JOURNALOF
Mdhematical
ELSEVIER
Journal of Mathematical Economics26 (1996) 133- 170
Incomplete markets over an infinite horizon:
Long-lived securities and speculative bub
Michael Magi11ap*, artine Quinzii b
a Department of Economics, Univemty of Southern California, Los Angeles, CA 9W8%0253, USA
b University of Cal$Qrnia, Davis, CA 95616, USA
Submitted September 1993; accepted August 1994
Abstract
This paperstudiessequence economies over an infinite horizon with general security
structures.Assumptionsaregiven underwhicha pseudo-equilibrium
existsfor all economies
andanequilibriumexistsfor a densesetof (appropriatelyparameterized)
economies.Under
theseassumptionstheindebtedness
of the agentsin equilibriumcan be limited eitherby an
explicit boundon theirdebtsor by a transversalityconditionlimiting the asymptoticgrowth
of their debts.The qualitativepropertiesof equilibriumpricesof infinite-lived securitiesare
studied:the prices of infinite-lived securitiesin zero net supply are shown to permit
speculativebubblesandthe existenceof bubblescan affect the equilibriumallocation.The
pricesof securitiesin positive supply (equitycontracts)cannothavespeculativebubbles:
the extentof speculationin this classof modelis thus severelylimited.
JEL classijkation: D52
Keywords: Speculative
bubbles; Incomplete markets; Infinite-lived securities
1. Introduction
Sequence
economiesover an infinite horizon form the basic framework for
modemtheoreticalmacroeconomics.
The modelsare of two kinds: in the first,
agents(families) are finitely lived and are succeededby their children in an
* Corresponding author.
Elsevier ScienceS.A.
SSDI 0304-4068(95)00746-6
134
M. Magill,
M. Quinzii /Journal
of Mathematical
Economics 26 (1996) 133- 170
infinite sequenceof overlappinggenerations(OLG); in the second,there are a
finite numberof families(dynasties)that are infinitely lived. The latter model,
which is the focus of this paper,has mainly beenexploredin the representative
agentcase:while this simplifiesthe analysisit alsolimits the insightsthat canbe
obtained;in a representative
agentmodel of an exchangeeconomy,the equilibrium allocationis givenby the initial resourcesandno changein the structureof
the marketsor in the monetaryenvironmentcan have real effects. It is for this
reasonthat the generalequilibriumanalysisof sequenceeconomieswith incompletemarketshasbeenthe subjectof muchrecentresearch- this classof models
beingreferredto more briefly as general equilibrium with incomplete markets
(GEI). I The analysisof the GE1 model, which promisesto provide a rich
frameworkfor the analysisof problemsin macroeconomics,
has,however,so far
beenrestrictedto the somewhatad hoc settingof a finite horizon.
In a companionpaper(Magi11andQuinzii, 1994)we showedhow the concept
of a GEI equilibriumcan be extendedto an infinite horizoneconomy.The new
elementthat rlecdsto be incorporatedinto the definition of an equilibriumis a
deviceto preventagentsfrom indefinitelypostponingthe repaymentof their debts
(so-calledPonzi schemes).We showedhow the two traditionalapproachesfor
deterministiceconomies
basedeitheron borrowing constraints or a transversality
condition can be extendedto a model with incompletemarkets.An equilibrium
with debt constraintsis a simpleand intuitively appeaiingdescriptiveconceptof
equilibrium:an equilibriumwith a transversalityconditionis, in principle,more
generalandis morenaturalfor theoretical(mathematical)analysis;however,some
economists
may find it hardto acceptasa positiveconceptof equilibriumsinceit
callsfor substantialrationalityon the part of agents.Under the assumption(8.4)
that agentshave a uniform lower bound on their degreeof impatienceat each
date-event- definedastheproportionof futureconsumptionthat anagentis ready
to give up in orderto haveonemoreunit of consumption(of the numerairegood)
at that date-event- thesetwo typesof equilibriawere shownto coincide.Thus,
equilibriawith transversalityconditions,which are more convenientfor shxiying
existenceand qualitativeproperties,can be usedto study the descriptivelymore
satisfactoryconceptof equilibriumwith debtconstraints.
The Magi11andQuinzii(1994)paperis restrictedto the simplestsettingof an
economywith short-livedsecuritiespayingdividendsin a numerairegood. The
objectof the presentpaperis to extendthe analysisto genera]securitystructures
with the emphasison the caseof infinite-livedsecurities.The first task is to find
conditionsunderwhich an equilibriumexists:even in the caseof finite hofizon
economiesit is well known since Hart’s (1975) paper that discontinuitiesin
’ See the Special Issue of the Journal of Mathematical Economics on General Equilibrium with
Incomplete Markets (vol. 19, no. I, 199Oi and the survey articles of Geanakoplos (1990) and Magill
and Shafer (1991).
M. Mugill, M. Quinzii/lournul
of Mathematicul Economics 26 (1996) 133-170
135
agents’demandscreatedby changesin the rank of the returnsmatricescanleadto
non-existenceof equilibrium.The methodused in the recentGEI literatureto
circumventth% problemconsistsof introducingthe concept of a pseudo-equilibrium: suchan equilibriumis shownto exist for all economies
andfor a generic
set of economiesevery pseudo-equilibrium
is shown to be an equilibrium (see
Duffie and Shafer,1985,1986).To extendthis approachto the infinite horizon
casewe adopta simplerand more intuitive approachto representa pseudo-equilibrium of an economywith long-livedsecuritiesthat amountsto adjoiningto the
economya family of potentially equivalentshort-livednumeraireassets.This
enablesus to draw on the argumentsin the earlier paper(Magi11and Quinzii,
1994)to establishthe existenceof a pseudo-equilibrium,
the proof beingobtained
by taking limits of pseudo-equilibria
of truncatedeconomies.It is here that the
techniquesintroducedby Bewley (1972) for establishingthe existenceof an
Arrow-Debreu equilibriumover an infinite horizon turn out to be most fruitf~tl.
For the price of an infinite-lived security(when it is priced at its fundamental
value)is the discountedvalueof its future dividendstream- an expressionwith
aninfinite numberof terms.The problemof convergence
is solvedby viewingthe
presentvaluepricesaselementsof thenorm dualof /% (the spaceba of bounded
finitely additive set functions), the convergencebeing taken in t%eweak star
topology. Mackey continuity of agents’preferenceorderingsis then used to
establishthat the limit pricesare summable.
The existenceof an equilibriumfor a denseset of securitypayoffsis obtained
by showing that whenevera pseudo-equilibrium
offers a larger subspaceof
income transfersthan the original securities,the commodity payoffs can be
perturbedso as to makethesesubspaces
coincide.This resultis weakerthan the
existenceresult for finite horizon economies,which establishes
existencefor a
genericset of economies.However, we have not found a way to use Smale’s
versionof Sard’stheoremfor infinite dimensionalspacesto establishexistencefor
a genericset of economies:this must be the subjectof laterresearch.
Since the equilibriaexhibitedin the existenceproof are obtainedby taking
limits of equilibria(pseudo-equilibria)
of truncatedeconomies,they have the
propertythat all securitiesare priced at &eir fundamentalvalues.It is naturalto
enquirewhetherthis propertyholds for all possibleequilibria:sinceno terminal
condition can automaticallybe attachedto the systemof stochasticdifference
equationsthat must be satisfiedby a securityprice (the first-orderconditionsfor
the portfolio choiceof eachagent), it is not a priori clear that the price of an
infinite-livedsecuritywill equalits fundamentalvalue.
The phenomenon
of speculative bubbles hasbeenthe subjectof much interest
in macroeconomics
(see Blanchardand Fischer, 1989,ch. 5). It is sometimes
arguedthat bubblescannotarisein an economywith a finite numberof infinitelived agents:in Section6 we showthat this statementneedsto be qualified.As in
Tirole (1982) and Santosand Woodford(1992) we find that there cannotbe a
speculativebubbleon infinite-livedsecuritiesin positive net supply:sinceequity
136
M. Magill, M. Q&ii/
Journui of Mathematical Economics 26 (1996) 133-I 70
contractsconstitutea significantsegmentof thecapitalmarket,this placesa bound
on theextentto which themodelpredictstheoccurrenceol”speculation.However,
the pricesof infinite-livedsecuritiesin zero net supplybehavequite differently:
they admitsubstantialamountsof speculation.
We show that a speculativebubble
can alwaysbe addedto the price of such a securitywithout affectingthe real
equilibriumallocation.Thereis thus a significantnominalindeterminacyin the
pricesof infinite-livedsecuritiesin zeronetsupply.However,thereis a qualitative
differencebetweenspeculative
bubblesthat canarisein an equilibriumdepending
on whetherthe marketsare completeor incomplete.If the marketsare complete,
then speculativebubblesonly introducea nominalindeterminacy;if the markets
are incomplete,then a speculative
bubblecanhavea real effect in the sensethat
the sameequilibriumallocationcannotbe supportedby a systemof pricessuch
that eachsecurityis pricedat its fundamentalvalue.
2. The infinite horizon economy
We consideran economywith time anduncertaintyover an infinite horizon.
Let T = (0, 1,. *. } denotethe set of time periodsand let S be a set of statesof
nature.Therevelationof informationis described
by a sequence
of partitionsof S,
lF=(lF,, IFI,..., IF,,... ), wherethe numberof subsetsin ff, is finite andff, is finer
than the partition ff,, I (i.e. Q E IF,, o’ E F,- 1* a C 0’ or a n CT’= 8) for all
t 2 1. At date 0 we assumethat there is no information so that ff 0= S. The
informationavailableat time t (for t E T) is assumed
to be the samefor all agents
in the economy(symmetric information) andis describedby the subset(T of the
partition IF, in which the stateof naturelies. A pair 6 = (t, a) with t E T and
crE IF,is calleda date-euentclr node and t( 6) = t is the date of node5. The set
D consistingof all dateevents(or nodes)is calledthe event tree inducedby IF,
D=U tET.aEl& a)*
A node5’ = (t’, a’) is saidto succeed(strictly) a node 5 = (t, a) if t’ 2 t( t’
> t) and Q’ C a; we write 5’ 2 e( 6’ > 5 ). Thesetof nodesthat succeeds
a node
&Discalledthesubtree D(&)and D+([)=(e’~D(e)Ir’>[)isthesetof
strict successors
of 5. The subsetof nodesof D( 5) at date T is denotedby
DT( 6) andthe subsetof nodesbetweendatest( 6) and T by DT( 5):
When 5 is the initial node (denotedby &,) the notation is simplified to
D+, DT, DT.
1) is the set of immediatesuccessors
of 5.
Thenumberof elementsof [+ is finite andis calledthe branchingnumberb( 5 >
~‘=(5’ED(4)1t(5’)=t(5)+
M. Magill, M. Quinzii/Journul
ofMathematical Economics 26 (1996) 133-170
137
atS(b(r)=#5+).IfT=(t,
a),withtrl,thentheuniquenodee-=(t1, a’)
with (Tc a’ is calledthe predecessor
of 6.
The economy consistsof a finite collection of infinitely-lived consumers
(families) I = (1, . . . , I} who purchasecommoditieson spot marketsand trade
securitiesat every node in the event tree D describedabove.There is a set
L=(l , . . . , L) of commoditiesat eachnode:the set consistingof all commodities
indexedover the eventtree is thus:
DxL=((& /y&D,
eq.
