Economics,
II, Amsterdam:
North-Holland,
1982,ch.
18.
,_
"On
the 'Law
277
of Demand',"
July1983,51,997-1019.
Econometrica,
ofIntegrability
Hurwicz,
L.,"On theProblem
in J. S. Chipman
of DemandFunctions,"
and DeUtility,
et al., eds., Preferences,
mand, New York: Harcourt Brace
1971,174-214.
Jovanovich,
of Brouwer's
Kakutani,
S., "A Generalization
Duke MathematiFixed PointTheorem,"
1941,8, 457-59.
cal Journal,
Y.,"Continuity
oftheCore
Properties
Kannai,
November
of a Market,"Econometrica,
1970,38, 791-815.
of Convex Pref"Approximation
Ecoerences,"Journalof Mathematical
1974,1, 101-06.
nomics,
ofProducT. C., Activity
Analysis
Koopmans,
tionand Allocation,
New York: Wiley&
Sons,1951.
ofWelfare
EcoLange,O.,"The Foundations
nomics," Econometrica,July-October
1942,10, 215-28.
Lemke,C. E. and Howson,J. T., Jr.,"Equi-
libriumPointsof BimatnxGames,"JourandApplied
nal oftheSocietyofIndustrial
1964,12,413-23.
Mathematics,
W. W., TheStructure
oftheAmerican
Leontief,
Economy,1919-1929, Cambridge:HarvardUniversity
Press,1941.
, (1947a) "A Note on the InterrelaVariables
tionof Subsetsof Independent
of a ContinuousFunctionwithContinuous First Derivatives,"Bulletinof the
Mathematical
Society,
1947,53,
American
343-50.
to a Theory
, (1947b)"Introduction
of the InternalStructure
of Functional
Relationships,"Econometrica,October
1974,15,361-73.
A. A., "On Completely
Additive
Lyapunov,
Vector-Functions,"
Izvestija Akademli
NaukSSSR, 1940,4, 465-78.
McKenzie, L. W., "On
Equilibrium in
Graham'sModel of World Trade and
Other CompetitiveSystems,"Econometrica,
April1954,22, 147-61.
of AgMantel,
R.,"On theCharacterization
ofEcogregateExcessDemand,"Journal
nomicTheory,
March1974,7, 348-53.
Mas-Colell,A., "Continuous and Smooth
Consumers:Approximation
Theorems,"
Journal
July1974,8,
ofEconomicTheory,
THE AMERICANECONOMIC REVIEW
278
JUNE1984
Shubik,M., " EdgeworthMarketGames,"
ofGames,Vol.
to theTheory
Contributions
Studies,40,
Mathematical
of
Annals
IV,
Approach,
Equilibrium;a Differentiable
Press,1959.
University
Princeton
Press,
Cambridge:CambridgeUniversity
Revisited,"
M.,"LagrangeMultipliers
Slater,
1984.
Cowles CommissionDiscussion Paper,
Pointsin N-Person
Nash,J.F.,"Equilibrium
403,1950.
Mathematics
Games," Proceedingsof the National
ofSciencesof theU.S.A., 1950, Smale,S., "Global Analysisand Economics,"
Academy
Economics,
IIA, Journalof Mathematical
36, 48-49.
1974,1, 1-14.
J. von,"Uber ein Okonomisches
Neumann,
, "Global Analysisand Economics,"
und eine VerallgeGleichungssystem
eds.,
in K. J.Arrowand M. D. Intriligator,
Fixpunktdes Brouwerschen
meinerung
Vol.
Economics,
ofMathematical
einesMathematischen Handbook
satzes,"Ergebnisse
1981,ch.8.
North-Holland,
I, Amsterdam:
1937,8, 73-83.
Kolloquiums,
A., An Inquiryintothe Natureand
Smith,
O., Theoryof Games
andMorgenstern,
Causesof the Wealthof Nations,2 vols.,
PrincePrinceton:
Behavior,
andEconomic
London: W. P. Strahanand T. Cadell,
Press,1944.
tonUniversity
1776.
Nikaido,
H., "On the ClassicalMultilateral
H., " MarketExcess Demand
Exchange Problem," Metroeconomica, Sonnenschein,
May 1972,40,
Econometrica,
Functions,"
August1956,8, 135-45.
549-63.
Pareto,V., Manuel d'EconomiePolitique,
, "Do Walras' Identityand ContinuParis:Giard,1909.
theClass of Community
ityCharacterize
of Economic
P. A., Foundations
Samuelson,
Journalof
Cambridge:HarvardUniversity Excess Demand Functions?,"
Analysis,
August1973,6, 345-54.
Theory,
Economic
Press,1947.
J.H. von,Der IsolierteStaat,HamSard,A.,"The MeasureoftheCriticalPoints Thunen,
1826.
burg:F. Perthes,
Maps," Bulletinof the
of Differentiable
Theoremand
1942,48,
Society,
Uzawa,H., "Walras'Existence
American
Mathematical
Brouwer'sFixed Point Theorem,"Eco883-90.
1962,8, 59-62.
nomicStudiesQuarterly,
Scarf,H., "Some Examplesof Global Instain an ExInter- Vind,K., "Edgeworth-allocations
Equilibrium,"
bilityof Competitive
In1960,
Economic
Review,
September
changeEconomywithManyTraders,"
national
May1964,5,
Economic
Review,
ternational
1, 157-72.
305-36.
,_
The Theoryof General Economic
, "An Analysis of Markets with a
165-77.
Recent Wald,A., "Uber die EindeutigePositive
Large Numberof Participants,"
The Princeton
Losbarkeitder Neuen ProduktionsgleiAdvancesin GameTheory,
einesMathematischen
1962.
Ergebnisse
Conference,
chungen,"
University
1935,6, 12-20.
Kolloquiums,
9 (with the collaborationof T.
of Economic
Hansen), The Computation
, (1936a) "Uber die ProduktionsWertlehre,"
derOkonomischen
Equilibria,New Haven: Yale University gleichungen
KolloErgebnisseeines Mathematischen
Press,1973.
1936,7, 1-6.
quiums,
, "The Computationof Equilibrium
in K. J. Arrow
Prices:An Exposition,"
, (1936b) "Uber einigeGleichungsOkonomie,"
derMathematischen
eds., Handbookof
Systeme
and M. D. Intriligator,
1936, 7,
Zeitschrift
furNationalokonomie,
Vol. II, AmsterMathematical
Economics,
637-70.
1982,ch. 21.
dam:North-Holland,
L. S., "An Exampleof a Slow-con- Walras,L., Elementsd'EconomiePolitique
Shapley,
Pure,Lausanne:L. Corbazand Company,
ReEconomic
Core,"International
verging
1874-77.
view,June1975,16, 345-51.
_