When Optimists Need Credit: Asymmetric Disciplining of
Optimism and Implications for Asset Prices
Alp Simsek
MIT May 30,
2010
Abst ract Heterogene ity of b e lief s h as b een su
gge sted as a ma j or contrib uting f actor to th e rec ent
n ancial cris is . T his p ap er th eoretic ally evaluates th is hyp oth esis. Similar to
Geanakoplos (2009), I assu me th at op timists have limite d wealth and take on leverage in
orde r to take p osition s in lin e with the ir b e lief s. To have a signi c ant e /ect on ass et
prices, the y nee d to b orrow from traders with mo d erate b elie fs u sing loan s collateraliz
ed by the as set its elf. Sin ce mo de rate len de rs do not value the collateral as much as
optimis ts do, th ey are re luctant to lend , which provide s an en dogen ou s con straint on op
timists ability to leverage and to in uen ce as set p rices . I de monstrate th at op timism c
once rn in g th e like liho o d of b ad eve nts h as no or little e/e ct on ass et prices b ec au se
th ese typ es of optim is m are discipline d by this con straint. Instead , optimism conc erning
th e re lative likeliho o d of go o d events cou ld have s ign i cant e /ects on as set p rices .
This asymmetric disciplining of optimism is rob us t to allowing f or s tate c ontingent loans
and s hort s ellin g of the ass et. Th ese res ults emp hasize that what inve stors d is agree ab
out matte rs f or as set pric es, to a greater exte nt than th e leve l of d is agree ment.
I th en us e a dyn amic extens ion to show how the asymmetric d is ciplining res ult inte
racts with th e sp e culative c om p onent of as set pric es ide nti ed in Harrison and Kreps
(1978). W hen optimis ts h ave lim ited we alth, b elief he te rogen eity can lead to sp e
culative asse t pric e bu bbles b ut only if it c once rn s th e re lative likeliho o d of go o d
events. The asymme tric disc ip lin in g re sult shows that th e s iz e of the bub ble d ep e nds
on the typ e of b elief heterogen eity, an d that b ubb les c an c om e to an e nd b ec au se of
a s hift in b elief heterogene ity towards th e likelih o o d of b ad eve nts.
C o nt a c t i n f o r m a t i o n : a l p s t e i n @ m i t . e d u . I a m g r a t e f u l t o my a d v i s o r , D a r o n A c e m o
g l u , f o r i nva l u a b l e g u i d a n c e i n t h i s p r o j e c t a n d nu m e r o u s s u g g e s t i o n s t h a t s i g n i c a
nt l y i m p r ove d t h e p a p e r . I t h a n k A y t e k E r d i l a n d M u h a m e t Y i l d i z f o r nu m e r o u s h e l p f u
l c o m m e nt s . I a l s o t h a n k M a r i o s A n g e l e t o s , R i c a r d o C a b a l l e r o , M e l i s s a D e l l , G u i d
o L o r e n z o n i , Fr e d e r i c M a l h e r b e , I va n We r n i n g a n d t h e s e m i n a r p a r t i c i p a nt s a t t h e U
n i ve r s i ty o f C a l i f o r n i a B e r ke l e y, B r ow n U n i ve r s i ty, t h e U n i ve r s i ty o f C h i c a g o , C o l u mb
i a U n i ve r s i ty, t h e E u r o p e a n U n i ve r s i ty I n s t i t u t e , H a r va r d U n i ve r s i ty, t h e I nt e r n a t i o n
a l M o n e t a r y Fu n d , M I T , N o r t hwe s t e r n U n i ve r s i ty, N e w Yo r k U n i ve r s i ty, t h e U n i ve r s i ty
o f Pe n n s y l va n i a , P r i n c e t o n U n i ve r s i ty, S t a n f o r d U n i ve r s i ty, To u l o u s e U n i ve r s i ty, t h e
U n i ve r s i ty o f Wa r w i ck , Ya l e U n i ve r s i ty f o r h e l p f u l c o m m e nt s . A l l r e m a i n i n g e r r o r s a r
emine.
1
1 Intro duction
Be lief he te rogen eity and optim is m have b ee n s uggeste d as contributing f actors to th e
rec ent n anc ial c risis. Sh iller (2005), Reinh art an d Rogo/ (2008) and Gorton (2008), along
with many other comme ntators, have identi ed th e optim is m of a fraction of inves tors as a p
ote ntial cau se f or the in cre ase in p rice s in the hous in g and th e comp le x s ecu rity
markets in th e run -up to the cris is . As n ote d by Gean akop los (2009), for th e optimism of a
fraction of inve stors to have a signi c ant e /ec t on asse t prices , they nee d to le verage the ir
inves tme nts by b orrowing f rom less optimis tic inves tors- f rom mo d erate len de rs . Most b
orrow in g in nan cial markets is collate ralized , and optimists often us e th e ass et itse lf as
collateral (e.g., mortgages, REPOs, or ass et p urch as es on margin). This re prese nts a p uzz
le b e caus e m o d erate lend ers do n ot value the collate ral (the ass et) as much as op tim is
ts d o, which might make th em re lu ctant to le nd . Pu t d i/ere ntly, b elie f hete rogen eity
implies an en dogen ou s cons traint on op timis ts ability to leverage an d to in uen ce asse t
pric es .
The pu rp os e of this pap er is to und erstand th e implication s of th is cons traint f or ass et
prices . I con stru ct an equilib riu m m o d el in th e asse t an d the loan marke t, an d I show th
at ce rtain typ e s of op timism, s p eci cally th os e conc ern ing the likelih o o d of bad events,
have n o or little e /ec t on ass et p rice s b e caus e th ey are disc ip lin ed by th e e ndogen
ous n ancial con straints . Ins tead, optimis m con cerning th e relative likelih o o d of go o d
events cou ld have signi c ant e/ects on as se t pric es, b e caus e th ese typ es of op timism are
un ch ecke d by thes e c onstraints.
1To
illus trate th e e/ect of di/e re nt typ es of optim is m, c on side r a simple exam ple in which
a single ris ky ass et is traded . The re are three futu re s tate s, go o d , normal an d b ad , in wh
ich the asse t pric e will res p ec tively b e high, average and low. M o d erate lend ers ass ign
an e qu al probab ility, 1 =3 , to each state , wh ile optimists have a greater exp e cted valuation
of the ass et. Optimis ts b orrow from m o d erate lend ers u sing loans collate ralized by the
asse t. M ore sp e ci cally, the ass et an d collate raliz ed loans are trade d in a comp e titive
marke t (as in Gean akop los , 2009), and loan s are n o-re course in the se nse that payme nt is
on ly e nf orce d by the c ollateral ple dged for th e loan. Loan s of di/erent sizes (p er un it c
ollateral p le dged ) are available f or trad e, and the loan to valu e ratio is end ogenou sly
determine d in equilib rium. For th e b as eline se tting, su pp ose that loans are n on-c
ontingent, th at is , the y promise the same payme nt in all f uture s tate s, an d th at the as se t
cann ot b e short sold. Thes e as sump tion s arguably p rovide a go o d starting p oint, b e caus
e c ollateralized loans (e .g., mortgages, REP Os ) typ ic ally do n ot have m any c ontinge ncies
in the ir payo/s ; an d short s ellin g of many ass ets other than sto cks (and some sto cks) is d i¢
cu lt an d c os tly.
In th is setting, th ere are two di/e re nt ways in which optimis ts can b e optimistic ab out the
ass et. For the rst case , s up p ose optimis ts as sign a probab ility less th an 1 =3 to the b ad
state , and e qu al probabilities to the n ormal and th e go o d s tate s. That is, optim is ts are op
timistic
1Fo
r a d e s c r i p t i o n o f t h e s h o r t m a r ke t , s e e , f o r e x a m p l e , J o n e s a n d L a m o nt ( 2 0 0 1 ) ,
D A vo l i o ( 2 0 0 2 ) , D u ¢ e , G a r l e a nu , a n d Pe d e r s e n ( 2 0 0 2 ) , O f e k a n d R i ch a r d s o n ( 2 0 0
2 ) , L a m o nt a n d S t e i n ( 2 0 0 4 ) , a n d A s q u i t h , Pa t h a k , a n d R i t t e r ( 2 0 0 5 ) .
2
2b
e caus e the y think b ad e ve nts are un likely. For the sec ond case , su pp os e op timists
agree ab ou t the probability, 1 =3 , of the bad state, but th ey think the go o d state is more
likely th an the normal state. M ore over, c on struct the two cas es s uch that optimists h ave the
same valu ation of the asse t, s o that th e level of the optimis m is the same bu t the ty pe of the
optimis m is di/e re nt.
My main re su lts, Theorem 1 and Th eorem 3, sh ow that th e asse t p rice in th e rst case
of this example is always lowe r than in the se cond case (and s trictly so for the ap propriate ran
ge of param eters). In oth er words, op timism is asymmetrical ly disciplined by e nd oge nous n
anc ial cons traints: op timism con cernin g the p rob ability of bad states is d is ciplin ed m ore
than optimis m c on ce rn in g the relative like lih o o d of go o d states .
More gene rally, The ore m 1 c onsid ers the ab ove se tting with a c ontinu um of s tate s (rath
er than th ree ), an d sh ows that th e asse t is p rice d according to a mixtu re of m o d erate
an d optimis tic b elie fs : the m o d erate b elie fs are u sed to ass ess the likelih o o d of def
ault state s, wh ile the op timistic b e lief s are us ed to asse ss the cond itional likeliho o d of
non -de fault states. M ore prec is ely, the ass et price can b e written as:
p= 11 + r (Prm o d e r [v< v] Em o d e r a [vj v< v] + Prm o d e r a t [v v] Eo p t i m i s [vj v v]) ,
ate
te
e
tic
(1)
wh ere ris th e interes t rate on a b en ch mark asse t, th e ran dom variable vcap tu re s the fu
ture value of the asse t, an d v is th e e nd oge nous ly dete rmine d def ault thres hold valu e,
th at is, collate ralized loans in this ec onomy de fault whe n the as set value vfalls b e low v.
The notation Prm o d e r a t e[v< v] cap tu re s the probab ility of the eve nt fv< vg with re sp ec t to
th e mo d erate b e lief s, an d Eo p t i m i s t i c[vj v v] cap tu re s the e xp ected value of th e asse
t cond itional on the eve nt f v vg with resp ect to th e optimistic b e lie fs.The express ion in (1)
fu rth er illustrate s that optimism is asymmetric ally discip line d. This asymme tric d is ciplining
res ult is robus t to allowing f or more gene ral c ollateralized loans an d sh ort se lling. In p
articular, S ections 5 an d 6 of this pap e r s how that the ass et price in the se more gen eral s
ettin gs can als o b e repres ented with an express ion similar to (1). While th e de tails of th e e
xp re ssions dep end on the typ e of th e contracts available f or trade, it remains tru e th at
optimism ab ou t bad s tates is discip line d more than optimis m c once rn in g the relative like
lih o o d of go o d states.
The intuition f or the as ymm etric d is ciplining re sult is related to the asymme try in the
shap e of the de bt c ontrac t payo/s . The se contracts make th e same fu ll p ayment in nondef ault states , bu t the y make los se s in d efau lt states. Conse qu ently, any disagreem ent
ab out the probab ility of d efau lt states tran slates into a disagre eme nt ab out how to valu e th
e de bt contracts, wh ich
2Fo
r a n e x a m p l e o f c a s e o n e ty p e o f o p t i m i s m , c o n s i d e r t h e l a s t q u a r t e r o f 2 0 0 8 , w h e
n a m a i n d i m e n s i o n o f d i s a g r e e m e nt wa s w h e t h e r t h e u p c o m i n g r e c e s s i o n wo u l d b e a
d e p r e s s i o n o r a g a r d e n va r i e ty r e c e s s i o n . Fo r a n e x a m p l e o f c a s e two ty p e o f o p t i m i s
m , c o n s i d e r t h e I nt e r n e t t e ch n o l o g y a n d t h e t e ch s t o ck s i n 1 9 9 0 s , w h e n a m a i n d i m e n
s i o n o f d i s a g r e e m e nt wa s h ow p r o t a b l e t h e I nt e r n e t t e ch n o l o g y wo u l d b e .
3
in turn tighte ns optimists nan cial cons traints . In contras t, disagreem ents ab ou t the relative
like lih o o d of n on -d efau lt s tate s d o not tighten the n an cial cons traints .