‘LetRDx b. denotethe vector spaceof all maps x : D X L + W andlet /%(D x L)
denotethe subspace
of lRDxL consistingof all bounded maps(sequences):
r,(DxL)=
I
XEIW~~~I (~ ,su,“,,,l4~~e)
1 < -)*
The norm 11 IIm of f%(E) X L) is definedby II x IIp = supfs.e)EDxL I As, a.
As in Bewley(1972)we take the commodityspaceto be /i( D X L). Eachagent
i E I has an initial endowment processgivenby ui = ( oi( 6, c@),(5, R’) E D x
L) which is assumedto lie in the non-negative
orthant c( D X L). Let d( 6) =
( oi( 6, /‘), &‘E L) E R L denotethe agent’sendowmentof the L goodsat node5.
Agent i choosesa consumption process xi = (x’( 6, e), (5, /‘) E D X L), which
must lie in his consumptionset Xi =c( D x L); x’( 5) = (x’( 5, k’), /‘E L) E
W$ denotesthe agent’sconsumptionat node6. Note that this descriptionof the
commodityspaceassumesthat eachgoodis perfectlydivisibleandis perishable
(no storableor durablegoods)andthat the supplyof goodsdoesnot grow without
bound.The agent’spreferenceamongconsumptionprocesses
in Xi is expressed
by a preference ordering 2 ,.
At each date event there are spot marketson which the L commoditiesare
traded.Let
l
P=(P(~,P),(~,$)EDXL)EIW~~~,
denotethe spot price process andlet p( 5 ) = ( p( 5, J’), /E L) denotethe vector
of spotpricesfor the L goodsat node5. At eachnode5, good1 playsthe role of
numerairegood:
p( 671) = 1, WED,
(2 . 1)
so that all paymentsare denominated
in unitsof good 1.
In this paperwe considerthe classof red1securities:a financialassetis saidto
be a real securityif its returnat eachnode6 (after its nodeof issue)is the value
underthe spot pricesat node5 of a specifiedbundleof the L goods.As is well
known,a modelwith nominalsecuritiescanbe convertedinto a family of models
with real securities(seeGeanakoplos
andMas-Colell,lC89). Let J denotethe set
of (real) securities.Securityj E J is issuedat node t(j) E and promisesto
delivera dividendprocess( p( 6 ) A( 5, j) E Ill I 5 E ‘( 5(j))) at all nodes strictly
138
M. Magill, M. Quinzii/.Iournd
of Mathematical Economics26 (1996) 133- I70
succeedingits node of issuec(j), which is the value of a bundle A( 6, j) =
(A( 5, R’, j), & L) E [wL of the L commoditiesunderthe spot prices p( 5 ). A
maturity nodeof a securityis a node(after its nodeof issue)beyondwhich it
makesno payment(i.e. 6 E D( s(j)) is a maturitynodeof securityj if A( 5, j) # 0
and A( t’, j) = 0, Vt’ > 5 ). A securityis first tradedat its nodeof issueand is
alwaysretradeduntil the predecessor
of a maturityncde:a securityis usuallynot
retradedafter a maturity nodeis reached,but in somecircumstances
(which are
discussed
later) this will be permitted.Securityj with nodeof issue6 = e(j) is
saidto be short-lived if it is tradedonly at its nodeof issueandpaysdividends
only at the immediatesuccessors
of this node,namelyif A( e’, j) = 0, k/t’ e
t’(j). In all other casessecurityj is said to be long-lived. A securitywhose
commoditypayoff is exclusivelyin good 1 ( A( 6, R’, j) = 0 if J’# 1) is calleda
numeraire security.A securitythat is tradedat everynodeafter its nodeof issueis
saidto be infinite-lived.
The set of traded securities at node 6 is denotedby J( 5): it includesall
securitiesissuedat or beforenode5 which havenot yet reacheda maturity node
J(S)z,(jEJI5(j)$~and35’>5
with A(&‘, j)+O).
P-2)
As indicatedaboveit is sometimesof interestto includein .I( 5) securitiesthat
have alreadymatured.In all circumstances,
however,we assumethat for each
node 5 E D, the number of traded securitiesj( 6 > = # J( 6 ) is finite. It is
convenientto extendthe definitionof the commoditypayoff of eachsecurityj
from its subtreeD+( t(j)) to the wholeeventtree D by defining A( 5, j) = 0 for
all 6 e D+( t(j)). Let
denotethe commoditypayoffs of all securitiesat node 6 and the commodity
payoff process of the securities.Note that A( 6 ) hasat most a finite numberof
non-zerocomponents.
Definition 2.1. A security structure JV= (J, ( S(j))j EJ, (J( 6 ))&ED, A) consists
of a setof securitiesJ, a nodeof issuer(j) for eachsecurityj E J, a setof traded
securitiesJ( 5) c J at eachnode 5 E D and a commoditypayoff processA E
RDXLX’ for the securities,which satisfythe following consistencyconditions:
6) A( 5, j) = 0 if 5E DY E(j));
(ii) &ED(&(j)>*j!EJ(();
(iii) t(jke
and j~J(s)~j~J(~‘),‘Js’25;
(iv) (2.2) holds.
Conditions(i) and (ii) assertthat a securitywhich has not yet been issued
cannotpaya dividendandcannotbe traded.(iii) assertsthat oncea securityceases
to be traded,it is neverretraded.(iv) statesthat all ‘active’ securitieswhich have
not yet reacheda maturitynodeare traded.
M. Magi& M. Quinzii/Journal of Mathematical Economics 26 (1996) 133-170
139
If j E J( 5 ), thenlet q( 6, j) denotetheprice of one unit of securityj after its
dividendat node5 hasbeenpaid. It is convenientto definethe price processof
eachsecurityj on .thewholeeventtree D by setting q( 6, j) = 0 if j e J( 5). Let
q(5) = (dh j19 jE Jb
s=(q(&
&-)
EQ
denotethe vector of securityprices at node 6 and the security price process,
where Q = (q E IwDxJ 1q( 6, j) = 0 if j GZ
J( 5 )) denotesthe space of security
prices.
Let z’( 5, j) E Iw denotethe numberof units of the jth securitypurchased(if
~‘(6, j) > 0) or sold (if ~‘(5, j) < 0) by agent i at node 6: each security is
assumed
to be perfectlydivisibleandcanbe boughtandsoldin unlimitedamounts
(no short-sale
s constraints).If securityj is not tradedat node6, thenwe adoptthe
conventionz’( 5, j) = 0. Let
z'(
5) =
(zi(
5, j), +J),
zi=(zi(S),
&D)(=
denotethe agent’sportfolioat node6 andhis por$iolio process (trading strategy),
whereZ= (Z’E lwDxJ1z’( 6, j) = 0 if j E J( c )} is the portfolio space.
If k=(k ,,..., ki) and o=(u’ ,..., w ’ ) denotethe profilesof preference
orderingsandendowmentsof the I agentsand & is the securitystructure,then
gm(D, 2 o, &) denotesthe associatedeconomyover the event tree D. The
economyZZZ(
D, 2 cr),&‘) satisfiesthe following assumptions.
Assumption BI (Event-tree).For eachnode6 E D the branchingnumberb( 6) =
#t+ is finite.
Assumption B2 (Endowments).There exist scalarsm and m’ with 0 < m < m’
suchthat V(&
Let /‘,( D
R’)EDXL,
X L)
~‘(5, k)>m, ViEI
and Cie,Wi(S, /‘)<m’.
denotethe subspaceof II?!
DxL consistingof all summable
sequences,
Recallthat the Mackey topology on !%(D X L) is the strongestlocally convex
topologysuchthat thedualof /%(D X L) underthis topologyis J’,(D X L). For a
discussionof this topologyseeBewley(1972)and Mas-Cole11
andZame(1991).
Assumption 83 (Preferences).
For i E I, 2 , is a transitive,reflexive, complete
preferenceorderingon Xi =c (D X L) which is strictly convexand continuous
in the Mackeytopology(i.e. for all Zi E Xi, (xi E Xi I xi * , Xi} is strictly convex
andclosedin the Mackeytopologyand(xi E Xi I xi + i Zi) is openin the Mackey
140
M. Magill, M. Quinzii/ Journal
ofMathematical Economics 26 (1996~133-170
topology). 2 i is monotonein the sensethat for each xi E Xi and for each
YE~(DXL),
x'+ykiX'.
To expressthe next assumption
on preferences
it is convenientto introducethe
following notation.Let
F= [y~c(DxLl
11ylkm’)
be a boundedset which includesas a subsetthe feasibleconsumptionplans of
eachagent.For a subsetE c D of nodeslet xE denotethe characteristic
function
of E:
if &E,
X,(5) = 0, if [4E,
1,
and for x E/~( D X L) define x& =(X(& 1)x&),(&
1kDxL).
Let +
/.%(D X L) denotethe processthat hasall components
0 exceptfor the component
of goodR’ at node 5 whichis 1:
ej( t’, k’) =
1,
( 0,
if(&‘,L’)=(&k),
if(&‘,tf’)#([,e).
Assumption B4 (Uniform lower boundon impatience).Thereexists p < 1 such
that for all i E I
x’X~\~+~~~ + /3xkD+cz, + i$;xi,
V‘E D, Vx’ E F.
Assumption B.5(Securities).Every securityj E J is a real securitywith bounded
commoditypayoff A(. , j) E/%(D X L) andthe numberof tradedsecuritiesj( 6)
is finite at eachnode6 E D.
Assumption B6 (Short-livednumerairebond). At each node 6 E D, a security
J’~E J( 6 ) is issuedwhich is tradedonly at this nodeandhasa commoditypayoff
of oneunit of good 1 at eachsuccessor:
AssumptionsB 1-B6 areessentiallythe sameasAssumptionsAl -A6 in Magi11
andQuinzii (1994).The maindifferenceis that A3 in Magi11andQuinzii (1994)
only requiresconvexityof agents’preferredsets,while AssumptionB3 requires
that these sets be strictly convex. The reasonfor this strengtheningof the
convexityrequirementis asfollows. The proof of existenceof an equilibriumfor
the infinite horizoneconomyis basedon taking limits of equilibriaof truncated
finite horizon economies.In the case of short-livednumerairesecurities,the
existenceof an equilibriumin a finite horizoneconomyonly requiresthe useof a
standardKakutanifixed-pointargument,which can be appliedto an economyin
which agents’demandsare expressedby correspondences.
For a finite horizon
M. Magill, M. Quinzii/Journul of Muthematical Economics26 (1996) 133-170
141
economywith a generalsecuritystructure,the existenceof a (pseudo-)equilibrium
(seeSection4) requiresthe use of degreeargumentsandthe currentlyavailable
results(Duffie andShafer,1985,1986;Husseiniet al., 1990;Hirsch et al., 1990)
usedegreetheoryfor functions.The proofsthusrequirethat agentshavewell-defineddemandfunctions- a propertyensuredby the strict convexityassumptionin
B3. It seemslikely that the existenceof a pseudo-equilibrium
for a finite horizon
economycanbe extendedto the caseof demandcorrespondences,
but we will not
enter into such refinementshere. B3 invokesonly the assumptionof weak
monotonicity,but recall that strict convexityandweak monotonicityimply strict
monotonicitywith respectto every goodat everynode.
Theassumptionthat thereis a uniform lower boundon the impatienceof each
agent(AssumptionB4) meansthat at eachnode5, eachagentis preparedto give
up at leastthe fraction 1 - p > 0 of his)future consumptionplan after node e in
exchange
for an additionalunit of commodity1 at that node,1 - /3 beinguniform
acrossall nodesof the event tree and acrossall feasibleconsumptionplans.B4
playsan importantrole in the analysisandis essentiallythe only new assumption
addedto thosemadeby Bewley(1972).Assumptions
B3 andB4 aresatisfiedby a
by an additivelyseparable
utility function,
preferenceorderingI? represented
U’( Xi) = C p( s)sil(~)ui(
X’( 5)),
(24
5ED
wherep( 5 ) is the probabilityof 6 (inducedby a probabilitymeasurep on the
measurablesubsetsof S), SiE (0, 1) is a discountfactor and Vi: R$ + R is a
continuous,increasingconcavefunctionwith Vi(O)= 0.
3. Definition of an equilibrium
The problem of defining an appropriateconceptof an equilibriumfor an
infinitehorizonsequence
economywith incompletemarketshasbeendiscussedin
Magi11and Quinzii (1994).The simplestconceptis that of an equilibriumwith
debtconstraints:to showthat suchan equilibriumexistswe introducedthe more
abstractconceptof an equilibriumwith a transversalitycondition.As we shallsee
below,this more abstractconceptprovidesa particularlypowerfultool to prove
the existenceof an equilibriumfor an economywith infinite lived securities.
In an exchangeeconomy,equilibriumconceptsdiffer only by the specification
of the agents’budgetsets.For agent i, facedwith the price procrss( p, 4) E
Ryx LX Q, let us considerthe following budgetsets:
(i) Thebudgetsetwith an explicitdebtconstraintM
EZ, suchthat V&D
-M
(~)(xi(~)-wi(5))=v(5)zi(5-)-4(5)zi(5)
,
142
M. Magill, M. Quinzii/Journal oj’iklathemutical Economics 26 (1996) 133-170
whereV( &) = p( 6 )A( 6) + q( 5) givesthe vectorof returnsat the successor5 of
the J t-) securitiestradedat c- (ignoringthe remainingzerocomponents).
(ii) The budgetset with an implicit debt constraint
s?“(p,
q, d
d)
3z’EZ, with (42) E&J D)
= x'EX' uchthatVeED
1
b (5)(xi(~)-w’(5))=v(s)zi(~-)-4(5)zi(~)
where (qz’) = (q(()z’@),
&D).
(iii) Thebudgetsetwith a transversality condition
gcy( p, 4, 7Ti, d
=
It
&fgx’
Jq
32’ EZ, suchthat V& D
lim C ?r’( e’)q( S’)zi( 5’) =0
--Z’ED#)
,
(~)(xi(5)-Oi(5‘))=V(5)zi(5-)-4(5)zi(~)
wherew i = (T ‘( 6 ), 5 E D) is a processof presentvalueprices.
M, leadsto the simplestconceptof an equilibriumwith
The first budgetset, LJI&
explicit debt constraintM. Sincethe objectof the debt constraintis to eliminate
Ponzischemes(the indefinitepostponement
of debt) and not to introducea new
imperfectionover and abovethe incompleteness
of the markets,we do not want
thedebtconstraintsto be bindingin anequilibrium.It is clearthat if M is chosen
independently
of the characteristics
of the economy,there will alwaysbe some
Gconomies
for which M is too smallandis bindingin equilibrium.Thebudgetset
(ii) avoidsthe bad choiceof a bound M by leaving the bound unspecified:
provingthat an equilibriumwith budgetset(ii) existsis thusa convenientway of
provingthat thereis an appropriateboundM for a given economysuchthat an
equilibriumwith budgetset(i) existsin whichthe bound M is neverbinding.The
requirementthat the constraintsthat preventPonzischemesbe non-bindingis also
emphasized
in the approachof LevineandZame(1996).
If the presentvalueprocess7~i is summable(r i EC,(D)), thencondition(iii),
that the asymptoticpresentvalueof debttendsto zeroon any subtree,is a weaker
restrictionthan a debt constraint.By showingthat an equilibriumwith budgetset
(iii) exists,where 7~~is the presentvalueprocessof agent i at equilibrium,we
will showthat an equilibriumwith budgetset(ii) exists.
If (xi, z’) is a consumption-portfolio
planfor agenti satisfyingthe constraints
in the budget set L&!=*(where an asterisktakes the place of the superscript
indicatingthe type of constraintinvolvedin the budgetset), then zi is said to
finance xi and we write (xi; z’) EgX*(p, q, wi, M). (?; 2’) is saidto be 2 i
maximal in 22?:(p, q, 02, M) if 2’ financesXi and 3’ 2 iXi for all (xi; z’) E
A&* ( p, q, fd, Jd.
M. Mugill, M. Quinzii / Journul oj’ Mutlremuticd Econamics 26 (1996) 133- I70
143
The debt constrained budget sets L%!!~and L%!!~ lead to the following equilibrium concepts.
Definition 3.1. An equilibrium with implicit debt constraint (resp. with explicit
debt constraint M) of the economy lEz(D 2 o, s’) is a pair
((X, ~),(j5,~))~~(DxLxI)XZ*XRDXLxQ,
where (X, Z) = (?, . . . , X’, t’, . . . , Z’), such that
(i) (Z’; 2’) is & i maximal in BZDc( F, 4, oi, ti)
LB/( j5, Zj, d, at)), for each i E I;
(ii) Ci, ,( Xi - Oi) = 0;
(iii) Ci, I 2’ = 0.
(resp. in
The transversality conditions in the budget set ~8’:~ require that on every
subtree the present value of an agent’s debt be asymptotically zero. However,
when markets are incomplete there is no objective present value vector (which can
be deducedfrom the market prices ( p, 4)) for evaluating an agent’s indebtedness.
For this reason we use agent i’s present value vector 5’ to evaluate the
asymptotic value of his debt. The implicit prices (Z ‘)i E, must thus be added to
the objective market prices ( p, q) to define an equilibrium with the transversality
condition.
Definition 3.2. An equilibrium with the transversality condition (a TC equilibrium) of the economy 2F,(D, 2 , o, JZ’) is a pair
((X9 Z), (jj9 S9 (zi)iE,))
~~(DxLxI)xZ’xlW~~~xQx~(
such that
(i) (E’; 2’) is 2 i maximal in &?!‘( i?, Q, ?‘, wi, B’), for each i E
(ii) for each i E I
(a) Z=‘(e)>O, V&Z
XL),
where F’=(p’(e),
6E
D)=(+(5)j%[
(b) P is 2 i maximal in S&(F’, oi) = (xi E Xi 1P’( xi - oi) 5 0),
(cl ?I tf >ij( 5, j) = Et@E (+-‘((‘)($
7~
T’)A@‘, j! + q((‘, j)), vi E
J(e), V&D;
(iii) Ci, ,(E-’ - Oi) = 0;
(iv) CiE,Zi=O*
Remark. Condition (ii) characterizes the equilibrium present value vector 5’ of
agent i. (b) and (c) expressthe fact that if T( 5 ) is the multiplier associatedwith
the budget equation for node 5 (for all 5 E D), then the first-order conditions with
respect to xi and zi are satisfied. Mackey continuity of the agent’s preference
ordering implies that Fi lies in 6,( X L), the condition required in (a).
144
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Journal
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Economics 26 (1996) 133-l 70
4. Existence of a pseudo-equilibrium
Equilibriawith the debtconstraintandtransversality
conditionsaretwo waysof
extendingthe notion of an equilibrium with incompletemarketsfor a finite
horizoneconomy(often calleda GE1equilibrium)to an infinite horizon. As is
well known, evenfor a finite horizoneconomya GE1equilibriummay not exist.
Theproblemarisesfrom thefact that at a node5 the dimensionof the subspace
of
Rb(() spannedby the columnsof the b( 5) Xx 5) returnsmatrix,
V(5’) = [V(5’, 81tk6+= IP(5’)W9 8 +4(5’, j)jtk~+y (4.1)
jEJ(5)
jEJ(()
canchangewhenprices( p( t’), q( r)>)tr E 5+ vary. This createsdiscontinuitiesin
agents’demandswhich may leadto the non-existence
of a GE1equilibrium.The
way to circumventthis problem,adoptedin the recentliterature,is to introducethe
conceptof a pseudo-equilibrium:
our objectiveis to show how this approachcan
be extendedto the infinite horizoncase.In this sectionwe showthat a pseudoequilibriumexistsfor all infinite horizoneconomies;in the next sectionit will be
shownthat ‘for most’ economiesa pseudo-equilibrium
is a TC equilibrium.The
statement‘for most’ economieswill meanl’ora denseratherthana geneiic set of
economies,so that the existenceresult is weakerthan in the finite horizon case.
The ideaof a pseudo-equilibrium
is asfollows: typically if j( 6) securitiesare
tradedat a node6 whichhas b( 6) successors,
thenthe rank of the returnsmatrix
V( 6’) in (4.1) is
~(5) =min(b(t),
j(6)).
(44
However,for someprices(p, q) the rank may fall. This statementpresupposes
that the securitiestradedat node 5 have non-trivial dividendsand/or capital
valuesat the immediatesuccessorsof 5. In this section we consideronly
equilibriain which securitieswith zero dividendshave a zero capital value
(securitiesare priced at their fundamentalvalues).There is thus no loss of
generalityin assumingthat once a securityhas matured,it is no longer traded.
Thisamountsto modifying(2.2) to
J(~)=(jEJI~(j)$Sand3~‘>~
with A(&‘, j)+O).
(4.3)
A pseudo-equilibrium
is an equilibriumof an economyin which agentsare given
an artfzcial subspace of incometransfersof dimensiona( 6 ) at node 5 which
containsthe subspaceof transfersachievablewith the existing securities- but
which is largerwhenthe matrix V( 5’) hasrank lessthan a( 6 ).
In the analysisof finite horizon economies,a pseudo-equilibrium
is defined
usinga vector of discounteddate0 pricesandan abstractsubspace
at eachnode.
The subspaces
are parameterized
in a way that is convenientto proveexistence.
Here we adopt an alternativerepresentation
which is more convenientfor the
passage
to the limit from the finite to the infinite case:this representation
consists
M. Mugill, M. Quinzii/ Journal oj’Mathemuticu1 Economics 26 (I 996) 1X3- I70
145
in definingthe artificial subspace
at eachnodeby an orthogonalbasiswhich may
be interpretedas the returns of a( 5) short-livednumerairesecuritiesissuedat
node5.
Definition 4.1. ST = (K, ( t(k)), E K, (K( 5 ))( E D, r ) is an artificial short-lived
numeraire security structure of subspace
dimensions(a( 5 ))& E D if
(i) K = IJ 5EDK( 6 ), where K( 5) consistsof a( 8 ) short-livednumeraire
securitiesissuedat node5:
and r([‘, /, k) =0, if &Ee+or F’# 1;
k-(5)=+(k)=5
(ii) at eachnode5 the returnsof the securitiesissuedat node 5 are pairwise
orthogonal:
c
r( (‘, 1, k)r(
t’, 1, k’) =0,
tlk, k’EK(
t), k+k’;
43 5“
(iii) the payoff on eachsecurityis non-zeroand is normalizedso that
max lr(t’,
0 6’
1, k)l = 1, VkEK(&),v[ED.
Using a short-livedsecuritystructureX to describethe artificial subspaces
of
income transfers availableto agents leads to the following definition of a
pseudo-equilibrium.
Note that if V is an n X k matrix, then (V > will denotethe
subspace
of UP spannedby the k columnsof V.