More sp e ci cally, in th e ab ove exam ple (f or the relevant ran ge of parame ters ) collate
ralized loans th at are trad ed in equilib rium def ault in the b ad s tate b ut n ot in th e normal or
the go o d states . Th is im plie s that th ese loan s always trade at an interes t rate with a
spread over the b enchmark rate , which comp en sates the le nde rs f or exp e cted loss es in
case of def ault. Moreove r, in a comp etitive loan market, th e spread on a loan is j ust enou gh
to comp e nsate the len de rs f or the ir exp ec ted losse s acc ordin g to their m o d erate b elie
fs . None theles s, in the rst cas e of the exam ple, th is spread ap p ears to o h igh to op
timists. This is b ecau se optimists ass ign a lowe r p rob ab ility to the bad s tate , and thus th
ey n d it m ore likely that the y will pay the sp re ad. The re fore , optimis ts b elieve th ey w ill p
ay a higher e xp ected interest rate than the b en ch mark rate , wh ich d iscou rages them f rom
b orrowing and leveragin g th eir inves tme nts . Th is lowers optimists dem and f or the asse t
an d lead s to an equilib rium pric e close r to the mo de rate valu ation. In contrast, in th e s
econ d c ase of the example , the spread app e ars fair to optimists b ec ause th ey agree ab out
th e probability of the bad s tate . This en cou rages optimists to b orrow and leverage their
investme nts , in crease s their d eman d for th e asse t, and lead s to an equilib rium pric e c los
er to th eir valu ation .
The asymmetric d isciplining charac terization of ass et prices len ds itse lf to a numb er of
comp arative statics res ults regard ing the e/e ct of a ch ange in the level and the ty pe of b elie
f he teroge neity. E arlier work by Mille r (1977) has sugges ted a lin k b e twee n the le ve l of b
elief he teroge neity and as set p rice s. Acc ord in g to th is mechanism , b e lie f he teroge neity
and limited sh ort s ellin g le ads to an ove rvaluation of the asse t (relative to the average
valuation of the p op ulation) b ecau se th e as set is he ld by th e mos t optimis tic inve stors.
This m ech anism has b e en rec ently emp hasize d an d em pirically tes ted by a growin g
literatu re in n ance , e.g., Ch en , Hong an d S te in (2002), Diethe r, Malloy an d S ch erbina
(2002 ) and Of ek and Rich ard son (2003). In c ontrast to th is literatu re , th e level of b e lief he
te rogen eity in th is mo del h as ambiguou s e/ects on th e as set p ric e. This is b ec ause , wh
ile an in crease in op timists op timism tend s to in cre ase the p rice , an inc re as e in mo de
rate lend ers p es simis m tend s to d ec reas e the p rice th rou gh the tighten in g of nan cial
con straints . Th is ob servation sugges ts th at the Miller mechan is m m ay not ap ply in
markets in which optimists nan ce th eir as set purchase s by b orrowing f rom le ss optimistic
inve stors .
In c ontras t, this mo del su gge sts that th e typ e of the b e lie f he te rogen eity may b e a
more rob us t de termin ant of as se t prices in thes e markets . To c apture th e e/e ct of d i/ere
nt typ es of b e lief he te rogen eity, I f ormally d e ne a n otion of right- skewe d (resp . leftskewe d) optimism as a s in gle -crossin g con dition on th e hazard rates of op tim is ts b elief
dis tributions . In the ab ove de scrib ed e xamp le with th re e states, th e optimis m in th e se
cond c ase is more right-s kewe d than the op timism in th e rst case . Th is is b ec au se , in th
e se cond c ase, th e optimis tic h azard rate at th e bad state is higher (since op timists are not
op timis tic ab out the p robab ility of th e bad state), bu t the hazard rate at the normal state is
lowe r (s in ce optimists are optimistic ab ou t
4
the re lative likelih o o d of the go o d and th e n ormal s tates ). Th eorem 3 s hows that an in
cre ase in th is typ e of right-s kewne ss of b e lief he te rogen eity un ambiguou sly inc re ases
the as set p rice , b e caus e a given level of op timism is d is ciplined les s by nanc ial c on
straints wh en it is m ore right-skewed . In ad dition , The orem 4 sh ows that th e level of b elief
h eterogene ity also has an un amb igu ou s e/ect on the ass et pric e if the typ e of th e he te
rogen eity is also accou nted for. In partic ular, in re sp on se to an in crease in b elief he te
rogen eity, the as set p ric e inc re ases if the add itional hete rogen eity con cerns th e re lative
likeliho o d of non -de fault states, wh ile it d ec reas es if th e ad dition al hete rogen eity conc
erns th e prob ability of de fau lt s tate s. Thes e res ults s ugges t that what investors disagree
about matters f or asse t prices , to a gre ate r extent than th e le vel of th eir disagreeme nt.
While the b aseline mo del with n on -contingent contracts an d n o short selling is a go o d
startin g p oint, it is imp ortant to ve rify th e rob ustn ess of the asymmetric d is ciplining ch
aracterization to more ge neral s ettin gs, esp e cially b ecau se allow in g f or a rich er s et of
contracts intro d uc es new econ om ic force s. Theorem 5 shows that a ve rs ion of th e as
ymme tric disciplining resu lt continues to ap ply in th e settin g in which d eb t contracts can b e
fu lly c ontingent. The optimal c ontingent c ontract (c ollateralized by on e un it of the as set) is s
uch that op timistic b orrowers give up th e ass et complete ly if the state re alization is b elow a
thres hold level, wh ile paying nothing if the state is ab ove the thresh old . While this thresh old
c ontrac t is d i/erent than a n on -contin gent c ontract, it has the same f eature of makin g a xe
d payme nt (name ly, ze ro) for all relatively go o d states. C on sequ ently, op timism ab ou t the
relative like lih o o d of go o d state s do es not le ad to h eteroge ne ity in the valuation of the op
tim al contin gent c ontrac t. It follows th at th ese typ es of optimis m do n ot tighte n op
timists nanc ial con straints, an d thus the y lead to a highe r asse t p rice . In contras t,
optimism ab out the relative like liho o d of states b e low the thres hold le vel tighten s optim is
ts nanc ial c onstraints and leads to a lower as se t price.
3The
setting with c ontingent contracts reveals a surprising re sult: the e qu ilibrium ass et price
can excee d th e valu ation of e ven the most optimistic investor.Intu itively, the ability to n e-tu
ne the ir b orrowing en ab les optimists to take loan s which they p erceive to b e even m ore f
avorable than b orrowing at th e b en ch mark inte res t rate. Optimis ts conc entrate all of th eir
payme nts at the bad s tates (wh ich th ey nd the leas t likely), and thus they exp ec t to make a
relatively sm all p ayme nt. Con seque ntly, optimists c ontinue to deman d th e ass et wh en the
price exce ed s th eir valuation (wh ich is calcu lated ac cord in g to the b e nchmark rate), b
ecau se the y n ance s ome of the p urch as e with continge nt contracts wh ich they p erce ive
to b e very attractive. Th is res ult creates a p res ump tion that ner leve ls of n ancial en gin
eering of loans can p otentially have a large imp ac t on as set p rice s.
Anothe r natural ques tion is wheth er the asymme tric discip linin g re su lt gene raliz es to
the
3A
s d i s c u s s e d b e l ow , i t i s we l l k n ow n t h a t t h e a s s e t p r i c e c a n e x c e e d t h e o p t i m i s t i c
va l u a t i o n i n a d y n a m i c s e t t i n g w i t h b e l i e f h e t e r o g e n e i ty ( t h r o u g h a d i /e r e nt m e ch a n
i s m ) . H owe ve r , w h e n c o nt i n g e nt c o nt r a c t s a r e ava i l a b l e , t h e a s s e t p r i c e c a n e x c e e d t
h e o p t i m i s t i c va l u a t i o n e ve n i n t h e s t a t i c s e t t i n g .
5
class of asse ts that can b e short s old (e.g., the m a jority of s to cks ). Wh ile s hort se lling
redu ce s the ge neral overvalu ation of the as set, Th eore m 6 sh ows that a version of as ymm
etric dis ciplin ing ap plies also in th is c ase. The c ritic al obs ervation is that m o d erate inve
stors th at wish to s hort se ll the asse t f ace an end ogenou s b orrowing cons traint similar to th
e one fac ed by optimis ts. In particu lar, to short s ell the asse t, mo de rates ne ed to b orrow
the ass et f rom optimists. Moreover, sinc e b orrowin g in th is e conomy is collate ralized , th ey
n eed to u se the ris kles s b on d that the y h old as collate ral in their sh ort contracts. But
optimists value the b on d (the collateral) re latively le ss, which m ight make th em relu ctant to
len d th e as set. He nc e, b elief he teroge neity repres ents an end oge nou s con straint on mo
d erates ability to sh ort s ell.
The seve rity of this con straint d ep e nds on the typ e of b elief h eterogene ity. If the fu tu
re state is ab ove a thres hold level, then the value of the asse t is greater than th e value of th e
p os ted collate ral, and mo d erate s de fault on the s hort contract. Hen ce, a sh ort contract
e/ectively promises the s ame payme nt in all s tate s ab ove a thres hold state. Con se qu ently,
if the b elief he teroge neity is ab ou t th e relative like lih o o d of s tates ab ove the th re shold,
the n mo derates can not b et on the ir p ess im is m by s elling the sh ort contract. Pu t di/e re
ntly, to b et on the se typ es of p es simism, m o d erate s ne ed to cho ose a sh ort contract with
a highe r leve l of collateral (that has a highe r de fau lt thresh old ). He nce , th ese typ e s of sh
ort sales are more di¢ cu lt to leve rage, which le ads to an ass et p rice close r to the optimistic
valu ation. In contras t, if the b e lie f hete rogen eity is ab out the probab ility of b ad state s, th
en mo de rates are ab le to make le verage d b ets on the ir p e ssimism, which lead s to an ass
et price closer to th e mo de rate valuation .
While the res ults de scrib ed so far conc ern a static se tting, th e asymme tric disc ip lin in g
me chan ism naturally inte racts with th e sp e culative c om p onent of as set pric es identi ed in
Harrison and Krep s (1978). I c onside r a dynamic extens ion of the bas elin e m o d el to
analyze this inte raction. In a dynamic e conomy in which the ide ntity of optim is ts ch an ges
ove r time, a sp ecu lative p hen omen on obtains as the current op timists pu rchase the as set n
ot only b ecau se the y b elie ve it will yie ld gre ate r d ivide nd retu rn s, bu t also b ecau se th
ey exp ec t to make capital gains by se lling th e as set to f uture op timis ts . The asse t pric e e
xc eed s th e p re sent disc ounted valuation of the as set with re sp ec t to the b e lie fs of any
trader b ec ause of the resale op tion value intro d uc ed by the sp ecu lative trading motive . As
S ch einkman and Xiong (2003) n ote , this res ale option valu e may b e reason ab ly called a s
p ecu lative b ub ble. This se tu p is the startin g p oint of the d yn amic exten sion, which intro
duc es the add itional elem ent of op timists n anc ial con straints . Th e d yn amic mo del
reveals th at, whe n optimists n eed to pu rchase the ass et by b orrowin g f rom mo de rate len
ders, b elie f he te rogen eity c an le ad to sp ecu lative as se t price bu bb le s, bu t only if it c
once rn s th e re lative likeliho o d of n on-d ef ault states . Wh en this is the case , however, the
res ale option value c an inc reas e the size of the sp e culative comp on ent of the ass et pric e
cons id erably b e caus e large p osition s can b e n ance d by cred it collateraliz ed by the sp ec
ulative as set. This is b e cause mo d erate le nde rs valuation, as well as op timists valuation
, features a sp e culative c omp one nt. Put di/erently, in a sp e culative e piso de , mo de rate
6
len ders agree to nanc e op timists pu rchase of the ass et by extend ing large loans b ec
ause the y think, sh ould the op timist de fault on the loan, the y can s ell the collateral (the as
set) to anothe r optimist in the n ext p erio d. T he asymme tric d is ciplin ing ch arac te rization
sh ows that the size of th e bub ble dep end s on the skewn es s of b elief he te rogen eity. Th is
res ult als o sh ows th at bu bb le s can c ome to an e nd b ecau se of a sh if t in b elie f he te
rogen eity towards th e like liho o d of de fau lt s tate s.
4The
close st work to my pap e r is by Gean akop los (2009), who c on side rs the dete rmination
of leverage an d ass et p rice s in a mo d el with two continuation states and trade rs with a c
ontinu um of b e lief typ e s. In c ontras t, I con sider a mo de l with a continu um of continuation
s tate s an d trad ers with two b elief typ e s (optim is ts and mo d erates ). My assu mptions are
relevant f or u nde rs tan ding a range of situations, includ ing th e e/e ct of d i/erent typ es of b e
lief disagre eme nts on asse t prices , leverage, and the de fault f re qu enc y of equ ilibrium loan
s. In p articular, wh ile Geanakoplos (2009) illustrate s that an in crease in b elief hete rogen eity
c an d ecrease asse t pric es con siderably, my p ap er shows that an increas e in the le ve l of b
elief heterogene ity gen erally has amb igu ous e/e cts on as set pric es , and identi es the s
kewne ss of b elief he teroge neity as an imp ortant de term in ant of ass et prices . In th e m o d
el con sidered by Ge an akop los (2009), the inc reas e in th e le ve l of he teroge neity d ecreas
es asse t p rice s b ecau se the add itional hete rogen eity is conc entrated on d ef ault s tates .