Definition 4.2. ((F, 71, (F, p, (F’Ii E 1)) is a pseudo-equilibrium of the economy
gW(D, 2 0, s’) if thereis anartificial short-livednumerairesecuritystructureLZY
with subspace
dimensions
(a( 5 ))( ED givenby (4.2) suchthat
‘is a TC equilibriumfor EW(
D, & , w, 3);
6) ((Z, y), ( jj9 Jj,CFi)ie111
(ii) thereexists 4’E RDxJ suchthat
1
C +( S’)F( 5’)A(5’9 j),
ij( t, j) = -7
Z’( 5) pD+(~)
(4-4)
QjEJ(~),Q@D,QiW
(iii)
F( 5’)A(t’, j) + 4( 5’, j)] efEt+c IQ 5’, 1,k)l l’Et+ 9
I(
jEJ(()
Q&zD.
kEK(()
W)
If, in a pseudoequilibrium,
the returnsmatrix of the originalsecuritystructure
JV hasmaximalrank a( 5) at eachnode 5 E , then the inclusionin (4.5) is an
equality.In this casetradingin the artificial securitiesgiveseachagentaccessto
the sameopportunityset as tradingin the originalsecurities.Thuswhenthe rank
M. Magill,
146
M. Quinzii/.?ournul
of Mathematical
Economics
26 (1996) 133-170
conditionin (4.6) below is satisfied,up to convertingthe portfoliosand security
prices from the artificial to the originalsecurities,a pseudo-equilibrium
is a TC
equilibrium.
Proposition 4.3. If (( F, 7). ( j5, p, (Z ‘)i E I )) is a pseudo-equilibrium of ZZ( D, 2 ,
o, M) that satisfies
where Zj is defined by (4.4), then there exists a vector of port$olios for the agents
2’) such that ((X, Z), ( j?, 4, (F ‘)i E ])) is a TC equilibrium of
z=(i’,...,
&,@, t 0, d.
Proof. By (4.4), Vj E J( 5 ), V( E D, Vi E I:
Fi( 5E(
T9 j)
= C +(t’)V(
efE6’
5’,
j),
which can be written as ?( 6 )q( 6) = F’( [‘)V( t+), where F’( 5’) =
($( t’), 6’ E &+), denotesagent i’s b( 5) row vector of presentvaluepricesat
the successors
&+. Let I’( 6”) = [ ZY5’, 1, k&t E(,, Kt6b,
thenby (4.5) and(4.6):
W( Es)> = (r( 6'))9
'@ED.
-
This impliesthat the .&budgetset .%!Jc(ii, ii, ?‘, oi, H) andthe X-budget set
c( J, ii, ljTI, CO’,J!?“)arethe same.Note that if z’( 5 :t and y ‘( 6 ) areportfolios
generatingthe sameincometransfer
v(e+)zi(s)
=r(f+W(~~9
(4 .7)
then
so that they have the samecost. Thus, sinceevery tradi:lg strategyy i in the
X-budget set has an equivalenttrading strategy zi in the &budget and conversely,thetwo budgetsetscoincide.Let 2’ be a tradingsttategyin the M-budget
setcorresponding
to 7’ in the X-budget set for i = 2,. . . 9I andlet 2’ = C:, 22’.
It is easyto checkthat ((Z, Z), ( F9& (~‘)i E,)) ISa TC equilibriumof ES(D, k 9
0, Jf).
Cl
M. Mugill. M. Quinzii/.Iournal
ofMathematical Economics 26 (1996) 133-170
147
We now provethe main theoremof uris section.
Theorem4.4. EacheconomyZCzt
D, 2 , cr),M) that satisfiesAssumptions
I31-B6
hasa pseudo-equilibrium.
Proof. The theoremwill be proved by taking limits of pseudo-equilibria
of
truncatedeconomiesin which tradestopsat somefinite date.
Let Z$(D, 2 , w, &‘)) be an infinite horizoneconomy:the associatedT-truncatedeconomyE&D, 2 , w, JV)) is the economywith the samecharacteristics
as ZX in which agentsare constrainedto stop trading at date T. If ( pr, qT) E
RDxL x Q is a commodityandsecuritypriceprocess,thenthebudgetset of agent
i in the truncatedeconomykF’*is definedby
3~~~2, zi(e)=O,
if t(e)zT
’
Tw(xi(wWi~~~)
X’EX’
=(PT(5jA(5j+qT(~))Zi(~-)‘*
--q~(~)zi(~)~
i
:
(t)=wi(5),
if
t(tjsT
ift(t)>T
/
Eventhough the consumption-portfolio
processof an agentis definedover the
wholeevent tree, a T-truncatedeconomyis essentiallya finite horizoneconomy
with T + 1 periodssincean agent’sconsumption-portfolio
processis fixed after
dateT.
Definition4.6. A GE1equilibriumof the truncatedeconomyZr( D, & , w, A%‘)is
a pair
K ET, ZT),(~T,~T))E~(DXLXI)XZ'XIW~~~XQ
suchthat
(i) ($.; 2;) is ,F maximalin &&( &, qr, ai d), tli E I;
(ii) Ci, I( Zk- Oi) = 0;
(iii) Ci E1 2: = 0;
(iv) jTjT([)=O, if t(t)> T, and qr(&)=O, if t(t)2 T.
Sinceonly the pricesof the commoditiesand securitiesthat are tradedin gr
arewell determined,(iv) is a naturalway to extendthesepricesto the wholeevent
tree. Since in an equilibriumof the truncatedeconomythe terminal condition
zi( 5) = 0 for all 5, with t( 5) 2 T replacingthe transversalityconditionof the
budgetset ~22~
TC,the presentvaluevectorsof the agentsdo not appearexplicitly in
Definition 4.6. Howevereachagenthasa well-definedpresentvaluevector in an
148
M. Magill,
M. Quinzii / Journal
of Mathematical
Economics 26 (I 996) 133-l 70
equilibriumof S?‘rdefinedby the first-orderconditionsof the agent’smaximization problem.When an agent’spreferenceorderingis represented
by a differentiable utility function, then the presentvalue vector is simply the vector of
Lagrangemultipliersgivenby the Kuhn-TuckerTheorem.Whenthe preference
orderingis assumednot to be differentiable(asis the casehere),the existenceof a
presentvaluevectorcanbe derivedby a directseparationargument.In Magi11and
Quinzii (1994, lemma 5.6) it is shown that under Assumption B3 if
((z,, Z,), ( &, &)) is a GE1equilibriumof JFr(D, 2 , cc),s’), theneachagent
i E I hasa presentvaluevector?ri E RD satisfying
(a)YF#)>O, if r([)sT, andZ#)=O,
if t(t)> TT;
(b) X; is i” maximalin
B,(i$,
w’)=
9
The following definitionis the finite horizonanalogof Definition 4.2.
Definition 4.7. (( ET, yT), ( &, &)) is a pseudo-equilibrium of the truncated
economy$Yr(D, 2 , o, s’) if thereis an artificial short-livednumerairesecurity
structureX,. with subspacedimensions(a( 6 ))* ED, where a( 6 ) = min(b( &?,
DT- ‘, suchthat
(i) (( Zr, TT), (&, j&)) is a GE1equilibriumfor &( D, 2 , cc),XT);
(ii) if nT is any positivevectorof nodepricessatisfying
QQ)iq5)
= ~TwYT(t+)~
andif & is definedby
t(5) ST-
1,
w9
1
C vT( s’)FT( 04
n,(5) gED+([)
qT( 6, 8 =
if jE J( t), SE DT-‘,
0,
otherwise,
I
5’, jL
VW
then
,
V&DT-‘.
(4.10)
M. Mugill, M. Quinzii/ Journul of Mathematical Economics 26 (I 996) 133-l 70
149
Remark. When (4.10) holds,the definitionof ZjTin (4.9) is independent
of the
choiceof the vector of node prices +. satisfying(4.8). This can be seenas
follows:
1
where ( ai, k E K( 5 )) are the coordinatesof the vector ( &( 5‘) A( 6 ‘, j) -IqT(tt, j)& 5’ on the basis(&( , 1, k), k E K( 5 >). The reverseCakUkitiOn
with any other vector of nodesprices(‘iir( 5 ), 5 E DT) satisfying(4.8) leadsto
(4.9) usingthe terminalconditionqr( 6, j) = 0 if t( 6) = T.
l
It follows from the extensionof the existenceresultof Hirschet al. (1990) to
the multiperiodcase2 (seeMagi11and Shafer,1991, theorem16) that a pseudoequilibrium((ET, yT), (&, &)) of the truncatedeconomygr(& k , w, d)
exists for every finite T. IA 2r denotethe associatedartificial short-lived
numerairesecuritystructureand let i=r denoteits (numeraire)commoditypayoff
process.since (( 3i-,,yT), ( FT9pT)) is a GE1equilibriumof &(D, & , WVzT),
it hasassociated
with it Presentvaluevectorsanddiscountedpricesfor the agents
<if;, FIJiE I satisfying(a) and (b) above,(c) being replacedby (4.8). Zk, and
hencePi, canbe normalizedby setting
F;(&/‘)=l,
WEIJTET,
(4.11)
F;n=
c
(E,C)ED”XL
where 1 =(l,..., l,... ) E/~( D x t) denotesthe vector all the componentsof
whichareequalto 1. Let & denotethe pricesof the originalsecuritiesdefinedby
(4.9). In view of the remarkfollowing Definition4.7 for eachi E I:
1
c
O~(54=+(~)
T
z~(~')~T(~t~A~~t9
jh
[‘ED+(e)
WjE J( t), &EUT-‘,
whichcanbe written as
1
qT(tr j) = .&FiA(-,
j)xD+f61, vjEJ(t)9&EDT-‘*
(4*12)
T
2 This result cannot be derived from the existenceresult of Duffic and Shafer (1986) for multiperiod
economies which depends on the differentiability of agents’ demand functions - a property which is
not required here.
150
M. Magill, M. Quinzii/hurnal
of Mathematical Economics 26 (1996) 133-170
bt eT= (ET, TTYPT9pT9(%:&a 1, Fr) denotethe actions,pricesandpayoff
processcharacterizingthe T-periodpseudo-equilibrium.
Since ETE R DxL’ ‘, &
E RDxJxf, etc. er (andhencethesequence
(eT)r& lies in a spaceY which is a
productof Euclideanspaces.
Thediscountedprices($ i E I)T Er canbe viewed
aselementsof ba( D x 1,) =&* (D X L), i.e. thenorm dualof f?,(D X L), consisting of boundedfinitely additiveset functionson D X L. Let 11 11
ba denotethe
normof ba( D X L) andlet c( ba, /%) denotethe weak * topologyof ba( D X L).
The ideais to take limits of eT in the producttopologyandof (P:, i E I) in
the a( ba, /%) topology.In Magi11and Quinzii (1994, step 1 in the proof of
theorem5.1) it is shownthat the actionsand prices(ET, TT, Jjr, &, (7Fk)iE,)
canbe boundedindependently
of T and that for each 6 E D thereexists cz> 0
such that ?7;(c) 2 cr, V’i E I, WE T. By property (iii) of Definition 4.1 the
commoditypayoffs rr are boundedindependently
of T, I Fr( 5, 1, k) I 6 1, vk
E K, ‘JsE D. It follows from Tychonov’sTheoremthat ( eT)Ter lies in a subset
of Y which is compactin the producttopology.Since(4.11) impliesII Pi 11ba=
1, Vi E I, V’TE T, thesepricesbelongto the unit spherein ba(D X L) which, by
Alaoglu’stheorem,is compactin the o(ba, Hm)topology.Thus there exists a
directednet ( A, 2 ) and a subnet((F& i E I, A E ( A, 2 )} suchthat Fi convergesto Fi in the a( ba, /=) topology,Vi E I. By extractingan appropriate
subnet,eT.Converges
to e = (z, 7, F, p, (%‘)i E f, T ) in the producttopology.
It is clearthat F satisfies(ii) and(iii) in Definition 4.2 andhas a( 5) linearly
independent
payoffsat eachnode( 6 E D: let Z( 6) denotethesesecuritiesandlet
K= U,.+(&).
Then p=(K,
&(k)kEK,(K(&))2ED, T) is an artificial
short-livednumerairesecuritystructurewith subspacedimensions(a( &I& E D. It
r?ollowsfrom Magi11andQuinzii (1994, step3 in the proof of theorem5.1) that
((k 71, (3, 3, (In’)i~ ,N is a TC equilibriumuf ZZZ(
D, k , CI),2): in particular
it is shownthat for each i E I, the limit price pi lies in /,( D X L) and that
Pitt, l’)=F’([)jj((,
k),V(& /kDXL.
SinceA(#, ~)x~+~~,E/~(DXL)
andsince$.. convergesto Fi in the u( ba, fW>topology,
l
mus (4. I T) impliesthat K E I, Vj E J( 5 ), v6 E D:
Let Zj(5, j) denotethis limit. Then q = ($5, j), 5 E D, j E J( 5 )) satisfies(4.4).
Passingto the limit in (4.10) in the obviousway gives (4.5), and the proof is
complete. Cl
M. Mugill, M. Quinzii/ Journal of Mutltematical Economics 26 (I 996) 133- I70
151
it is the fact that, for eachagent i E I, the personalized
prices
em)topologyand that the limit price satisfies
5?(~)~([, k’),V(& /')EDXZ+
which enablesus to showthateachsecurityprice ZjT,<
, j) convergesto the discountedsum
of its future dividendsfor eachagent.This result seemsdifficult to obtainusing
convergence
of the personalized
pricesin the producttopology.
l
5. Existence of an equilibrium
In this sectionwe showthat for an economywith givenuncertaintyandagent
characteristics(D, 2 , o) and for given nodesof issueand maturity for the
(J( 5 >>s.
ED) thereis a densesetof securitypayoffs A *
securities(,!, ( &j>&
such that for all A E A * the economy 8’&3, 2 , (I), &‘), with JX?
=
A),
has
a
pseudo-equilibrium
which
is
a
TC
equilib(J, (S(j))j,=,, (J(5))5ED9
rium. To establishsucha resultwe first definea setof admissiblesecuritypayoffs
compatiblewith a given set of nodesof issueand maturity. More precisely,let
(J, (scj>>iEJ, (J( 5 >>rED) be a collectionof securities,nodesof issue and
traded securitiessuch that J( 5) is finite for all @ED and such that the
consistencyconditions(ii) and (iii) in Definition 2.1 are satisfied.The set A of
admissible payofi for the securitiesis definedby
J9
The definitionof A ensuresthat eachpayoff processA E A is compatiblewith
(J, l(j)i,J, (J(&)gED) - in the sensethat (i) and (iv) in Definition 2.1 are
satisfied- andthat AssumptionsB5 andB6 hold.
SinceA Cr,( D X L X J), it is naturalto endowA with the norm topology,the
normof a payoff processA beingdefinedby
11A 11
r, =
sup
IA(W
(&f.jkDxLXJ
Note that A is a closedsubsetof dA(D x L
x
J).
Theorem 5.1. Under Assumptions B 1-B6 there exists a densesubsetA * c A such
then the infinite horizon economy em
t ‘)
that if A E A*,
w, M) has an equilibrium with the transversality condition.
152
M. Mugill, M. Quinzii / Journal
of Matkematicul Economics 26 (I 996) 133- I70
Proof. Seethe appendix.
-
The ideaof the proof is to show that if we pick a payoff processAE A for
whichthe rank condition
is not satisfiedin a pseudo-equilibrium
((X, y), (p, p, (Z’)i, ]) of ZTZ(
D, 2 ,
W, 2) thenfor all E> 0, thereexistsa payoffprocessA E A in theball of radius
E around x such that (( 2, T), ( F, j5,(~‘)i E,) is a pseudo-equilibriumfor
&Z=(
D, & , o, M)) in which the rank conditionis satisfiedfor A.
5.1. Equilibriumwith a debtconstraint and the transversality condition
For a deterministicsequence
economyoveraninfinite horizonin whichthereis
a market for short-termborrowingand lendingat each date, it has long been
recognized
that a conceptof equilibriumbasedon a transversalityconditionon the
indebtedness
of eachagent- requiringthat an agent’sdebt growsasymptotically
slower than the rate of interest - permits the link to be madebetween the
equilibriaof the sequenceeconomyand the Arrow-Debreu equilibria(see, for
example,Kehoe, 1989).The conceptof an equilibriumwith the transversality
conditionin Definition 3.2 providesan extensionof this conceptto the caseof
stochastic
economieswith incompletemarkets.Whenmarketsareincompletethere
is no uniquepresentvaluevector (representingimplicit prices of incomein the
future) which links the pricesof the securitiesto their future dividends,i.e. which
satisfies
For this reasonthe indebtedness
of eachagentis evaluatedusinghis own present
valuevector5’ at the equilibrium.Someeconomistsmay arguethat this makes
the conceptof a TC equilibriumsomewhatquestionable
as a descriptiveconcept
of an equilibrium:sinceagentsare assumedto assesstheir capacityto postpone
the repaymentof their debts(i.e. their capacityto borrow from the market)using
their own present value vector, the conceptof an equilibrium is based on
self-monitoringrather than someobjectivemarket-based
schemefor controlling
the indebtedness
of agents.
Evenfor deterministiceconomies,macroeconomists
havetypically beenmore
comfortablewith a conceptof equilibriumbasedon explicit debt constraints.It is
thusof someinterestthat underAssumption
B4 the equilibriaof an economywith
implicit debt constraintscan be shown to coincidewith the equilibria with the
[email protected], in turn, impliesthat the equilibriacanbe obtained
by imposingan explicit boundM on the indebtedness
of the agentsat eachnode,
M. Magill, M. Quinzii/Journal OfMathematical Economics 26 (1996) 133-170
153
whereit4 canbe chosenso that it is neverbinding.Proposition5.3 below permits
the advantagesof each concept of equilibrium to be retained, since for a
theoretical(mathematical)analysisa TC equilibriumis more natural,while an
equilibriumwith a debt constraintis more plausibleas a descriptiveconceptof
equilibrium.
Proposition
equilibrium
5.2. Under Assumption B4, if ((X, Z), ( ii, Zj, (5 ‘)i E *))
of Z$(D, 2 , o, &), then (42’) E/~(D) for al! i E I.
is a TC
1)) be a TC equilibrium.Pick any node $E D
andconsideran agentwho is a net lenderat 5, i.e. q( 5 )Z’( z> 2 0. Agent i can
considerscalingdownhis portfolio from node$ onwards,i.e. to change2’ to 2’
definedby
Proot Let ((F, 3, (p9 q, Gi)i,
zi(
5)
=
”
‘w ifWD(c),
’
pZ’(
0,
if
&D(E),
where/3 < 1 is the factordefinedby AssumptionB4. It is easyto checkthat this
new portfolio satisfiesthe transversalityconditions
lim C Zi( &‘)g( t’)Zi( 5’) =0
T’“r’ED,(5)
for all -5~ D. With the trading strategy 7’, agent i can consumeX’( 6) if
& D( 0, at least/3X’(5) if 6~ D+( 5) since
PbS)Wi(6)
+Pv(s)z’(~-)-Ps(5)2’(5)
=-PF(5pi(5)
+(1 -P)F(W(s)
ePF(w(o
and can increasehis consumptionof good 1 at node 2 by (1 - p )q( 8) Z’(t)
since
By AssumptionB4 the increment( 1 - p)Zj( 8) 2‘( g> to his consumption
of good
1 mustbe lessthan 1 so that
SinceXi, I q( 8) z’( 8) = 0, agentswho are net borrowersmust find net lenders.
Thus
q(g)qf)so*
- (3- 1gj(S)z’(5)
M. Magill, M. Quinzii/Journal cfMathematica1 Economics 26 (1996) 133-170
154
so that
-(E 1sa(S)ii(S)q+j.
ViEI.
-
Thusfor eachagenti E I, (qZ’) E~C,(
D).
q
Proposition 5.3. Under Asslmlptions B1 -B4, (( F, Z), ( p’, q)) is an equilibrium
with an implicit debt constraint of &CD, 2 , o, d) if and only if there exist
present value vectors G ‘Ii E I such that ((Z, Z), ( ~YJ,4, (Fiji E ,)> is a TC equilibrium.
(e=) The fact that a TC equilibrium((E, Z), (F, Zj,(Z ‘)i E,)) is also an
equilibriumwith implicit debtconstraintsfollows from Proposition5.2. Sincefor
all i E I, (Z’; ii) ES!mDc( j5, q, d, at’) CL&fc( jj, q, F’, d, at’) and since
( Ei; 2’) is 2 i maximalin the largerbudgetset G!Jc, it is 2 i maximalin 9:‘.
Thus((E, Z), ( ii, ?j)) is an equilibriumwith an implicit debt constraint.
( * ) The main ideasof the proof are as follows: for a completeproof see
Magi11and Quinzii (1993, theorem5.2). Let ((Z, Z), ( p, 4’)) be an equilibrium
with an implicit debtconstraint.SinceEi is & i maximalin BWDc(F,Zj, wi, H),
a separationargumentgives the existenceof a price Pi E ba( D X t) which
separatesthe preferredset V’ = (xi EC( D X L) 1xi 2 i E’), which has a nonempty interior in the norm topology, from L@~. Mackey continuity of 2 i
impliesthat Pi E/‘,( D X L). SinceL@mW
is definedby linearinequalities,Farkas’
Lemmaappliedto a sequence
of truncatedbudgetsetsleadsto the existenceof a
presentvaluevectori? E 1,(D) which is strictly positiveby the monotonicityof
2 i and satisfies(a)-(c) in Definition 3.2(ii). Sincefor each i E I, 5E’EL@:’ c
’ C B( P’, CO’)and sinceby the separationargumentXi is L: i maximalin the
larger set B( Pi, oi), it is also & i maximal in 92’.
Thus
((E, Z), (j3, & (Y?‘)iE,)) is a TC equilibrium. 0
ProoJ
Corollary 5.4. Under Assumption B 1-B6 there exists a dense subsetA * c A such
that if AEA*,
then the infinite horizon economy ZEX(D, & , w, ..& has an
equilibrium with an explicit debt constraint M which is never binding.
Proof. Let A* be the densesubsetin Theorem5.1. If the commoditypayoff
TrocessA lies in A*, thentheeconomyZz(D, 2 , o, ti) hasa TC equilibrium
((X, Z), ( j3, q, (~‘)i E,)), which by Proposition5.3 is an equilibriumwith an
implicit debt constraint. Any number M > g, where M = maxiE I
sup&ED 1q( 6 ) Z’(5 ) I, providesa debt constraintthat is neverbinding. 0
M. Magill,
M. Quinzii/
Journal
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Economics 26 (1996) 133-170
155
5.2. Generalization to securities in positive supply
Although the model introducedin Section2 is restrictedto an exchange
economyin which securitiesarein zeronetsupplyit canbe generalized
to include
securitiesin positivesupply (such as equity contracts)that arisenaturally in a
productioneconomy.