An increas e in th e level of hete rogen eity in that mo d el would rath er inc re ase asse t p rice s
if th e ad dition al he teroge neity were conc entrated on go o d states . Moreove r, in the two s
tate mo del an alyze d in Ge an akoplos (2009), loans that are traded in equilibriu m are always f
ully s ecu re d with re sp e ct to the wors t c ase s cen ario, i.e., th ere is no de fau lt. Th is fe atu
re makes it imp os sible to an alyze the e/ec t of b elie f h eterogene ity on the de fau lt f re qu
enc y an d riskin es s of equilib rium loans , wh ich is one of the top ics th at I cons id er. In add
ition, my p ap er exten ds the mo d el in Gean akop los (2009) by allowin g for s hort s ellin g,
and charac te rize s th e e /ect of b e lief he teroge neity in th is more gen eral setting.
5The
relations hip of my pap e r to the lite ratures in itiated by Miller (1977) and Harris on and
Kre ps (1978) have already b ee n disc us sed .A re lated literatu re c once rn s the p lau sibility
of the he te rogen eous priors assu mption in nan cial markets . Th e m arket s election hyp oth
es is , wh ich go es b ack to Alch ian (1950) and Fried man (1953), p osits that inve stors w ith
inc orrec t b e lief s s hou ld b e drive n ou t of th e market as the y would cons is tently lose
money. Thus, this
4O
t h e r r e l a t e d p a p e r s t h a t c o n c e r n t h e e /e c t o f c o l l a t e r a l c o n s t r a i nt s o n l e ve r a g e o
r a s s e t p r i c e s i n c l u d e H a r t a n d M o o r e ( 1 9 9 4 ) , G e a n a ko p l o s ( 1 9 9 7 , 2 0 0 3 ) , K i yo t a k i
a n d M o o r e ( 1 9 9 7 ) , C a b a l l e r o a n d K r i s h n a mu r ty ( 2 0 0 1 ) , G r o mb a n d Vaya n o s ( 2 0 0 2 ) ,
Fo s t e l a n d G e a n a ko p l o s ( 2 0 0 8 ) , B r u n n e r m e i e r a n d Pe d e r s e n ( 2 0 0 9 ) , B r u n n e r m e i
e r a n d S a n n i kov ( 2 0 0 9 ) , A s h c r a f t , G a r l e a nu , a n d Pe d e r s e n ( 2 0 1 0 ) , a n d H e a n d X i o n
g(2010).Mypaperisalsorelatedtoalargeliteraturethatconcernstheendogenou
s d e t e r m i n a t i o n o f l e ve r a g e . I n a d d i t i o n t o s o m e o f t h e a b ove p a p e r s , a n i n c o m p l e t e
l i s t i n c l u d e s Tow n s e n d ( 1 9 7 9 ) , M ye r s a n d M a j l u f ( 1 9 8 4 ) , B e r n a n ke a n d G e r t l e r ( 1 9 8
9 ) , S h l e i f e r a n d V i s h ny ( 1 9 9 2 ) , H o l m s t r o m a n d T i r o l e ( 1 9 9 7 ) , B e r n a n ke , G e r t l e r , a
n d G i l ch r i s t ( 1 9 9 8 ) . O n t h e e m p i r i c a l s i d e , a nu mb e r o f r e c e nt s t u d i e s d o c u m e nt t h e
va r i a t i o n i n l e ve r a g e a n d i t s e /e c t o n a s s e t p r i c e s ( s e e , f o r e x a m p l e , A d r i a n a n d S h i
n,2009).
5T h e r e i s a l a r g e l i t e r a t u r e t h a t a n a l y z e s t h e e /e c t o f h e t e r o g e n e o u s b e l i e f s o n a s s
e t p r i c e s o r b u b b l e s . I n a d d i t i o n t o t h e a b ove m e nt i o n e d p a p e r s , a n i n c o m p l e t e l i s t i
n c l u d e s Va r i a n ( 1 9 8 5 , 1 9 8 9 ) , H a r r i s a n d R av i v ( 1 9 9 3 ) , A l l e n , M o r r i s , a n d Po s t l e wa i
t e ( 1 9 9 3 ) , D e t e m p l e a n d M u r t hy ( 1 9 9 4 ) , Wa n g ( 1 9 9 4 ) , M o r r i s ( 1 9 9 6 ) , K a n d e l a n d Pe
arson(1995),Zapatero(1998),Basak(2000),HongandStein(2003),Gollier(200
7),Banerjee(2007).
7
6hyp
oth esis su gge sts that investors th at re main in the long ru n shou ld h ave ac cu rate (an
d common) b elie fs. Rec ent res earch has e mph asized that the market se le ction hyp oth esis
d o e s n ot app ly for inc om plete markets , th at is, trad ers with inacc urate b elie fs may h ave
a p e rman ent pres enc e w hen ass et markets are in complete .7Of p articular interes t for my
p ap er is th e work by Cao (2009), who c on side rs a s imilar econ omy in which marke ts are e
nd oge nou sly inc omplete b e caus e of collate ral c onstraints. Cao (2009) s hows that b e lie f
h eterogene ity in this ec on omy remains in the long run, thus provid in g theoretical su pp ort
for my ce ntral assu mptions. Anothe r strand of literatu re con cerns wh ether investors Baye
sian learning dynamics would eve ntually lead to ac curate, an d thus common, b elie fs . Rece
nt work (e.g., by Acem oglu, Chernoz hu kov and Yild iz , 2009) h as em phas iz ed th e
limitations of Baye sian learning in ge neratin g lon g ru n agre eme nt.
The organ ization of the rest of th is pap er is as f ollows. S ection 2 intro d uce s th e base
line version of the mo de l an d de ne s the collate ral equilib rium. Se ction 3 charac te rize s the
collate ral e qu ilibriu m an d p re sents th e asymme tric d is ciplining res ult. Sec tion 4 es tab
lish es the comp arative static s of the c ollateral equ ilibriu m with re sp ec t to the typ e an d the
le ve l of b elief he teroge neity. S ections 5 and 6 gen eralize th e asymmetric disc ip lin in g res
ult to se ttings with resp e ctively contin gent d ebt contracts an d sh ort selling of the ass et. S
ec tion 7 intro d uce s the dyn am ic e xten sion and p re sents the res ults for s p ecu lative bu
bb le s. Sec tion 8 conc lu de s. The pap er end s w ith seve ral app end ic es th at pres ent th e
p ro ofs omitte d f rom th e m ain text.
2 Environment and Equilibrium
Cons id er a two p e rio d ec onomy with a s in gle c onsu mption go o d in wh ich a continuu m
of ris k ne utral trad ers have en dowm ents in p erio d 0 bu t th ey n eed to con sum e in p erio d
1 . The resou rc es can b e tran sf erred b etwe en p e rio d s by inves tin g eithe r in a ris k- free
b ond , den oted by B, or a ris ky asse t, den oted by A. Bond Bis su pplied e las tically at a
normaliz ed price 1 in p e rio d 0. Each un it of the b on d yields 1 + run its of the cons ump tion
go o d in p erio d 1 . As set Ais in xed su pply, which is normalize d to 1. Th e asse t p ays divid
end only on ce (in u nits of th e con sum ption go o d ), an d it pays it in p e rio d 1. T he d ivide
nd payment of each u nit of the asse t is d enoted by v(s). Taking th e s et of all p os sible state s
as S =ass ume th at the f un ction v: S ! R++ smi n;sma x8is s trictly inc re asing and continuou sly
di/erentiab le . R , II de note th e p rice of th e as set by p.
6S
e e , f o r e x a m p l e , D e L o n g , S h l e i f e r , S u m m e r s a n d Wa l d m a n ( 1 9 9 0 , 1 9 9 1 ) , B l u m e
a n d E a s l e y ( 1 9 9 2 , 2 0 0 6 ) , S a n d r o n i ( 2 0 0 0 ) , K o g a n , R o s s , Wa n g , a n d We s t e r e l d ( 2 0
0 6 ) , B e ke r a n d E s p i n o ( 2 0 0 8 ) , a n d C a o ( 2 0 0 9 ) .
87Fo r f u r t h e r d i s c u s s i o n o n t h e m e r i t s o f t h e c o m m o n p r i o r a s s u m p t i o n i n e c o n o m i c t
h e o r y, s e e B e r n h e i m ( 1 9 8 6 ) , A u m a n n ( 1 9 8 6 , 1 9 9 8 ) , Va r i a n ( 1 9 8 9 ) , M o r r i s ( 1 9 9 5 ) ,
a n d G u l ( 1 9 9 8 ) .N o t e t h a t t h e s t a t e s p a c e c o u l d b e e q u i va l e nt l y d e n e d a s v (S ) = v
smin ; v (smax) ove r a s s e t p ayo /s ,s o t h e va l u e f u n c t i o n v ( ) i s r e d u n d a nt i n t h i s s e c t i o n . P u
t d i /e r e nt l y, w i t h o u t l o s s o f g e n e r a l i ty, t h e va l u e f u n c t i o n c a n b e t a ke n t o b e t h e i d e nt i
ty f u n c t i o n v (s) = s. I i nt r o d u c e t h e va l u e f u n c t i o n v ( ) b e c a u s e t h i s w i l l c o n s i d e r a b l y s i
m p l i f y t h e a n a l y s i s o f t h e d y n a m i c m o d e l i n S e c t i o n 7 , i n w h i ch t h e va l u e f u n c t i o n w i l
lbeendogenouslydetermined.
8
Trad ers h ave heterogen eous priors ab out th e retu rn of the ass et. In p articu lar, th ere ~ h(s)
are two typ es of trad ers, optimists and moderates, re sp e ctive ly den ote d by sub script i2
f1 ;0 g, with corre sp on ding p rior b elief ab out th e n ext p erio d state given by th e p rob
ability d is tribu tion Fiover S. Trad ers know each oth ers priors, and thu s op timis ts and
mo de rates agre e to dis agree . I n orm alize the p opu lation measu re of each typ e of trad
ers to 1, an d I le t (re sp . wii) de note typ e itrad ers p erio d 0 en dowme nt of the asse t
(re sp. th e c on su mption go o d ). Th e asse t e ndowments satisfy 0>0 and i= 1. An ec
onomy is den oted by th e tuple E = ( S ; v( ) ; f Fgi; fwigi; f igi0). I ad op t the followin g notion
of optimism.+ 1
1
~
H
(
s
)
De ni tion 1 (Optimi sm Or der). Consider two probability distributions H; ~ H overS = mi
ns;sma x with density funct ions h;~ h that are continuous and positive over S . T hedist
ribution ~ His more optimistic than H, denoted by~ H OH, if1 ~ H( s)1 H( s)is strictly
increasin g; equivalentl y, if t he fol l owing hazard rate inequal it y is satis ed for al l s2
The distribution ~ His weakly more optimistic than H, denoted by ~ H O
Assumption (O). Th e prob ability distrib ution s F1 and
have d en sity f unc tion s
F0
f1;f0
O
1 O F0
i
H,
if
(2
)
is
sa
tis
e
d
as
1 H(s) . (2)
0
ma x
f1( s) f0( s)
a
w
e
ak
in
e
q
u
ali
ty.
hat
are
cont
inu
ou s
an d
p os
itive
over
S,
and
the
ys
atisf
y F.
The
op
timi
sm
ord
er,
,
con
cern
s op
timi
sts
re
lativ
e
pro
b
abili
ty
as
ses
sme
nt
for
the
upp
erth
resh
t
ol d
eve
nts
[s;s]
S
, an
d it
p
osit
s th
at
op
tim
is ts
are
incr
eas
in
gly
opti
mist
ic f
or
the
se
eve
nts
as
the
thre
sh
old
leve
l sis
inc
re
ase
d . It
cap
tu
re s
the
ide
a th
at,
th
e b
e
tter
the
eve
nt b
e
com
es ,
th e
gre
ater
the
op
timi
sm
is f
or
the
eve
nt.
This
ord
e r,
also
kno
wn
as
th e
haz
ard
rate
ord
er,
is
relat
ed
to
som
e
we
ll
kno
wn
reg
ulari
ty
con
d
ition
s . It
is
stro
n ge
r
than
th
e r
st
ord
er
sto
ch
asti
c
ord
er,
th at
is ,
F1 O
F0i
mpli
es
F1d
omi
n
ate
s
Fin
th
e r
st
ord
er s
to
cha
stic
sen
se.
Ho
we
ve r,
it is
we
aker
than
th e
mon
oton
e
likeli
ho
od
ratio
pro
p
erty,
that
is,
if1is
s
trictl
y
inc
reas
in g
over
S,
the
n
FO
F0(c
f.