Let Z?$D, 2 , cd,6, M) denotean economywhich is identicalin all respects
to that consideredin Section2 exceptthat a subsetJo C J( &) of the securities
issuedat date 0 can have positive initial supply S = ( ?jj, j E J,), where $ =
CiE, Si and 6; is agenti’s initial holdingof securityj. (Securitiesissuedafter
date0 couldalsobe permittedto be in positiveinitial supply,but this complicates
the notation unnecessarily.)A TC equilibrium of the economy Em@, 2 ,
O, 8, &) is given by Definition 3.2 with the following modifications:the new
budgetset a:‘( p, q, 7r‘, oi, a’, @‘) of agent i is identicalto that definedin
Section3 exceptfor the equationat the initial nodewhich becomes
(5.1)
andthe market-clearingcondition(iv) becomes
CZ’(S, j)=si,
(54
W&D,VjEJ,
iEI
where~j = 0 if j GE
Jo.
Securitiesin positivesupplymodelownershiprights to the income(dividend)
streamscreatedby productiveassetssuchas firms, land or other durablecapital
goods.An economyE,( D, t , U, S, &‘) in which therearesecuritiesin positive
(initial) supply thus servesas a modelof a productioneconomyin which all
decisionsregardingthe useof the productiveassets(i.e. the productionplans)are
fixed: for suchan economywe makethe following additionalassumption.
Assumption B7. If ~j > 0, then A( , j) EC( D
Sj = 0, then 6; = 0 for all i E I.
l
X
L) and 6; )=0 for all i E I. If
Thussecuritiesin positivesupplyrepresentproductiveassetswith non-negative
payoffs and agentsonly inherit non-negativeinitial sharesof such assets.For
securitiesin zero supplyagentsdo not inheritany initial debt(or credit).
Let us showhow the aboveexistenceresultscanbeextendedto sucheconomies.
To this end let 3
(m’,..,,
a=1WC
0’) E~(DXLXZ)Io’~ml,
id)
M. Magill,
156
M. Quinzii/Journal
ofMathematical
Economics 26 (1996) 133-170
denotethe spaceof initial endowmentsof the agentssatisfyingAssumptionB2.
Theorem5.1 leadsto the following result.
Theorem 5.5. Under Assumptions BP -B7 there exists a dense subset A c 0 X A
such that if (0, A) E A, then the infinite hor!zon economy a( D, 2 , co, 8, d)
has a TC equilibrium.
Proofi With an economy&Cz(D, 2 , cr),S, H) in which somesecuritiesare in
positiveinitial supplywe may associatean economyg%(D, 2 , o, s’) in which
meritshavethe modifiedendowments
all securitiesare in zeroinitial supplyandas
ei(tj)=wi([)+
cF$A(&
j),
V&D.
iEJJ,
If &?!(D, &, W, JY’)doesnot havea TC equilibrium,thenby (the constructionof
theequilibriumin the proof of) Theorem5.1 thereexists,for eachE> 0, A’ CA *
such that 11A’ - A 11
z < E and @!!(D, 2 , w, JZ”) has a TC equilibrium
(( E, E), ( p, 4, (77‘)i E& where 4 satisfiesthe pricing equations(4.4). Let us
showthat if for eachi E I
zi(*, j) =F’(*,
j) +a;,
VjE J,
and o’ is suchthat for all i E
di( 5) + c $4’( 6, j) = $( 6) =d(&)+
6 Jo
c$A(&
j),
QED,
jEJo
W)
then(( X921,( fj, ?j,(*‘)i, 1)) is a TC equilibriumof ZZ(D, 2 , cc)‘,6, H”). To
seethis it sufficesto checkthat if z’(. , j) = Z’(. , j) + 6; for all j E J, then
(
xi;
fi
-
a:“(
(xi;
z’)
j5,
if,
F?&zqC(
z’,
p,
g,
if,
J3q
if’,
di,
s’,
a’).
The budgetequationsat eachnodeare clearlythe same,andif the securityprices
satisfy(4.4), thenfor all j E J, i E I and 6 E D:
;$+
F&w
‘E
m?(
5’)
T
= lim
*i(5’)F(t’)A’(5’,
j) =O,
c
T--5’eD(~)\0Ttg)
sinceF’ E/,( D X L) and A’(. , j) E/~( D X L). Thusthe transversalityconditions
for the two budgetsetsare equivalentsincefor all 5 E D:
lim C Zi( [‘)q( 5’) zi( 5’)
-(‘EDT(~)
= lim c Fi( S’)S( (‘)( ?( 5’) + si)
‘-(‘EDiP(S)
= Tjxg
lim
M. Mugill, M. Quinzii/hurnal
ojMathemtctica1 Economics 26 (1996) 133-170
!57
Since(5.3) implies 11o - W’11
p s (Cj, J,sj) 11A - A’ II=, co’, A’) can be chosen
arbitrarily close to (0, A). Thus the set A of parameters( W, A), such that
Kc@, k ,
O, S, &) hasa TC equilibrium,is densein &?X A. CI
SincePropositions5.2 and5.3 canclearlybe appliedto aneconomyZX(0, 2 ,
O, 8, Ccs),it follows that if (w, A) E A, thenany TC equilibriumof the economy
Zm(D, 2 , o, 8, &‘) canbe obtainedby imposingan explicit debt constraintM
that is neverbinding.
6. Equilibrium
prices of infinite lived securities
In any equilibriumthe pricesof the securities4 andeachagent’spresentvalue
vectorZ’ must satisfythe equations
j) = C ~i(S’)(P(S’)A(~‘~j)+~(5’,j)),
0 5’
~WS
V@D,VXI,
(64
which expressthe first-order conditionsfor the agent’sportfolio. In a finite
horizoneconomy,or as here in a T-truncatedeconomyET, ‘integrating’these
equations(by successive
substitution)andusingthe terminalcondition
(6.2)
&( 5) = 0, t’5E&9
gives
&(55 ,)
1
= *it
c
n:( 6’)&(
5) <‘ED+(()
!$‘)A( 5’, j),
&ED,
vjiII,
(6.3)
so that the equilibriumprice of eachsecurityis equalto the presentvalueof its
future incomestreamfor eachagent.The expressionon the right-handside of
(6.3) is calledthe jhhzmental oalue for agent i of securityj (at node6).
It is evidentthat in an infinitehorizoneconomy(6.3) holdsfor anyfinitely-lived
security.However, for an infinite-lived securitythere is no terminal condition
(6.2) that can be addedto Eq. (6.1) that would force the equilibriumprice of the
securityto equalits fundamentalvalue‘foreachagent.This leadsto the following
definition.
Definition 6.1. Let (( X, Z), ( p, 4, (*‘)i EI )) be a TC equilibriumof the economy
ZZZ(
D, & , OJ, A?). Securityj E J is saidto be pricedat its fundamental value if,
for all agentsi E I,
1
C +( 5’)ii( 5’)A( 5’, j),
pal’(~)
QED(
SO)
(6 .4)
158
M. Mugill, M. Quinzii/ Journal of Mathematicul Economics 26 (1996) 133-l 70
Securityj is saidto havea speculutive bubble if (6.4) is not satisfiedfor some
agenti E I.
It is a naturalconsequence
of the methodusedin Sections4 and5 to construct
equilibriaof an infinite horizoneconomyZI that in theseequilibriaall securities
are pricedat their fundamentalvalues:sincein the truncatedeconomiesET (6.3)
is satisfied,this propertyis transmittedto theequilibriumpricesof the securitiesin
the limit. To what extentare suchequilibriatypical?Do thereexist equilibriaot
8% in which someof the securitieshavespeculative
bubbles?
Theanswersdependcruciallyon the typeof infinite-livedsecuritiesavailablein
the economy:securitiesin positivesupplycan never have speculativebubbles,
while infinite-livedsecuritiesin zero net supplyalwaysadmit the possibilityof
speculativebubbles.
Proposition 6.2. Under Assumptions B4 and B7, if ((Z, Z), ( p, 4, (F’)i, ,)) is a
TC equilibrium of Z,( D, 2 , o, S, M), then the price of eveby security in
positive supply (aj > 0) is equal to its fundamental value.
Proof. This result could be deducedfrom Santosand Woodford (1992), but a
titularly simple proof can be given in the present context. Let
, 3, ( p, q, W) iE ,)) be a TC equilibrium.For all 6~ D, xi, ,?$ e)Z’( 5) =
jE J,q( 5, j)aj. SinceAssumptionB4 holds,by Proposition5.2, (@‘) E/I(D)
for all i E 1. Thus Cj EJ , j)aj &I?(D). By AssumptionB7, 4 > 0 implies
Zi(6, j) 2 0, V[ E D. Thus every security in positivl?supply (j E J,) satisfies
y’(. ,j) E!%(D). Integratingthe first-ordercondition(6.1) >f agenti up to date T
gives
l
We now show that theequilibriumpricesof infinite-livedsecuritiesin zeronet
supplyare not tied to their fundamentalvalues.Proposition6.3(i) below shows
that it is alwayspossibleto adda bubblecomponentto the equilibriumprice of
sucha infinite-livedsecurityso that the resultingprice remainsan equilibrium
price. However,thereis a differencebetweenspeculativebubbleswith complete
and incompletemarkets.If, in an equilibrium,the financialmarketsare complete
even without securitieswith speculativebubbles,then the same equilibrium
allocationcan be supportedby pricing every securityat its fundamentalvalue;
M. Magill.
M. Quinzii/
Journal
of iV¶athematical Economics 26 (1996) 133-I 70
159
removing the bubble component in the price of any infinite-lived security does not
alter the span of the markets and hence does not affect the real equilibrium
allocation (Proposition 6.3(ii)). When markets are incomplete there exist equilibria
in which infinite-lived securities have speculative bubbles that are non-trivial in
the sensethat the same equilibrium allocation cannot be obtained if securities are
priced at their fundamental values: the bubble components in the prices of the
infinitely-lived securities affect the span of the markets in such a way that they
cannot be removed without altering the real equilibrium allocation (Proposition
63(iii)).
The following preliminary remark will be helpful before stating Proposition
6.3. Suppose((Ti, Z), ( jj, q, (z i)ie 1)) is a TC equilibrium of an economy
gE(D, 2 , U, 8, JV) in which the price d(. , j) of security j has a bubble. If
q(. , j) is the fundamental value of the security defined by (6.4), then both q(. , j)
and q(. , j) satisfy the first-order condition (6.1). Thus the difference p(. , j)
between the two prices,
~(6,
j) =q(&
j) -9(
6, j)*
(6 .5)
QED,
called the bubble component 4 of q(. , j), satisfies the following homogeneous
equation:
+([)p(t)=
C
0
Fi(f)p(t’),
ViiZ.
(6 .6)
5’
In a deterministic economy (b( 6 ) = 1, Wt E D) in which there is short-term
borrowing and lending at each date, markets are complete (5 i = ?r, Vi E I) and
the equilibrium short-term interest rate satisfies
1
7fI+ I
-=-,
VttT.
1 +‘;,
5,
In this case (6.6) becomes p, = [ l/( 1 -t F,)] pI+ I, V t E T, which implies p,+ l =
a!nA=,( 1 + &,) and leads to the well-known conclusion that in a deterministic
economy a bubble must grow at the rate of interest.
This property generalizes to stochastic economies in which markets are not
necessarily complete, as follows. The equilibrium short-term interest rate 7( 5) at
node 6 satisfies
4 Since a bubble assignsa non-zero value to a zero dividend stream, when the price of a security has
a bubble component, the price ceasesto be a linear functional on the apaceof income streams k?=(D).
Thus the bubbles that arise in the model of this paper are not the same as the bubbles studied by Gilles
and LeRoy (1992) which come from a pure charge component of a continuous linear functional on
L(D).
160
M. Mugill, M. QGn=ii/ Journul of Mathernuticul Economics26 (1996) I33- I70
and the product p( 5 ) of the randominterestrateson the path [ to, 5-1 from the
initial node&, to the predecessor
c$- of 5,
satisfieshomogeneous
equation(6.6) since,for all 5 E D and for all i E I,
Proposition 6.3. Let 8=(D, 2 , ct), 8, &) be an economy satisfying Assumption
B6, with at least one infinite-lived security (j E J( &,)I in zero net supply
6;=0, WEI).
6) If ((Z, 21, (jj, q, (*‘)iE ,)I is a TC equilibrium of the economyand if G is
defined by
(si=O,
q(g, j) =ij(*,
j) +iW
q(-, J)=Zj(-,
j),
J+h
where p = ( p( 5 ), 5 E D) is the bubble component given by (6.7), then q is also
an equilibrium security price vector; i. e. there exist portfolios ? = ( i’, . . . , z’ 1
SUC~Ithat ((X, i), ( j?, q+(*‘)i,
t)) ,is a TC equilibrium of the economy.
(ii) Conversely, [f (( E, z), ( a, q, (YFi)i E ,)) is a TC equilibrium of Z,( D, 2 ,
W, 8, &) and if the financial markets are complete even without the infinite-lived
securities whose prices exhibit a bubble, then there exists a vector of port$olios
Z==(?,..., 2 ’ ) and a vector of security prices q, under which every security is
priced at its fundamental value, such that ((F, 21, ( F, & (5’& E ,)> is a TC
equilibrium.
(iii) Zf the hypothesis in (ii) is not satisJed, then the existence of bubble
componentsin the security prices can have real effects, i.e. there exist equilibria
in which the price of some infinite-lived security has a speculative bubble, such
that the same real allocation cannot be supported by a vector of security prices
under which every security is priced at itsfGndamenta1value.
Proof. (i) For a vector of securityprices q, let V(q, 6’) = [ i;( 5’) A( t’, j) +
of
q(e’)J*‘E b.jE J(*) denotethe b( 5 ) x j( 5 ) matrix of returnsat the successors
5. Since ii;(6’) has the same value for each successor5’ E t+, the vector
( jX 5’1, 5’ E e+) is collinearto (1,..., l).SincebyAssumptionB6,(1,..., l)T~
(V(q’, e+)>, addingthe bubblecomponentto the price of securityj does not
M. Mugill, M. Quintii / Journul
of Muthemuticul Economics 26 (1996) 133-I 70
161
changethe subspace
of returnsat the successors
of e, (V( g, [+)) = (V( 9, 5’)).
Sinceq and q satisfy(6.1),
VW
sothat the subspace
of incometransfersis unchanged
at eachnode.Sinceno agent
inheritsdebt at date0 in the infinite-livedsecurityj, q( &)S i = q( &# i, so that
the agents’ wealth at date 0 is unchanged.Thus Z@(p, & 5 i, oi, Si, A) =
&?(p, 4, 5 i, oi, 6 i, A). For all agentsi E I, i # 1, we definethe new portfolio
Z’ by
1
--q(5) 1-i
v(q, ,,,1"5'=
-4(r) F(S), KED.
[v(tL 6') I
(6.9)
Let Z’(E)= - cl= 2Z’(5), VeE D. Thespotmarket-clearing
equationsimply that
Z’( 0) satisfies(6.9). Thus (Z’; Z’) is 2 i maximalin B( p, 2, ?i’, oi, 6 i, A), k/i
E il. Since Ci, , Z’ = 0 and “7
o ‘) = 0 it follows that
LiE
I (Xi ((Z, z)( p, q, (Z i)i E,)) is a TC equilibrium.
(ii) Let Z’= 5, Vi E I, denotethe commonpresentvaluevectorof the agents
andlet 4’ denotethe vector of securitypricesdefinedby (6.4). Then q( , j) =
$9, j) except possiblyfor some infinite-!ived assetsin zero net supply. By
assumption,(V(& l+ 3) = (V(Zj, 5’ )) = R!‘(‘? Sinceboth 4 and q satisfy(6.l),
it follows that (6.8) holds. Since $ = 0 for securitiesin zero net supply,
.53(,5, q, ?, oi, S’, A)=S??(p, 4, Z’, oi, S’, .A). By the sameargumentas in
(i) thereexists Z suchthat (( X, Z), ( F, Zj,(~i)i E,)) is a TC cr;quihbrium.
(iii) When the subspaces
(V(q, 5’)) dependon the bubblecomponentsof
in
the sensethat replacingprices with speculative
infinite-lived securitiesbubblesq by the fundamentalvalues Q of the securitiesleads to different
subspaces
(V( & [+)) - then the equilibrium(( F, Z), ( p, q, (5 ‘)i EI) cannotbe
achievedwith prices q and a changein portfolios.This will be shown by two
examples.The first hasthe merit of simplicity,but doesnot satisfyAssumption
B6: it illustrateshow a speculativebubbleon an intrinsicallyworthlesssecurity
canenableagentsto carry out mutually advantageous
borrowingand lendingin
the absenceof a bond. Constructingan examplesatisfyingAssumptionB6 is
necessarilymorecomplicatedsince,as shownin (i), when the short-termbond is
traded,the simplestbubblegivenby (6.7) cannothaverealeffects.ThusExample
2, which satisfiesAssumptionB6, exhibitsa more sophisticated
bubblewhich has
realeffects.
l
Example 1. Supposethat the eventtree D is suchthat the only uncertaintyis at
date 1, the future after date 1 beinginfinite but certain.Let ( &, 6;) denotethe
two nodesat date 1,the nodefollowing 5, (resp.6;) at datet beingdenotedby 6,
(resp. &‘) for i 2 1 (seeFig. 1). We assumethat the two nodesat date 1 are
162
M. Mugill,
M. Quirtzii/
Journul of Mathematical
Economics 26 (1996) 133-170
Fig. 1. Event tree and agents’ endowments for Example 1.
equallyprobable( p( & )) = ( p( 6;) = l/2). Supposethe economyhas an equal
numberof agentsof two typesandone good(income).Every agenthas the same
additivelyseparableutility function(2.3) with
The endowments
of the two typesof agentsare shownin Fig. 1: both typeshave
oneunit of incomeat nodeto, andif naturechoosesnode6, , thentype 1 agents
have1 - E for ever andtype2 agentshaveE for ever; if naturechoosesnode&;,
thenthe incomesof the two typesarereversed.The financialstructureconsistsof
two securitiesissuedat nodeto. Security1 (2) paysoneunit of incomeat node6,
(6;) and0 elsewhereand is retradedat all nodeson the upper(lower) branch
(S,(s:), t& 0.
If the securitiesare pricedat their fundamentalvalues,then the equilibrium
pricesare
andtheequilibriumallocationis
P(&)=Xi(5;)=f,
P(&)=E*([;)=l-E,
q&)=1,
I?( 5:) =Z2( &) = e, tz 2.
Tradingthe securitiesonly permitsincometransfersat date 1, the agentsremaining at their initial endowmentsthereafter.
If there is an equilibriumin which both securitieshave speculativebubbles,
then the financial marketsare complete;the allocationmust correspondto the
Arrow-Debreuequilibriumallocation
?( 50) = 1,
?(&,)=5’(5;‘)=&
tzl,i=l,2,
(6.10)
and the agents’presentvaluevectors(normalizedby F’( to) = 1) coincidewith
the Arrow-Debreu pricesZ givenby
q 60)= 1,
ts
qet)=z(5:)=21/2’
tr 1.
M. Magill, M. Quinzii / Journal of Mathematical Economics 26 (1’996)133-170
163
Sincethe pricesof the securitiesq = (q,, y2; must satisfy(6.11,we must have
at+I
6
94 Qo)= 21/2(1 +si( m,
g&d 6) = 2’/241( &+A tz 1,
S’
6 t+1
q*( 50) = $(1+4*(~~))~
21/2q2( 5/J = 21/2q2(LL
e 19
in which the initial prices of the securities(ql( &,), q2(&)) are arbitrary.5
Choosing
1+6
1
il,( 43 = &( 50) = - p/2
gives4(sI>=!T2(5/)
= 6’9
e 1. (6=11)
Theportfolios( 2’, z2) that financetheallocation(6.10)mustsatisfy z2= - z‘,
where z’ = ( zf , zl) is a solutionof
l= 1 +Q1(&Jzf&J
+~2(s&:&)~
+= 1--E+(1+4”*(t,))zf(50)-6l(h)zf(51)~
+=E+(l+~2(sI))z:(~o)-~2(51)z:(5;)1
andfor t 2 2:
3= 1 --+4”1(&)zt(5;-1)--~,(5;)zt(~t)~
f=E+iT2(5:)z:(S:-,)
so that
-~2(~:ME),
21(5,)=(1+s)zl(So)+s(f-E)1S
l-6’
-
z~(5:)=(l+s)z:(5,)-s(f-E),,.
1-S’
-
Since +( @j,C 5,) = Z( &‘)?j2(&‘>= I/ fi,
t 1 I, the transversalityconditions
are satisfied if and only if lim,,, z,‘(&)= lim,,, &i>=
0. The unique
initial portfolio for which the transversalityconditionsare satisfiedis thus given
bY
s
1
(6.12)
z”f(&-J)= -Zj(&))= --l-G2 ( 2 e1
so that the equilibriumportfolio of agent1 is given by (6.12) for t = 0 and for
tzl by
s t+ I 1
$(Et) = --= -z;( r:)*
1-s ( 2 e i
5 This is a nominul indeterminacy which has no real effects, since a cL:ge in ‘7i(&,) can always be
compensated for by resealing the portfolios zj, i = 1, 2.
M. Magill, M. Quinzii/ Journal of Muthemuticd Economics 26 11996) 133-170
164
is small (E < +), then agentsof type 1 are rich on the upperandpoor on
the lower branchof the eventtree, with the conversefor agentsof type 2. To
financea transferof incomefrom the upperto the lower branchat datezero,type
1 agentstake a long positionin security2 coveredby a short positionin security
1: they are thus debtorson the upper and creditorson the lower branch.The
situationis reversedwhen E is large(E > f), and when E= i there is no trade
sincethe initial endowmentis Paretooptimal.The securitiesenablethe agentsto
achieverisk-sharingonly if they have a positivevalueafter date 1: this in turn
arisesonly if eachagentbelievesthat other agentsbelievethat they havevalue.
Theequilibriumthusdependson the beliefsof theagents,not on the fundamentals
of the securities.
Note that the transversalityconditionsdeterminethe only valueof z’( &,) for
whichthe asymptoticpresentvalueof the debtis zero.If 2:( I&) > - [ S/(1 - 6 *)I
(3 - E), thenagentsof type 1 pay off their debtson the upperbranchtoo fast and
becomecreditorsat infinity - which cannotbe optimal; similarly, if S,‘(&) <
-[S/(1 - S2)](i - E), then agentsof type 2 becomelendersat infinity (on the
upperbranch).