Ap
p en
dix
A.1)
. Let
E[ ]
de
note
the
exp
ec
tatio
n op
erat
or
corr
e sp
on
ding
to th
eb
e
lief
of a
typ
e
itrad
er.
Ass
ump
tion
(O)
also
imp
lie s
E0[v
(s)]
<E1[
v(s)]
, th
at is
, mo
de
rate
s
valu
e
the
ass
et
less
than
opti
mist
s
do.
Th
is fu
rthe
r
impl
ies
th at
mo
de
rate
s
wou
ld
like
to s
hort
se ll
the
as
set
in
this
ec
ono
my,
whi
ch
is
rule
d
out
by
ass
ump
tion.
Ass
ump
tion
(S).
Ass
et
Aca
n
not
be
sh
ort
sold
. Th
is
ass
ump
tion
will
be
mai
ntai
ne d
for
mos
t of
the
p ap
er
(u
ntil
Se
ctio
n
6).
In
reali
ty,
man
y
ass
ets
othe
r th
an
sto
cks,
an d
also
s
om
e
sto
cks,
are
di¢
cu lt
and
cos
tly
to
shor
t
sell
(see
,
e.g.,
Jon
es
and
Lam
ont,
200
1).
G
iven
as
su
mpti
on
(S ),
if
ther
e
we
re
no
n
anci
al f
ric
tion
s,
i.e .,
if op
timi
sts
c ou
ld
freel
yb
orro
w
and
len
d at
the
goin
g
inter
e st
rate
1+
r, th
ey
wou
ld b
id
up
the
pric
e of
the
ass
et
9
E [ v( s)]
1+ r
; E1[v(s)] 1 ,
+r
9
E0[v(s)] 1 + r
+
1
0
10
9
10
to the op timistic valu ation 1,. Howe ve r, nanc ial fric tions may prevent op timists f rom inc reas in g th e asse
this leve l. With nan cial f rictions, the as set (in the b as eline se tting) trades at a
interval
the exact lo c ation b eing d etermined by op timists wealth and the typ e of th e n
rictions . Th e nanc ial f rictions are microf oun ded through a collateraliz ed loan m
2.1 Financial Frictions and Collateral Equilibrium
I make a nu mb e r of in stitu
;’) . (3)
tion al assu mptions for th e
loan marke t. First, I assu I analyze the loan marke t u sing a comp etitive equ ilibriu m notion, col l ateral equ
me th at loans in this e
originally de velop ed by Gean akop los an d Zame (1997, 2009). In p articu lar, e a
conomy mu st b e s ecu re contrac
d
t ’is traded in an anonymous market at a comp e titive price q(’). Note that t
by c ollateral own ed by th eonymity
b orrower, and th e court
T h e r e i s a p o t e nt i a l q u e s t i o n o f w h o h o l d s t h e c o l l a t e r a l t h r o u g h o u t t
system en forces th e tran
f
t
h
e l o a n c o nt r a c t , i . e . s h o u l d t h e c o l l a t e r a l b e l o cke d i n a wa r e h o u s e , h e
sfe r of collate ral to the le
e n d e r , o r t h e b o r r owe r . I n r e a l i ty ( e . g . , i n m o r t g a g e s o r R E P O s ) , d i /e r e nt
nde r in cas e th e b orrowerr e u s e d i nt u i t i ve l y d e p e n d i n g o n w h e t h e r t h e b o r r owe r o r t h e l e n d e r b e n e
d o e s not pay.Se cond , I r o m h o l d i n g t h e c o nt r a c t d u r i n g t h e l o a n p e r i o d . A c o m m o n a s p e c t o f a l l v
as su me th at loans are n c o l l a t e r a l i z e d l e n d i n g r e l a t i o n s h i p s i s t h a t t h e b o r r owe r mu s t ow n t h e a s
on-rec ourse, that is , th e be t i m e o f t h e l o a n p ay m e nt . T h i s a s p e c t i s n e c e s s a r y b e c a u s e o t h e r w i s e
r wo u l d n o t h ave a ny i n c e nt i ve t o p ay b a ck t h e l o a n a n d c o l l a t e r a l wo u l d n
orrowe r do es n ot get f owe
e p ay m e nt .
urther pu nishm ent than p I n t h i s m o d e l , t r a d e r s r e c e i ve n o u t i l i ty f r o m h o l d i n g t h e c o l l a t e r a l i n p
ote ntial los s of c ollateral. h i ch i m p l i e s t h a t t h e d i /e r e nt va r i a nt s o f c o l l a t e r a l i z e d l e n d i n g a r e e s s e
Third , I als o assu me th atu i va l e nt . T h e r e f o r e , w i t h o u t l o s s o f g e n e r a l i ty, t h e b o r r owe r i s r e q u i r e d t
the loan s are non -contin c o l l a t e r a l t h a t s h e p l e d g e s .
T h e a s s u m p t i o n t h a t e a ch c o nt r a c t p l e d g e s o n e u n i t o f t h e a s s e t i s a n o r m
gent, th at is, the y promiseo n w i t h o u t l o s s o f g e n e r a l i ty. N o t e a l s o t h a t , i n p r i n c i p l e , b o t h t h e b o n d B
the s am e payment in all f e a s s e t A c o u l d b e u s e d a s c o l l a t e r a l . H owe ve r , i n t h i s m d o e l , t h e a s s u m p
uture states. T hes e assu t o n l y t h e a s s e t c a n b e u s e d a s c o l l a t e r a l i s a l s o w i t h o u t l o s s o f g e n e r a l i
mptions argu ably p rovide pa r e c i s e l y, t h e e q u i l i b r i u m d e s c r i b e d b e l ow i n T h e o r e m 2 c o nt i nu e s t o b e
go o d s tartin g p oint, b e e nt i a l l y u n i q u e e q u i l i b r i u m i n t h e m o r e g e n e r a l s e t t i n g i n w h i ch t h e b o n d
o b e u s e d a s c o l l a t e r a l . T h i s i s b e c a u s e o p t i m i s t i c b o r r owe r s d o n o t h o l d
caus e most REP O loan s d i n e q u i l i b r i u m ( e x c e p t f o r t h e c o r n e r c a s e i n w h i ch t h e i r we a l t h i s m o r e t
an d some mortgages are ¢ c i e nt t o p u r ch a s e t h e e nt i r e a s s e t s u p p l y ) .
non -re course , an d the y
do not h ave m any contin
genc ie s.
Formally, a unit debt cont
ract , d en ote d by ’2 R, is a
promis e of ’un its of th e con
su mption go o d in p erio d
1 by the b orrower, c
ollateralize d by 1 un it of the
asse t A(which th e b
orrower owns ).In p erio d 1,
the b orrowe r def aults on th
e u nit d ebt c ontrac t ’if an d
on ly if th e as set s value is
les s th an the promised
amount. Thus, e ach
contract ’pays
m
i
n
(
v
(
s
)
pxA i+ xB i
q(’) d + i
R+
d
i
q(’) d
i
(’)
wi
+ p i. (5)
Ai
ZR+
of th e market is e nsu re d by collate ral: each lend er knows that re payment is on ly se cu
red by collate ral, an d th at sh e will get the payme nt in (3) regard le ss of th e id entity of the
b orrowe r in th e tran saction.I re fer to a d ebt contract ’= v( s) 2 v smi n ;v(sma x) as a
loan with riskiness s, sinc ethis contract d ef au lts if and only if th e realized s tate is b e
low s. I re fe r to th e pric e of the de bt contract, q(v( s)), as the l oan size , s in ce this is th
e am ou nt of that the b orrower rec eives by collateraliz in g one un it of th e as set. In equilib
rium, q(v( s)) will b e in creasing in s, h enc e l arger loans are also riskier l oans. Moreove
r, I d e n e th e interest rate on th e loan as the ratio of th e p romis ed interest p ayment to th
e loan size:
v( s) q(v( s)) q(v( s)). (4)
Give n th es e de nition s, an interpretation of th e mo del is th at loans of d i/erent size s (and
thus di/e re nt riskin ess levels) are b e ing trad ed in a comp etitive e qu ilibrium at th eir c
orresp on ding inte res t rates .
To
formaliz
e trad
ers p
ortfolio
choice s,
I assu
me th at
th e p
(’)
Th
at is
,f
or e
ach
un it
deb
t
cont
ract
trad
ers
sell,
th
ey
ne
ed
to
set
as
id e
on e
u nit
of
the
as
se t
the
y
own
as
coll
ater
al.
T
yp
e
itr
a
d
er
s
ch
ond
holdings,
xi= xA
i;xB i 2
R2 +. The
ir b ud
ge t con
straint is
give n
by:
x
. (6)
Note that short s ellin g d eb
en ables the m to inves t mo
Howeve r, sh ort s elling is s
o
os
e
th
e
ir
p
or
tf
oli
o
to
m
ax
im
iz
e
th
e
ir
ex
p
ec
te
d
u
tili
ty,
i.e
.,
th
ey
s
ol
ve
th
e
pr
o
bl
e
m
:
[v
(s
)]
+
xB
i
+
i
R+
(’) RR+
su b je ct to (5) and (6)
.
xi 0;
max
11
+ i; i2
M( R+)
Ei [min ( v(s) ;’)] d
" xA iEi
i
(’)
Ei[min
( v(s)
;’)]
d (1 +
r) + R
11
Id
e
n
et
ra
d
er
s
p
or
tf
ol
io
s
u
si
n
g
m
e
a
s
ur
e
s,
ra
th
er
th
a
n
m
e
a
s
ur
a
bl
ef
u
n
ct
io
n
s,
b
e
c
a
u
s
et
h
e
o
pt
im
al
p
or
tf
ol
io
ch
ar
a
# ,(7)
ct
er
iz
e
d
b
el
ow
wi
ll
b
e
a
Di
ra
c
m
e
a
s
ur
e(
w
hi
ch
d
o
e
s
n
ot
c
or
re
s
p
o
n
dt
o
a
m
e
a
s
ur
a
bl
ef
u
n
ct
io
n)
.
Market c le aring for each unit d eb t contract ’require s th e s um of th e long p osition s to b e
e qu al to the su m of th e s hort p osition s, that is:
Xi2f 1 ;0 d + i (’) = Xi2f 1 ;0 Z d i (’) f or e ach Bore l set C R+. (8)
g
C
gZC
De ni tion 2 (C ollater al Equil ibrium). Given an economy E with assumptions (O ) and (S ), a col
lateral equilibrium is a col lection of prices (p;[q( )]) and portfolios xA i;xB i; + i; i i2f 1 ;0 gsuch that:
the portfol io of t ype itraders solves P roblem (7) for each i2 f 1 ;0 g, the asset market clears,Pi2f
1 ;0 gxA i= 1, and the debt market clears [cf. Eq. (8)].
3 Characterization of Collateral Equilibrium
Th is s ection provid es a ch aracterization of c ollateral equilib rium an d prese nts the main
resu lt wh ich ch aracterizes the e /ect of b elie f hete rogen eity on th e as set pric e. The equilib
riu m will intu itively h ave th e form th at mo d erates hold th e b ond an d long p osition s on
collateraliz ed de bt contracts (i.e., the y len d to optimis ts), wh ile op timists m ake le ve raged
inves tments in the ass et by sellin g collate raliz ed de bt contrac ts .
To charac te rize the e qu ilibriu m, it is u sef ul to d e ne the notion of a quasi-equil ibrium,
which is a c ollection of prices (p;[q( )]) and p ortfolios xA i;xB i; + i; i su ch th at marke ts c le ar
and the p ortf olio of typ e i2 f 1 ;0 g trad ers solves Problem (7) with the add itional re qu ire
ment + 1= 01 2= 0 .i2f 1 ;0 gThat is, in a qu asi-equilib rium , op timists are re stricte d n ot to b uy
de bt contracts , an d mo derates are res tricted n ot to sell d eb t contrac ts . For exp osition al
re asons , I will rst con stru ct a quas i- equilib rium. Th eorem 2 b elow es tab lis hes th at the
cons truc ted quasi-e qu ilibrium corres p ond s to a c ollateral equilib riu m with the sam e allo c
ations an d the same as set price (and with p otentially d i/ere nt de bt contract prices ). Th e
same the ore m also es tab lis hes that the asse t price in a collate ral equ ilibriu m is u niquely
de termin ed .
To con stru ct a quasi-e qu ilibrium, c on side r d eb t contract p rice s
q(’) = E0 [min ( v(s) ;’)] for e ach ’2 R+, (9)
1+r
that make mo d erate s ind i/erent b etween holding th e b on d an d any de bt contrac t ’2 RE0[ v(
s)] 1+ r+. Give n th e prices in (9) and th e as set price p E0, mo de rates
optimal d ecision in a
quas iequilibriu m is comp le tely ch aracte rize d: the y are ind i/erent b etween h old ing th e b
ond and any de bt c ontrac t, and the y always weakly p re fe r th ese option s to h old in g the
asse t (an d s trictly so whe ne ve r p>[ v( s)] 1+ r). Moreover, th ese p rice s ens ure that market
clearin g in de bt contracts will b e au tomatic, as mo d erate s will absorb any su pp ly of de bt
contracts from optimists.
The quasi-e qu ilibriu m ass et price an d allo c ations are then de term in ed by optimists p
ortf olio choice. I ne xt an alyze optimists problem for a given asse t price p, an d I then
combine
1 2H
1
= 0 ( s i m i l a r l = 0) d e n o t e s t h e 0 m e a s u r e (C
, i ) = 0 f o r e a ch B o r e l s e t
e r e , y+ 0
. e . , +i
C R+.
12
this an alysis with ass et marke t clearin g to solve f or th e quas i- equilib rium.
3.1 Main Result: Asymmetric Disciplining of Optimism
Th e n ext res ult, which is also th e main re sult, ch arac te rize s op timists p ortfolio ch oic
e.
Theor em 1 (Opt imal C ontr act Choice and Asymm et ric F il teri ng). Suppose assumptions
(O) and (S) hold, debt prices are given by (9) and the asset price satis es p2 E0[ v( s)] 1+ r;E1[ v( s)]
1+ r . In a quasi-equilibrium: i(i) T here exists s2 S such that is a Dirac measure that put s w
eight onl y at the contract ’= v( s), i.e., optimists borrow according to a single loan w ith
riskiness s. Optimists col lateral constraint (6) is binding, i.e., they borrow as much as
possible according to the opt imal loan. O ptimists choose xB 1= 0, i.e., they invest al l of their l
everaged w eal th in the asset A. (ii) The riskiness sof t he optimal loan is charact erized as
the unique solution to the folsmin
1
s
l owing equation over S( : s)
1
1+
r Z
p= popt
= 11 + r (F0 ( s) [v(s) j s< s] + (1 F0
E0
s
v(s) dF0
+ (1 F0 ( s))
Z
( s)) [v(s) j s
E1
smax
v(s) dF1
F1
( s)
s]) .
E1The
riskin ess sof the optimal l oan is decreasing in the price l evel p. If instead t he asset
price satis es p=[ v( s)] 1+ r, t hen optimists are indi/ erent between making a leveraged investmen t
by sel ling any safe debt contract ’ v1 3 smi n or invest ing in the bon d.
optI will s hortly provid e a s ketch pro of of th is re su lt along with an intuition . Bef ore d oin g
so, I n ote a coup le of imp ortant asp ects of the fu nction p( s). First, th e fun ction poptopt( s)
is similar to an inve rs e d eman d f un ction : it de scrib es the asse t price pf or wh ich the
riskine ss level sis optimal. Assu mption (O) implies popt( s) is stric tly de creasing and
continuou s (cf. App e nd ix A.1). Sin ce p smi n =E1[ v( s)] 1+ rand poptma x(s) =E0[ v( s)] 1+ r, this fu
rthe r im plie s th at the re is a u niqu e s olu tion to Eq. (10), and that the solution is strictly de
cre asing in p.
Se cond , note that poptopt( s) also des crib es the e qu ilibrium asse t p rice cond itional on th
e equilibriu m loan riskin ess s. He nce , The orem 1 is the main resu lt of this pap er, as it sh
ows that op timism will b e asymme trically d is ciplined in equilib rium. In p articular, the sec
on d lin e of (10) replicates Eq. (1) f rom th e Intro d uction and sh ows th at the ass et is pric
ed with a mixtu re of m o d erate an d optimis tic b elie fs . Th e mo d erate b elie f is us ed to
as ses s the like lih o o d of d efau lt states s< s, alon g with the value of the as set con dition
al on thes e states, while the optimistic b elief is us ed to as ses s the like lih o o d of n on -d
efau lt states s> s. C on se qu ently, the f unc tion p( s) will discip line any optimism ab out
th e prob ability of d ef ault states ,
1 3T
E0[v ( s )] 1+i s
r
( s )]
o m i t t e d , s i n c e t h e e q u i l i b r i u m a s s e t p r i c e aEl [v
1+ r
way s s a t i s e s p >
hecasep
=
13
0( c f . T h e o r e m 2 ) .
(10
)
2
1
2
0.6 0.7 0.8 0.9 1 1 .1 1.2 1.3 1.4 1.5 0
1
0.6 0.7 0.8 0.9 1 1 .1 1.2 1.3 1.4 1.5 0
1.02 1.04
1.06 1.08
1.1
0.6 0.7 0.8 0.9 1 1 .1 1.2 1.3 1.4 1.5 1
optFigu
re 1: The top two pan els disp lay the probab ility d ens ity fu nctions f or trade rs priors
in the two s cen arios of Example 1. Th e b ottom p anel d is plays the corre sp on ding c urve s
p( s), the inverse of which give s th e optimal loan ris kin ess sfor a given price leve l p.
wh ile inc orp oratin g any optim is m ab out the re lative probab ility of states c onditional on
no de fau lt. The f ollowing example de scrib es two cas es that d i/er ab ou t the typ e of
optimism an d illu strates the asymmetric disc ip lin in g prop erty.
Example 1 ( Asymmetr ic Fi lter ing of Opt imism). Consider the stat e space S = [1 =2 ;3 =2]
and the val ue function v(s) = s. A s the rst case, suppose moderates and optimists have the
prior distributions F0and F1 ;Bwith density fun ct ions:
(s) = 1 for each s2 S ,
B
f0
(
8 0:4 if s2 SN [2=3 1=6;2=3 + 1 =6) 1:3
s
s2 SG [1 1=6;1 + 1 =6) 1:3 if s2 S [4=
and f1 ;B
)
> 1=6;4=3 + 1 =6] ,
1 4intuitivel y capture bad, normal and
< good events, respect ively . I n words
alwl states equal l y likely, =w hil e optimists are optimistic because they believ
h
event,
that is, a real ization around> the bad state 2 =3, is less likely t han a n
e event (which they n d equal: l y likely).
good
r
e
S
B
,
S
N
,
a
n
d
S
G
Consider a second case in which moderates have the same prior, but optimists prior is
1 4N
otethatthebeliefdistributionsofthisexampledonotexactlysatisfytheregula
price
14
oti
p miti
sc
pdf
m
oderate
pdf
r i ty a s s u m p t i o n ( O ) . I n p a r t i c u l a r , t h e d e n s i ty f u n c t i o n s a r e n o t c o nt i nu o u s , a n d F1 ;Bi
s o n l y we a k l y m o r e o p t i m i s t i c t h a n F0. T h e s e d i s t r i b u t i o n s a r e u s e d f o r i l l u s t r a t i o n
p u r p o s e s b e c a u s e t h e y p r ov i d e a c l e a r i nt u i t i o n . T h e f o r m a l r e s u l t s u s e t h e s t r i c t e
r a s s u m p t i o n ( O ) f o r a n a l y t i c a l t r a c t a b i l i ty.
B
1
:
9
i
f
s
2
S
G
That is, optimist s are optimistic not because t hey think the
bad event is l ess likel y, but because they believe the good
event is more l ikely t han the normal event. Not e al so that
optimist s are equal ly opt imistic in both cases, i.e., E1
;G[v(s)] = E1 ;B[v(s)]. The bottom panel of Figure 1 displ ays
the opt imality curves, p( s), in both cases. Not e that, for
any l evel of loan riskiness s, the asset price is higher in the
second case t han in t he rst case. Equivalent ly, for any
price p, optimists choose a larger and riskier l oan in t he
second case than in t he rst case.
I next provide a ske tch pro of of The orem 1, which is u
sef ul to un de rstan d th e intu ition. Th e pro of in App end
ix A.2 shows that optimis ts b orrow u sing a loan with ris
kin ess s2 Sthat maxim iz es th e leve raged return:
[v(s)] E1 1+ rE01[min ( v(s) ;v( s))] : (11)
E0[m in( v( s) ;v( ~s))]
1+ r
R
U
1
= E1
U
opt
[
v
(
s
)
]
p
L
1
changed t o the distribution F1
;G
f1 ;G = 8 with density fun ct ion
>
<
>:
1 if s2 S0 :1 if s2 SN
.
opt
1
(
[min ( v(s) ;v( s))] p
Th is e xp re ssion is the exp ecte d re turn of optimists who b uy on e un it of th e asse t an d
who n anc e part of the purchase u sing a loan with ris kine ss s. The d en omin ator is th e
s
downpay ment optimists make f or th e leve raged inve stment: the y pay the p rice pof the
)
asse t bu t they c an b orrow q( s) =from mo de rates (given th e contract p rices (9)). The nu
merator is optimists exp ec te d payo/ f rom the le verage d inves tment: they exp ec t to re
ceive E[v(s)] f rom th e ass et and the y als o exp ec t to p ay E[min ( v(s) ;v( s))] on th eir
loan . The relation p= p( s) is th e rst orde r op timality cond ition c orres p ondin g to th e
E
maximization of the leverage1 d return, R( s). The le ve raged re turn h as a un ique maximum
ove r S , ch arac te rize d by the rst order con dition , which comp letes th e s ketch p ro of of
R for the th eore m, it is u se ful to fu rth er break down
The ore m 3.To un derstand the intuition
L re ssion (11) into two comp one nts . Firs t cons id er the le ft h an
the leve raged return e xp
d side te rms in the nume1 rator an d th e d en om in ator of (11), which c onstitute th e
unleveraged return :
.
Th is expres sion is the exp ec te d retu rn of optimis ts if they b uy th e ass et with their own
we alth (with out b orrowin g). Optimis ts b e lie ve the re turn on inves tin g in the ass et is
greater than the b e nchmark rate, i.e., R>1 + r, wh ich creates a f orc e th at pu she s toward
s le veraging. In partic ular, if optimists could b orrow at th e b en ch mark rate rwithout c
onstraints, the y would b orrow in n itely to leverage this un le verage d return.
15
Howe ve r, optimists have to b orrow with a collateraliz ed loan with riskin ess s, which re
prese nts a s econd force th at p ush es towards deleve raging. This f orc e is re lated to the
right h an d sid e terms in th e nu merator an d the de nominator of (11), wh ich con stitu te op
timists perceived interest rate on th e loan:
E0
E1 1+[min (v(s) ;v( s))][min (v(s) ;v( s))] . (12)
1 1 + rper 1(r1 s)
1+
E0
r
per 1
Optimis ts b orrow
1
per 1
opt
per 1
per 1
per 1
=
E1[ v( s)] p
U
per 1
opt
B.
In th e rs t c as e of E xamp le 1, optim is ts n d the b ad eve
nt SB
B
per 1
16
[min (v(s) ;v( s))] on th e loan, but th ey exp ec t to p ay E[min ( v(s)
;v( s))], wh ich le ads to the p erce ive d inte re st rate 1 + r( s). Assu mption (O) implies that
r( s) is always weakly gre ate r than the b enchm ark rate r, and that it is increasing in s[cf .
App end ix (A:1)]. The intuition for th is obs ervation is two fold. First, collateraliz ed loans always
trade at a spread over the b en ch mark rate (i.e., th e inte re st rate on the loan, (4), is always
gre ate r than th e b e nchmark rate ), b ec ause mo d erate le nde rs re qu ire comp ens ation f
or the ir exp e cted los ses in cas e of def ault. In p articular, since the loan market is comp
etitive , the sp re ad on a loan is j ust en ough to comp en sate the len de rs acc ord in g to the ir
mo d erate b elie fs . Se cond , op timists b e lieve that th e loan will de fault le ss often th an mo
d erate s b elieve, h en ce the y think th ey will end u p paying the s pread more of ten. Cons
equently, optimists b elieve the y will pay a gre ate r inte re st rate th an the b en ch mark rate,
i.e ., r( s) >r. More over, f or gre ater leve ls of s, th e sc op e of d is agree ment for d ef ault is
greater, wh ich imp lies th at r( s) is inc re asing in s.