If E
2. Supposethe event tret D is such that at eachnodethere are two
possibleevents,(u, d) (‘up’ and‘down’), whichcanfollow, eachwith probability
$ Theinitial nodeto is eventd. Therearetwo (typesof) agents(I = 2) andone
good(L = 1) at eachnode.Thepreferences
of agenti arerepresented
by a utility
function
Example
U’( Xi) = C (fs)l”‘ui( X'( 5 )), i = 1) 29
&D
where u, : R + -+ R, ui > 0, ul< 0, o,(O)= 0, and3 < S < 1. The endowmentof
agent1 is givenby
d-a(~),
d-p([),
if ~‘=d,Q’~[&,,
51,
o’(T)=
if t=uand ~‘=d,t/S’E[&-,,
e-1,
otherwise,
1a’,
where[ to, 5 ] denotesthepathfrom the initial node&, to node5 andwherea( )
and p(e) are givenby
l
M. Magill,
M. Quiri;.‘i/
Journul ofMathematical
Economics 26 (1996) 133-170
165
the endowmentof agent2 being given by o*( 5) = CY*+ QC*
- or( 0, V& Lb.
cu’ and ac* are chosenso that o’( ) and o*(- ) are uniformly positive.
If there is a singleinfinite-livedsecurity(a consol)with payoff A( 6, 1) = 1,
Vt E D+, then it is easyto checkthat ((Z, Z), (j5, q, (5 & ,)) is a TC equilibrium where
l
Z’(tj)=d
and Z*(t)=a*,
F(S)=l,
if5’=d,W’~[509
v@D,
51,
otherwise,
otherwise,
z’(5) = -Z’(t)
and F’(e) =7?*(c) =
, V&D.
Thestructureof theeventtree andthe priceof the consolaretakenfrom Santos
andWoodford(1993,example6), the price q being a solutionof the stochastic
differenceequation
qr=
SE(1
+4r+1
Iq),
whereF, denotesthe informationavailableat date t, which canalsobe written in
eventtree notation:
q( 6,) =$(I
+9(5‘1’:1))
wq
+4(cLI)
(6.13)
( 6,‘: I, 6:: , ) denotingthe successorsof 6,. The price q is the sum of two
components:the first is the findamen,+al value y’(5) = S/( 1 - S), Vt E D, and
the secondis a stochastic bubble that growsat the rate (2/8) on the lower branch
[d, d, . . . j and burststhe first time that event u occurs.To compensatefor the
probabilityof bursting, the return on the lower branch [cd,d, . . . ] exceedsthe
implicit risklessreturn 1/S.
Note that sincethe allocation(Xl, X2)is Paretooptimal,the equilibriumwould
not be affectedif the securitystructurewere modified to includethe short-lived
risklessbond at eachnode (AssumptionB6): the sameequilibriumwould be
obtainedwith zero tradein the risklessbond.However,the equilibriumallocation
(El, X2)cannotbe obtainedif the consolis pricedat its fundamentalvalue4: the
only portfolio 2 = ( Z’, 2*) that would permit the agentsto financethe allocation
(X’, X2) under price Zj can be computedby forward inductionand it is easyto
checkthat the transversalityconditions(for example,on the subtreethat begins
with event u at date 1) do not hold for Z.
M. Magill, M. Quinzii/Journal of Muthemutical Economics 26 (1996) 133-170
166
Acknowledgements
We aregrateful to ManuelSantosandMike Woodfordfor helpfuldiscussions.
Researchsupport from the DeutscheFiirschungsgemeinschaft,
G.W. Leibnitz
Forderpreis,duringBoWo 1992andfrom the NationalScienceFoundationGrant
SS-8911877are gratefullyacknowledged.
Appendix
Proof of Theorem 5.1 Let (D, 2, cr),(J, (S(j)>j,
(J(t$)& ,& be the fixed
characteristics
of the economyand let A * CA be the set of commoditypayoff
processes
for which the economyhas a TC equilibrium.For 5 E D let j( 5 ) =
# J( 5 ) andlet a( 5 ) = min(j( 5 ), b( 5 )). Let us considera payoff processAE A
of Zm(D, 2 , W, 2))
and let ((X, T), (p, ;li, (%i)i,I >) be a pseudo-equilibrium
generated by an artificial security structure (Z = (K, ( t(k)), E K,
J)
(BANKED,
f), with
ranl#‘YS’,
1, k)] 5’Ee+ =a(5),
Q&D.
Let 4’ be the vector of pricesof the original securitiesdefinedby (4.4). There
exists a vector of portfolios LE 2 such that ((E, Z), ( j& q, (5 ‘)i E,)) is a TC
equilibriumof Z$(D, 2 , w, M) if andonly if
rank[p([‘)~(&
j)+q(&,
j)],f,,+
=a([),
Q&ED.
(Al)
jEJ(t)
To provethe theoremwe showthat if (A 1) is not satisfiedfor x, thenfor every
E> 0 thereexistsa commoditypayoff processA E A with 11A - All rr:< E such
that (Al) is satisfiedfor A, so that A E A * . We show that this can be done
without changingthe underlyingpseudoequilibrium
((I& p), ( j?, p, (~‘)i EI).
Let H CA denotethe subsetof commoditypayoff processesA such that
(( 2, T), ( j!, i?, (?ji)iE ,)) is a pseudo-equilibrium
of Ea(D, & , o, ~‘1). The
payoffsin H must satisfy(4.4), i.e. for all 5 E D,
&
c ~i(Sr)F(5t)A(S’)
[‘E!l’(~)
mustbe independent
of i, andif y( 6) denotesthis commonvalue,then(4.5) must
be satisfiedwith 4 replacedby 4. Thus A E H if A E A and if for all [E D,
M. Magill.
=
M. Quinzii/Journd
of Mathematical
Economics 26 (1996) 133-170
167
5’91,k)] 5tE5+.
These two equationscan be written with date 0 discounted prices as: for all 6 E D,
c
p’( (“)A(
[“E D(6’)
1 I(
t”, j) rrE5+ c [ P’( S’)r(
(I, k)] SIEr+ .
kEK(Z)
jEJ(5)
To simplify notation, if x E/%( D X L) and P EL’,@ Y L), then we define
p**x=
c
PWb(5’)
(‘ED(()
and let
denotethe b( 5) vector of date 0 discounted values of x from all the successorsof
6 onwards. For A E A we define
P**A=(PD*A(=,j),jEJ(r)),
P 0 (+A = [ P l t0 A( , j)] gEt+
l
l
Then H can be written as
H=
I@$+A)
cp(
0,
@ED
where
Since 6~ H, H is not empty. Since for each 5 E D the second condition
( - ) Cp( 8) can be expressed by a system of linear equations. H is the
intersection of a countable collection of closed hyperplanes. Thus H is a closed
subset of A. Since A is a closed subset of the Banach space lz( D X L X J), it
follows that H is a Baire space (Rudin, 1973, p. 42, theorem 2.2).
168
M. Mugill, M. Quinzii / Journal
of Mathematical Economics 26 (1996) 133-170
For eachnode 5 let J^(6 ) be a subsetof J( 5 > consistingof a( 5 ) securitiet
includingthe short-livednumerairebfnd. For a payoff process A E A let A
denotethepayoffsof the securitiesin J( 5). Let & be a subsetof a( 5) nodesof
6’ (to be chosenbelow).Considerthe subsetof H:
Ht=
AEHldet[p’O&]
=O}.
We show that Ht has an empty interior in H. It sufficesto show that we can
perturbanyAEH*, A+A+AA,with
A+AAEHinsuchawaythat
det[P’O$(A+dA*)]
#O.
(A21
ConsiderchangesAA in the commodity payoff processconsistingsolely of
changesin the amountsof commodity1 which canbe decomposed
asfollows:
AA(-,
j)
c cd5’et’1 + C @ef’+ yjef,
= 5’~j
0 s’\s^’
if jE?(
t),
W)
if jEP( 5).
’
10,
whereJ^I<6 ) is the subsetof J^(5 ) which excludesthe short-livednumerairebond.
Note that the commoditypayoffsare perturbedonly at node 6 and its immediate
successors5’ and only securitiesin .?( 6 ) have their payoffs perturbed.For
securityj E J^,(6 >, its payoff in good 1 is perturbedby (Y/, at node6’ E &, by
#. at node5’ E ef\ p andby yj at node6. For brevity we write
j&(
a!= ( (y/,,
.
0,
5 E (+) E ~(a(t)-l)a(4),
pzz (&,,
t),
(f’E S+\&) E ~("(l)-IXb(t)-40)
jE$(
y= (yj, jES(
6)) E lf?P)-'.
ar canbe chosenso that (A2) is satisfied.To seethis, let g : lR(“(E)-‘M* ) + k! be
definedby
g(a)=det[P’O&+AA?],
03
where A A is defined by (A3). While AA is a function of (cy, p, y 1, the
determinantin (A4) dependsonly on CLA straightforwardbut tediouscalculation
showsthat g haspartialderivativesof ordera( 6 ) - 1 evaluatej at ac= 0, which
are not zero since P( t’, 1) > 0, Vt’ E &. Thus g is not locally constantin a
neighborhood
of ar= 0. It follows that thereexists a! arbitrarily small suchthat
gWf O.
If a( 5 ) < b( 6 ), then /3 is chosento ensurethat
M. Magill,
M. Quinzii/Journul
of Mathemutical
Economics 26 (1996) 133-170
169
Since p( c ) is a non-trivial subspaceof IL!b(6), there is a choice of a( 6 ) nodes
PCs’
and an appropriate ordering of the nodes such that L?( 6 ) can be
representedby a system of b( 6 ) - a( 0 equations of the form
The nodes are ordered so that the subset [+\ & constitutes the first b( 5) - a( 5 )
nodes. I is the (b( 6 ) - a( 5 )) X (b( p ) - a( 5 )) identity matrix and E is a
(b( 5 ) - a( 5 )) X a( 5 ) matrix (see Magi11and Shafer, 1991, p. 1544). For each
j E .?( 5 ), once (x has been chosen, there is a unique vector pi = ( @rT 5’ E c+\
p)ER b(*)- ‘(6) such that the vector
satisfies the equation [I 1E]d = 0.
If a(&)= b(t), p(t)=
Rib(*) so that (A5) is automatically satisfied.
We have to ensure that ( P’ 0 z+( A + AA)) CL?( g) is satisfied for all nodes
[S 5. By construction, this inclusion holds for all nodes 6 which are not on the
path from &-, to 5. If yj is chosen so that
for j&‘(t)nJ(t-),andif
yj = 0 for j GE.?( 6) n J( r-), then (A5) is satisfied
for all nodes & D. Thus A + AA satisfies the subspacerequirements(the second
condition) in the definition of . To show that the first condition in the definition
of H is satisfied it only remains to show that if Aq is defined by
then for all i E I,
q
1
C
i?‘( (‘)jj( [‘)AA( 5’, j) = Aq( 6, $
0 pEo+< I>
VjEJ([),V&D,
so that all agents agreeon the induced changes in the security prices. Since AA
has only a finite number of non-zero terms this follows from ( P’ 0 g+A A) C
2?( f>, v& D.
Since cr can be chosen to be arbitrarily small and since ( p, y ) are deduced
from a! by linear relations with bounded coefficients, the perturbation AA can be
made arbitrarily small. Thus for all 5 E c”, Hs is closed and has an empty interior
is a Baire space, the countable union IJ 5 E DH6 has an empty
in H. Since
170
M. Magill, M. Quinzii / Journal of Mathematicd Economics 26 (1996) 133-l 70
interior in H. Thus for all E > 0 there exists an E-perturbation A of x such that
AEHand
(Pq+A)
=ei?( t),
which completes the proof.
WED
J
,
0
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