It f ollows th at, while a larger loan with a greater ris kine ss le vel sen ables optimis ts to
leverage the unleve raged re tu rn more , it als o c omes at a greater p erceived interest rate ,
r( s). Optimis ts op timal loan choice balan ce s th es e two forces , as c aptured by th e
maximization of the leverage d retu rn expres sion (11).
optThis
breakdown of the two forc es als o p rovides an intuition f or th e asymmetric disc ip lin
in g prop erty of th e pricing fu nction p( s). Firs t c onside r the intuition f or the s im pler p rop
e rty that p( s) is de creasing in s. Th at is, cons id er why optimis ts ch o os e a larger an d
ris kier loan whe n th e p rice pis lowe r. This is b ec au se a lower ass et price in crease s the
un le verage d return, R, wh ich tilts op timists trade- o/ towards larger loan s. When the un le
verage d return is gre ate r, optimists have a greater in centive to leverage this return by takin
g a large r (and riskier) loan , agree in g to pay a greater exp e cted interest rate r( s) at the
margin. To se e th e intuition for the d is ciplining p rop erty of p( s), x a loan with ris kin
ess s, and con sider how much the p rice s hould d rop (f rom the op timistic valuation ) to
entic e optimis ts to take th is p articular loan . Cons id er this ques tion in th e conte xt of E
xamp le 1 for a riskin es s leve l s= 0:8 2 Sun like ly. Henc e, give n a loan with ris kin ess s2
S, th ere is disagre eme nt ab ou t the prob ability of de fault, which implies r( s) >r. As this
loan ap p ears exp en sive to optimists, th e asse t price sh ould drop con siderably to entice
optimis ts to un dertake a leverage d inve stment with this loan. Con side r ins te ad the s econ
d cas e of E xamp le 1 in which optimists are optimis tic b ecau se th ey nd the
go o d eve nt m ore likely than the normal eve nt. In this case , for a loan with ris kine ss s2 S,
the re is no d is agree ment ab ou t the probability of def ault, which imp lie s rper 1B( s) = r. As th
e loan app e ars ch eap to optimists, the as set p rice do es not nee d to f all to entic e them to
take the loan (see Figu re 1).
In oth er words, the asymme tric d is ciplining resu lt op erate s through optim is ts p erce
ived nan cial cons traints. Dis agree ment ab out d ef au lt state s tighte ns op timis ts nan cial
cons traints (captu re d by a higher rper 1( s)), wh ich lowers the ir d emand for the le verage d
inve stment and leads to an as set price c lose r to the mo de rate valuation. In contrast,
disagreem ent ab ou t n onde fau lt state s do es not tighte n the n ancial c on straints, and lead
s to an asse t price close r to the optimis tic valuation .
3.2 Asset Market Clearing and Collateral Equilibrium
Th eore m 1 ch aracterizes th e riskin es s sof the optimal contrac t as a fu nc tion of the ass et
p rice p. I n ext cons id er the marke t clearin g price pand solve f or th e e qu ilibrium.
Su pp os e op timis ts cho ose to b orrow us in g a loan with riskin ess sand c on side r the
price that clears the as set marke t. Th is p rice d ep e nd s on th e m aximu m rs t p e rio d
cons umption go o d that optimists c an ob tain:
wma ( s) = + 11 + r [min ( v(s) ;v( s))] . (13)
x1
w1
E0
Optimis ts are end owe d with w1un its of the cons ump tion go o d, and if the y h old th e e
ntire ass et su pp ly, they c an b orrow up to1 1+ r[min (v(s) ;v( s))] un its of th e con su mption
go o d f rom mo de rates , leadin g to th e e xp re ssion in (13). The asse t pric e d ep en ds on
th e comp arison of wma x 1E0( s) with the valu e of the as set in th e han ds of mo de rates
, 1 0p, wh ich op timis ts see k to pu rchas e. In particu lar:
[ v( s)]
max 1
if w
1+ r
0
> E [ v( s)]
[case
E
1+ r
(i)]
wmax
p= pmc ( s)
00
1( s)
wmax
if
2
(
E0 [ v( s)]
; E [ v( s)] ] [case (ii)]
[ v( s)]
1(
s)
1+ r
1+ r
1+ r
E0
if
w max
1( s)
0
E0[ v(
s)] 1+ r
( s)
8
>
<
>:
1
E1[ v( s)][case
1+ r
(iii)] . (14)
( s) = p 0
1
ma x 1
.
E0[ v(
s)] 1+
r
In c ase (i), optimists h ave acc ess to a s u¢ cient amount of con sum ption go o d in th e rs t p
e rio d th at they purchase all of th e asse t in th e hand s of mo d erate len ders (and they have
some cons ump tion go o d lef t over, which the y invest in th e b on d). In th is cas e, op timists
are margin al holde rs of the ass et an d th e pric e is given by their valuation,. In case (ii),
optimists still purchase all of the ass et f rom mo derate le nde rs , bu t the y can not b id u p the
ass et p rice to th eir valuation. In this cas e, th e market clearin g p rice is de term in ed by se
tting optimists con sum ption go o d equal to th e value of m o d erates asse ts , i.e., w. In cas
e (iii), op timists have acce ss to so little rs t p erio d con sum ption go o d that the y cann ot pu
rchas e all of the asse t in th e h ands of m o d erate le nd ers . In th is cas e, m o d erate le nd
ers hold some of th e as set, and the price is give n by their valuation ,
17
Note that E q. (14) de scrib es an in creasing relation b e twee n th e as set p rice and the loan
riskine ss s. Intuitive ly, wh en optimists take a larger and ris kier loan, they h ave ac ces s to
a greater amount of rs t p erio d c on su mption go o d , wh ich e nables them to b id up th e
ass et price h igh er. Comb in ing Theorem 1 and E q. (14), the e qu ilibriu m pric e and loan
ris kine ss p air, (p; s opt), is de termin ed as th e un ique intersec tion of th e stric tly dec re as
in g fun ction pmc( s) and th e we akly inc reas in g fu nction p1 5( s) (se e Figu re 2). This
analysis c om pletes th e charac teriz ation of th e quasi-e qu ilibrium. The an alysis in Ap p
end ix A.3 e stablishe s that th is qu asi-equilib riu m is a collate ral equ ilibriu m with m o d
i ed de bt contract p ric es given by:
q(’) = m ax E0 [min ( v(s) ;’)] ; E1 [min ( v(s) ;’)] . (15)
1+r
RL 1( s )
Th e following re su lt su mmariz es th is disc uss ion and p rove s the ess ential un iqu ene ss of
the collate ral equ ilibriu m.
Theor em 2 (E xistence, Character ization, E ssential Uniqueness). Consider the above
described economy w ith assumption s (O) and (S). T here exists a col l ateral equil ibrium in
which contract prices are given by (15), moderate ty pes are indi/ eren t between buy ing
bonds and lending to optimists, and opt imists make leveraged investmen ts in t he asset by
borrowing through a sin gl e l oan with riskiness s 2 S . The asset price pand riskiness soptof
l oans in this equilibrium are determined as the unique sol ution to p= p( s) = pmc( s)
over s2 S . In any col l ateral equil ibrium, t he asset price, p, and the price of the optimal
debt contract ,
E [ v( s)]
q(v( s )), are uniquel y determined. Except for the corner case in which p=
1+ r
), are not uniquely determined.
L1
) ( c f . E q . (11)) i s o p t i m i s t s e x p e c t e d r e t u r n o n c a p i t a l i n e q u i l i b r i u
m . T hu s , t h e e x p r e s s i o n
18
15
( s
E1
, traders al locations are al so uniquely det ermined. However, prices of the remaining debt
contracts, q(’) for ’6 = v( s
1
In other word s, most of th e equilib rium is un ique ly dete rmine d, excep t f or th e price of
de bt contracts th at are not trade d in equilibriu m. App end ix A.3 e stablis he s that, f or each
contrac t ’= v( ~s) 6 = v(s), the re e xists a c ontinu um of p rice s that can s up p ort the e qu
ilibrium with no- trade in the se contracts. Th is c om pletes the ch aracterization of th e c
ollateral e qu ilibrium.
Figu re 2 illus trates the equilib rium, an d s hows the e/e ct of a de clin e in optimis ts initial
en dowm ent of the cons ump tion go o d . Wh en optimists wealth d ec line s, the p rice f alls
towards the mo d erate valuation. Note also th at the equilibrium loans also b ecome larger an
d riskier. Th is is b ec ause , as the p rice f alls, op tim is ts s ee more of a b argain in th e
asse t p rice wh ich en courages them to leverage more . Hen ce, e qu ilibrium le verage resp
on ds in a way to ameliorate th e d rop the initial wealth s ho ck to op timists . Th ese comp
arative s tatics are s imilar to th e resu lts in Gean akop los (2009). I next tu rn to th e fo cu s
of this p ap er, and e stablish
Notethat
R
0
=0and
1
+
=0arerela
xed.
[min( v ( s ) ;’)] RL
1( s )
i s o p t i m i s t s va l u a t i o n o f t h e d e b t c o nt r a c t ’ i n e q u i l i b r i u m . U n l i ke i n a q u
a s i - e q u i l i b r i u m , o p t i m i s t s c a n d e m a n d d e b t c o nt r a c t s i n a c o l l a t e r a l e q u i l i b r i u m .
H e n c e , t h e p r i c e o f a d e b t c o nt r a c t i s g i ve n
by t h e u p p e r - e nve l o p e o f t h e m o d e r a t e a n d t h e o p t i m i s t i c va l u a t i o n s , a s c a p t u r e d
by (15). T h e a n a l y s i s i n A p p e n d i x A . 3 e s t a b l i s h e s t h a t o p t i m i s t s a n d m o d e r a t e l e
n d e r s a l l o c a t i o n s c o nt i nu e t o b e o p t i m a l w h e n t h e p r i c e s a r e g i ve n by (15) a n d w h e
n t h e c o n s t r a i nt s
1.1
1.08
1.06
pric e
1.04
1.02
1
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
Figu re 2: Th e gu re d isplays the c ollateral e qu ilibriu m, and th e resp on se of th e equilib riu
m to a d ecline in optimists initial e nd owment of th e c onsu mption go o d, w1.
the comparative statics of th e equilibriu m with re sp e ct to th e typ e an d th e level of b elief he
teroge neity.
4 Comparative Statics with Resp ect to Belief Heterogeneity
In add ition to the equilib rium loan riskin ess s E0[ v( s) ;v( s )] 1+ rand the ass et p rice p, I c
onsid er the comp arative statics of th e l everage ratio f or optimists as se t p urch ase, d
enoted by L. Re call th at optimists bu y on e u nit of the asse t by paying pout of th eir wealth
and nan cing the re st of the purchase by b orrowing from mo d erate s. Thus , the leverage
ratio for th e asse t p urch ase is given by
L pp E0
[min (v(s)
))] =(1 + r) . (16)
;v( s
1 Land the hairc ut on a REP O loan is equal to1 LTh e leve rage ratio has cou nterparts in real n
ancial markets: th e loan- to- valu e ratio of a mortgage loan is equ al to 1 . The n ext d e n ition
formaliz es the typ e of b elie f he te rogen eity that is u sed to s tate the
comp arative statics re sults. De ni tion 3 ( Skewed Optimi sm). Consider two probability dist
ributions H;S = mi ns;sma x with den sity functions h;~ H over~ hthat are continuous and
positive over S , and con-sider a continuousl y di/ erentiable and strictl y increasing asset
value function v: S ! R++. The optimism of distribution~ Habout the asset is we akly more
right-s ke wed than H, denoted
by ~H R H, if and only if:
19
~ Hi =
~
Hand
, which is the case if and onl y if the hazard rates ofHsatisfy the (weak) singl e
crossing condition: 8< :~ h( s)1 ~ H( s)~ h( s)1 ~ H( s) h( s) 1 H( s)h( s) 1 H( s)Rif s<sRif s>s,
denoted by ~ H R ~s.R;~s
Rs;sma x . Thu s, cond itional on s sR ~ H has a lowe r h az ard rate than H over the
, ~ His m ore optimis tic than Hin th e sen se
of
region
m
in
R
(a) T
he
dist
ributio
ns y
iel d t
he The optimism of distributio
same
valuat
ion ofhe conditions (a)-(b) are s
t he
asset,
To interpret th is de n
that
is, E h
ared
v(s)
;E[v(s
) ; H].acc ordin g to th e op timi
(b)
ordered . In add ition, th e
There
exists
is, the y have the s ame
sR2 S
such
thatw
eakly
increa
sing
over
Rs;sma
x1 ~ H(
s)1 H(
s)is
weakl
y
decre
asing
over
smi
n;sR
its op timism is c once ntrate d m ore on th is region. Henc e, th e optim is m of ~ His right-s
kewe d in
the se nse that it is con centrated more on re latively go o d state s. Note th at th e probability d
is tribu tions F1 ;Band Fof E xamp le 1 s atisf y c ondition (17). That is , F1 ;Gand F1 ;B1 ;Glead to
the s am e valu ation for the as set but the optimism of Foptis we akly more right skewe d, as illu
strate d in Figu re 3. The sam e gure also plots th e op timality relation p( s) f rom Figure 1
togethe r with th e market cle aring c urve pmc1 ;G( s), and illus trates that the equilib riu m price
pand loan ris kin ess s are highe r when optimists optimism is more right-skewed. The next
resu lt shows that this obse rvation is gene rally true .
Theor em 3 (Typ e of Heterogeneity). Consider the col l ateral equilibrium characterized in
Theorem 2 an d l et s denote the equil ibrium l oan riskiness. (i) I f optimists optimism
becomes weakly more right-skewed, i.e., if their prior is changed
to ~F1poptthat satis es ~ F1R F1 and ~ F1O F0
0
R
;
~
F0
20
0
De n ition 1. In contrast, Hhas a lower hazard rate than ~ Hover th e re
gion
s
~
F0
s
;s
, and
thus
(so t hat assumpt ion (O) continues to hold), then:
the asset price p, the loan riskiness s, and the leverage ratio Lweakly increase. (ii) I f
moderates optimism becomes weakly more skew ed to the left of s, i.e., if t heir prior is
changed tothat satis es F and F1 O~ F0, then: the asset price pweakly increases.
I provide a sketch pro of of this re sult, which is comp leted in Ap p en dix A.4. Firs t obs erve
that E q. (10) can b e written as :
( s) E[v(s)] = 11 + r (1 F0 ( s)) ( [v(s) j s s] E0 [v(s) j s s]) . (18)
1+r
E1
5
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0
5
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0
1.1
1
.
0
2
1
.
0
4
1
.
0
6
1
.
0
8
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1
Figu re 3: The top two pan els display th e hazard rates f or trad ers p riors in
the two cas es analyzed in E xamp le 1. Th e b ottom pan el plots th e c orresp
on ding equ ilibria.
In view of th e asymme tric disc iplin in g re sult, th e di/e ren ce b etwe en th
e ass et pric e and the mo de rate valuation d ep e nds on th e mo de rate p
rob ability of n o de fault, and trade rs valu ation di/e re nce s c ondition al
on n o de fault. For part (i), App end ix A.4 sh ows th at
[v(s) j s s] E1 [v(s) j s s] for e ach s2 smi n;sma x , (19)
i
~
E1
mc
opt
. Suppose the moderate and the optimist ic bel iefs are
21
wh ere ~ Ei
optimistic
oderat
m
e
haz
ardrat e haz
ard rat e
ice
pr
[ ] de
th e loan riskine ss s
notes theinc reas e, which f urther implies th e remaining c omparative s tatics. App end ix A.4 use s a
exp e s imilar argu ment to prove p art (ii).
ctation op
erator with The ore m 3 p oints to the imp ortance of the skewness of b elie f h eteroge neity for the
re sp e ctas se t p rice. A re lated qu estion is wh ethe r the l evel of b elief h eteroge ne ity has similar
to d is robu st pred iction s regard in g the p rice of th e asse t. Th e ans wer is no, as illus trated in
tribu tion the
~ f ollowing example .
Example
2 (Ambiguous Pr ice E /ect of Increased Bel ief Heter ogenei ty). Con sider the rst
F. That is,
wh en case of E xample 1 in w hich optimists are opt imistic because they nd the bad event
optimistsunlikely, i.e., they have the prior F1 ;B
optimism
b e comes
more
right-s
kewe d, th
eir valu
ation of th
e ass et
cond
itional on
any u pp e
r-th re
shold e
vent in
creases ,
e ven th
ough th eir
u ncon
dition al
valuation
is th e
same . It
follows, by
Eq. (19),
th at th e
optimality
cu rve
p( s) sh
ifts u p p
ointwis e.
As the m
arket
clearing
cu rve
p( s)
remains c
onstant, th
e equ
ilibriu m
asse t p
rice pand
4
5
2
0.6 0.8 1 1.2 1.4 0
0.6 0.8 1 1.2 1.4 0
5
4
2
0.6 0.8 1 1.2 1.4 0
0.6 0.8 1 1.2 1.4 0
1.1
1.1
1
0.6 0.8 1 1.2 1.4 0.9
1
0.6 0.8 1 1.2 1.4 0.9
Figu re 4: The lef t pane l p lots th e equilib riu m in th e rst s cen ario cons id ered in
E xamp le 2: the inc re as e in b elie f he te roge neity is con centrated to th e left of
state s and it dec re as es the ass et p rice. The right pan el p lots th e equilib rium
in the s econ d s cen ario con side re d in Example 2: the in crease in b elie f h
eterogene ity is to th e right of s and it in crease s the ass et price.
changed to ~
and ~ = F0 ;G
= F1 ;BG with density fun ct ions given
F0
F1
by
f0 8 1 if s2 S1 + 0 :5 if ; f1 ;BG
=8
~ F0 0 ;B, ~ F1
0N
f ;G ><
>: s2 SBN1 0:5 if s2
><
:4
SG
>:
if
< >:
s2G
S1
:3
0:5
if
s2
SB
N1
:3
+0
:5
if
s2
SG
1 ;GB
N
22
G
oderat
m
e
hazardrate
oderat
m
e
hazardrate
opt
imistic
hazard rate
opt
imistic
hazard rate
price
price
: 1 ;G
That is, moderates prior probability for the normal event increases and their
probability for the good event decreases, while the opposite happen s to
optimists prior. As the right panel of Figure 4 shows, in this case, the increase in bel
ief heterogeneity leads to an increase in the asset price.
Consider the second case in Exampl e 1 in which optimists are optimistic because t
hey nd the good event more likely than the normal event , i.e., they have the prior F.
Suppose the moderate and t he optimist ic beliefs are changed to= F= F1 ;GBwith
density functions given by
0
= 8 1 + 0 :5 if s2 S1
;f
= 8 1(1 0:5) if s2 S0 :1(1 .
;B
> 0 :25 if s2 SB1 0
> + 0 :25) if s2 SB1 :9(1
:25 if s2 S
< + 0 :25) if s2 S
>:
That is, moderates prior probability for t he bad event increases and their rel ative
probability
for good and t he normal event remains constant, while optimist s prior probabil ity for the bad
event decreases. A s Figure 4 show s, in t his case, the increase in belief heterogeneity leads to
a decrease in the asset price.
optExample
2 illu strate s th at the in crease in b elief he te rogen eity has n o rob us t pred ic
tions f or th e as set pric e. In partic ular, th e sec on d part provid es an example in w hich
optimists b e come more optimistic bu t th e ass et price d ecline s, which is in contrast with the
Mille r (1977) hyp oth esis. In view of th e asymme tric dis ciplin ing prop erty of p( s), b oth the
op tim is tic an d mo de rate b e lief s p lay a p art in the dete rmination of th e ass et price. Wh
ile th e increas e of optimists optimis m tend s to in crease th e asse t p rice , the dec re as e in
mo de rates p e ssimism ten ds to de crease it by tighten in g th e n ancial cons traints, an d
the n et e/e ct is amb igu ou s. This obs ervation su gge sts th at the Miller hyp othe sis may not
ap ply in markets in which optimists n anc e the ir p urch as es by b orrowin g from les s op
timistic investors.
I next show th at increas ed b e lief heterogen eity has robu st predictions regarding the ass
et price if the typ e of the ad dition al increas e is also take n into ac count. In the rst c ase of
Examp le 2, the b elie f he teroge neity is con centrated to the lef t of th e def ault th re shold s ,
and the ass et p rice d ecreas es. In the se cond cas e of th e e xamp le , the b elief he teroge
neity is con centrated to th e right of the d efau lt th re sh old s , and the ass et p rice inc reas
es (see also Figu re 4). The n ext res ult shows that thes e prop ertie s are gen eral: b elief hete
rogen eity h as an un amb igu ou s e /ec t on the ass et price if it is con centrated to the le ft, or
to th e right, of the equilibriu m d efau lt th re shold s . Theor em 4 (Level of Heter ogeneity) .
Consider the col lateral equil ibrium characterized in
Theorem 2 and let s denote t he equilibrium l oan riskiness, w hich is al so the threshold state
below which l oans defaul t. Consider a (weak) increase in bel ief heterogeneity, in t he sense
that beliefs are changed to~ F1and ~ F0that satisfy ~ F1 OF1and F0 O~ F0: (i) Suppose the
increase in belief heterogeneity is concentrated to the right of s 1 ~ F1, that is, suppose1( s) 1 F(
s)l oan riskiness s and1 ~ F00( s) 1 F( s)are con stan t over the set smi n; s . T hen the asset
price p, the, and the leverage ratio Lweakly increase. (ii) Suppose the increase in bel ief
heterogeneity is concen trated to the left of s 1 ~ F1, that is, suppose1( s) 1 F( s)and1 ~ F00( s) 1 F(
s)are constant over the set ( s ;sma x). Then the asset price pweakly decreases.
Take n toge ther with th e earlier res ults, this resu lt de monstrates that th e typ e of the b e
lief he teroge neity is a m ore rob us t de term in ant of as set prices than the leve l of b elief h
eteroge ne ity. With en dogen ous n ancial c on straints, wh at inve stors dis agree ab out matte
rs f or asse t prices , to a greater extent th an th e level of the ir dis agree ment.
5 Collateral Equilibrium with Contingent Contracts
Th e analysis in th e p re vious se ction s h as c on ce rn ed th e b aseline setting in which loans
are restric ted to b e non- continge nt and s hort se lling is not allowe d. Wh ile the b aseline mo
de l is
23
a go o d startin g p oint, it is imp ortant to verif y th e rob ustn ess of th e re sults to more gen
eral se ttings , es p ecially b ecau se allowing for a riche r set of c ontracts intro du ces ne w e
conomic f orce s. The an alysis in th is sec tion c onside rs an extens ion in which de bt
contracts c an b e f ully contin gent on the c ontinu ation state s2 S , and it e stablishe s three
res ults. First, the op timal contin gent contract is n ot a simple deb t contract. Rath er it is a
thresh old contract: optimists promise to make a ze ro payme nt in all s tate s ab ove a thresh
old , b ut they promise make a payme nt equal to the asse t value in the s tates b elow th e th re
sh old . Sec ond , th is th re shold contract is su¢ c iently s imilar to a simple de bt contract th at
a ve rs ion of the as ymm etric dis ciplin ing res ult (c f. Th eorem 1) also ap plies in this setting.
Th ird, u nlike th e case with simp le deb t c ontrac ts, th e asse t p rice in th is setting may e xc
eed the valuation of eve n the mos t optimistic inve stor.
5.1 De nition of Equilibrium with Contingent Contracts
A unit contin gent d ebt contract, d en ote d by ’ :S ! R+1 6, is a collection of p romise s of ’ (s) 0
un its in e ach state s2 S , collateraliz ed by 1 un it of the ass et.The b orrower de faults on the
contract if an d on ly if the value of th e as set is less than the promise on th e c ontract. Thus ,
the contract p ays min ( v(s) ;’ (s)) un its . Le t D de note th e s et of all un it deb t contracts. As b
e fore , each d ebt contract ’ is trade d in an anonymou s market at a comp etitive pric e q(’ ),
whe re q( ) is a B orel m easurable fu nction ove r D .Let + i; i2 M(D ) resp e ctively de note typ
e itrad ers long and s hort de bt p ortf olios, whe re M(D ) de notes the se t of Bore l meas ures
over D . Typ e itrad ers s olve an analogu e of problem (7): the y cho ose th eir p ortfolio xA i;xB
i; + i; i to maximize th eir exp e cted p ayo/su b je ct to a b ud get an d a c ollateral con straint
(se e prob lem (A:45) in the app end ix). Given this p rob le m and th e exten ded contract sp
ace, th e equilibriu m is de n ed s imilarly to S ection 2.1.
The ch arac te rization of equilib rium closely follows the analys is in S ection 3. In particu lar,
con sider rst a qu as i-equilib rium by re stric ting trad ers choices with th e cons traint +
0= 1= 0 . To con stru ct a quasi-e qu ilibrium, c on side r deb t contract p rice s:
q(’ ) = E0 [min (v(s) ;’ (s))] , (20)
1+r
wh ich m ake mo derates in di/erent b etween p urch as in g the b ond and any d ebt c ontract ’ .
Th e equilibriu m will b e d etermined by optim is ts p ortf olio choice given the se pric es.
1 6I
a s s u m e t h a t c o nt r a c t s mu s t m a ke n o n - n e g a t i ve p r o m i s e s i n a l l c o nt i nu a t i o n s t a t e
s , b e c a u s e a n e g a t i ve p r o m i s e by t h e b o r r owe r ( w h i ch i s e s s e nt i a l l y a p r o m i s e by t h e l
e n d e r ) wo u l d n o t b e e n f o r c e d by t h e c o u r t s y s t e m i n t h i s e c o n o my s i n c e l e n d e r s d o n o
t s e t a s i d e a ny c o l l a t e r a l . T h i s i s w i t h o u t l o s s o f g e n e r a l i ty, b e c a u s e i f t h e y w i s h , l e n
d e r s c a n a l s o m a ke p r o m i s e s by s e l l i n g a s e p a r a t e c o l l a t e r a l i z e d d e b t c o nt r a c t .
24
5.2 Asymmetric Disciplining with Contingent Contracts
Cons id er optim is ts p ortfolio choice for a given p rice p. The same an alysis f or The ore
m 1 (cf . App e ndix A.2) show s that op timists b orrow by se lling a c ontingent d ebt c ontrac
t, [’ (s) 2 [0;v(s)]]s2S, that maxim iz es th e leve raged return:
[v(s)] E1 1+ rE01[min ( v(s) ;’ (s))] . (21)
1 and
have d ens ity fu
;f0
F
nctions f
(
R [min ( v(s) ;’ (s))] p
L
Th e c ontrac t that maximiz
es
’ this express ion can b e charac te rize d und er the f
;
ollowing as su mption c, which is slightly s tronge r th an ass ump tion (O):
o
)
n
t
=
1
E
1
0
1
Assumption (MLRP). The p rob ab ility d is tribu tions F
f1wh ich are continuou s and p ositive over S , and which satis fy the m on oton e like lih o o d
ratio prop erty: that is,( s) f0( s)is s trictly in creasing ove r S . The an alysis in Ap p en dix A.5 es
tab lis hes that, un de r assu mption (M LRP), the op timal
v(s) if s< s 0
(22
contract is a threshol d contract :
if
s
s,
)
’ s(s) (
f or a th re shold s tate s2 S . That is , op timists make as large a p romise as p ossible f or
states s< s(they give up th e as set in th ese states ), wh ile p romising ze ro for states s s(they
ke ep the as set in th ese state s). Intuitive ly, optimists n d b ad states th e least like ly, an d
thu s the y con centrate all of their payme nts b elow a th re shold state .
E1The ne xt resu lt, which is the an alogu e of Th eorem 1 for contin gent loans , charac terize
s the thres hold state s2 S of th e op timal contract given pric e p. The resu lt also s hows
that, u nlike the case w ith s imple d ebt contrac ts , the maximum price at wh ich op timists d
eman d the as se t is greater than the op tim is tic valu ation,[ v( s)] 1+ r. This maximum p rice
level is given by:
pma = 11 + r Zsscr o v(s) dF0 +
v(s) dF1 , (23)
ssmin
x
Zssmaxcr o
ss
ere 2 S is th e u niqu e s tate su ch f0( scr o ss) = 1.
f1( scr o ss)
s
that
Theor em 5 (A symmetr ic Discipli ning wi th Contingent Contracts). Suppose assumptions (MLR
P) and (S) hold, debt prices are given by (20) and the asset price satis es p2 E0[ v( s)] 1+ r;pma x ,
w here pma xcross;sma x] such that iis given by (23). I n a quasi-equilibrium: (i) There exists s2
[sis a Dirac measure that puts weight onl y at the threshol d cont ract ’ sin (22). Opt imists col lat
eral constraint is binding, i.e., they borrow as much as possible according to the optimal contract
. Opt imists choose xB 1= 0 , i.e., they invest al l of their l everaged weal th in the asset A.
cross(ii) The threshol d state s2 [s;sma x] of the optimal cont ract is characterized as t he
25
crosswh
unique sol ution to:
( s) 11 +
v(s) dF0 + f0( s)
v(s) dF1 . (24)
p= popt;cont
r Zs smin
f1( s) Z ssmax
If instead the asset price sat is es p= pma x, then optimists are indi/ erent between making a l
everaged investment in t he asset by sel ling the safe debt contract ’ scr o ss or in vesting in the
bond.
Note that th e fun ction
( s) is th e an alogu e of the f unc
popt;cont
tion popt
s
> E [ v( s)] opt, give n
v(s) dF1 .
1+ r
s
s
by:
s
opt;cont
17
ma x
1 + rper;cont 1
( s): it d esc rib e s th
e ass et price cond itional on op tim is ts ch oic e of the th re sh old state s. M oreove r, the
form of p( s) is very s imilar to the f orm of p( s), wh ich s ugges ts th at op timism is as ymme
trically disc ip lin ed also in this se tting. In p articular, optimis m ab ou t the relative like lih o o d
of states ab ove sinc reas es th e asse t pric e, while the op timism ab ou t th e relative likelih o
o d of states b elow sdo es n ot in crease the price. The intu ition for this res ult can b e gleane
d f rom the s hap e of the op timal d eb t contract ’. Th is th res hold contract makes the same p
ayment (name ly, ze ro) in all s tate s ab ove the thresh old s, wh ile it has an in creasing p
ayment sche dule in the states b e low th e thres hold s. Hen ce, any op timism ab ou t th e
relative likelih o o d of go o d states d o e s not in cre ase optimists p erceived interest rate, an
d thu s thes e typ es of optimism increas e th e ass et pric e. Howe ver, op timis m ab ou t th e
re lative like liho o d of b ad states increas es optimists p erce ived interes t rate . Thus , th es
e typ es of op timism are re e cted le ss in the as se t price.
1Note als o th at optimists d emand th e ass et even if th e price is gre ate r than th eir
valuation, cap tu re d by the f act that p. To se e the intuition for this res ult, cons id er optimis
ts p e rce ive d intere st rate on th e c ontract ’
s
E0 [min ( v(s) ;’ s
v(s) dF0
s
( s) = E1 [min (v(s) ;’ s
1+ r1
min
(s))]
R(s))] = (1 + r)
min
E
R
Unlike the case with n on-c ontingent loan s,
rper;cont 1
[ v( s)]
1+ r
( s) is n ot nec ess arily greater than r. In partic ular, the a
orrowing e nable s optimists to take loan s which the y p e r
le than b orrowin g at the b en ch mark interes t rate. Cons
asse t even if the price exce eds the ir valu ation,, b e caus
the p urch ase with the se loan s which th ey p e rce ive to b
A c om plementary intuition for this re sult c omes from the f orm of pma xin (23). Th e availability
of f ully contin gent loans e nables op tim is ts to sp lit the as set in a way th at each typ e trad
ers hold th e ass et in the s tate s which the y ass ign a gre ate r p robab ility. Cons equently, the
maximum p rice at which optimists d emand the as set is c alc ulated ac cord ing to an u pp
erenvelop e of th e mo d erate an d the op timistic b e lief s, wh ich e xc eed s the optimistic valu
ation .
1 7T
1
h i s i nt u i t i o n a l s o i l l u s t r a t e s t h e l i m i t a t i o n o f t h e a s y m m e t r i c l t e r i n g r e s u l t w h e
n l o a n s a r e f u l l y c o nt i n g e nt . U n l i ke r e g u l a r d e b t c o nt r a c t s , a c o nt i n g e nt d e b t c o nt r a c
t , ’ s, m a ke s a l owe r p ay m e nt i n s t a t e s a b ove s r e l a t i ve t o s t a t e s b e l ow s. H e n c e , i f o p t i m
i s t s o p t i m i s m i s ch a n g e d i n a way t o a s s i g n a l owe r p r o b a b i l i ty t o s t a t e s b e l ow s, t h e n
t h e a s s e t p r i c e i n c r e a s e s ( u n l i ke t h e c a s e w i t h n o n - c o nt i n g e nt c o nt r a c t s ) .
26
1.5
1
.
5
0
1
1
.
5
0
.
5
1.2
1.15
1.1
1.05
0
.
5
1
1
.
5
1
Figu re 5: The top pan el dis plays the
probability d en sities . The solid (resp . dash
ed) line s in the b ottom pan el illus trate th e
equ ilibriu m with (res p. without) contin gent
contracts.
Th is res ult c re ate s a p re su mption
that ne r levels of nanc ial en gin eering of
loan s can p ote ntially have a large imp act
on as se t pric es.
5.3 Equilibrium Asset Price with
Contingent Contracts
Similar to Sec tion 5, the equ ilibriu m as
set p rice is determine d by combining op
tim is ts optimal contract choice with asse
t m arket clearin g. Th e marke t cle aring
cond ition is analogous to Eq. (14), and is
given by:
pma 8 ( s) p= pmc;cont
wma x w( s) 0if(
x
>
r0;pma x]
>
<
0
E0[ v(
E [ v( s)]
if
s)] 1+ r
max ;cont
1
1+ r
wmax ;cont
1
0
>p
s)
0max ;cont 1
2 ((
s) E0[ v( s)] 1+
wmax ;cont
1( s)
if
.
pric e
densti
i es
( s) = w1 + > > :1 1+
minv(s) ds de notes optim is ts
maximu m rst p e rio d c on su mption go o d given that th e
rR s s ch o os e to b orrow with the c ontingent deb t contract s. Th e equilibriu m as set pric e pan
the th res hold le vel of the optimal c ontrac t s opt;contare characteriz ed by con sidering th e
interse ction of the s trictly de cre asing curve pmc;contcross( s) and th e weakly in creasing cu
Here, wma x;cont 1
rve p( s) over the range s2 [s;sma xma x]. Figure 5 disp lays th e equilib riu m with contin gen
and non- continge nt c ontrac ts . Sinc e p>E1[ v( s)] 1+ r, th e e qu ilibrium as set p rice with c
ontingent c ontrac ts e xce ed s th e op timistic valu ation when ever th e op timistic wealth i
su ¢ ciently large .
27
6 Collateral Equilibrium with Short Selling
Th is sec tion cons id ers an extens ion of th e bas elin e s ettin g in wh ich s hort sellin g is
allowed , wh ich is relevant to un derstand th e data f or the fraction of th e asse ts th at can b e
s hort sold (e.g., for th e ma jority of sto cks). Th e analysis in th is sec tion establish es that a
version of the asymme tric d is ciplining res ult (cf . The ore m 1) app lie s in this s ettin g. I rs t
gen eralize the de n ition of equilib riu m to allow for short s ellin g. I th en charac teriz e trade
rs p ortf olio ch oic es f or any given ass et p rice , p. I nally c ombin e this analys is with ass et
market clearin g to solve f or the equilibriu m p rice.
+A
p otential sh ort se ller of an ass et ne eds to b orrow the asse t from another trad er. B ut sin
ce b orrowing in th is ec onomy is c ollateralize d, s hort selling also n eed s to b e collate ralized
. Formally, a unit short contract , den oted by 2 R 1+ r, is a prom is e of v(s) un its of the con su
mption go o d c ond itional on state s2 S , collate ralized byun its of th e b ond (so that de notes
the valu e of the collateral in the ne xt p e rio d). A trader se lling th e u nit sh ort contrac t can b
e interpreted as b orrowin g th e as set f rom a len de r, an d p os tin g 1+ r1 8un its of th e b on d
as collate ral in a m argin accou nt. In reality, th e lend er of the s ecu rity will as k f or a sh ort fe
e.
In th e mo de l, the s hort fee is implicitly captu re d by th e price of the short contract, qshort( ),
with the lowe r p rice corre sp on ding to a h igh er sh ort f ee.
As in th e b aseline s ettin g, the re are als o non -contin gent un it de bt c ontrac ts, ’2 Rdebt,
each of which is trade d at p rice q(’). I als o assu me th at only a fraction short+debt2 [0;1] of
trade rs can s ell sh ort c ontracts, wh ile only a fraction short2 [0;1] can s ell de bt c ontrac ts
an d le verage . Th ese assu mptions are made to simplify the analys is , bu t they are n ot
unreas on able b ecau se sh ort s elling in n an cial markets (an d to s ome extent, leverage)
is con n ed to a s mall f raction of inves tors. I de note th e s hort sellin g ability of a trad er
with tdebt2 f 0;1g , and the leverage ability with t2 f0 ;1 g. Taking the b e lief hete rogen eity
also into ac count, there are 8 typ es of trade rs , where a typ e is d en ote d by T=Let short;+
T; short; Tcontracts , an d d e n e
i;tshort;tdebt . de note meas ures that rep re se nt typ e T
trad ers p ortfolio of s hortdebt;+ Tf ormaliz ed by as sumin g that; debt; T similarly for deb t
contracts. The ab ove res triction is
short; T = 0 for e ach T= ;tshort
= 0; , and debt; T = 0 for each T= ; ;tdebt =
0 .
Th e de n ition of equilib rium follows close ly De n ition 2 w ith min or chan ges th at take into
acc ou nt th e ad dition al res triction.
As b efore, I rst con side r a quas i- equilib rium in which optimists are res tricted to ch o
ose debt;+ T= 0 (so th ey are not allowe d to bu y de bt c ontracts) w hile mo d erates are re
stricte d to cho ose short;+ T= 0 (so the y are not allowed to bu y s hort contracts). Similar to b
efore, the se
1 8Fo
r d e t a i l e d d e s c r i p t i o n s o f t h e s h o r t i n g m a r ke t , s e e , f o r e x a m p l e , J o n e s a n d L a
m o nt ( 2 0 0 1 ) , D A vo l i o ( 2 0 0 2 ) , a n d D u ¢ e , G a r l e a nu a n d Pe d e r s e n ( 2 0 0 2 ) .
28
restric tion s will n ot b e b indin g in equilibriu m and the qu asi-equilibriu m will c orresp ond to a
collate ral e qu ilibrium. To ch aracte rize the quasi-e qu ilibriu m, I rst c onj ecture an
equilibrium of a p articular f orm in wh ich trade rs are en dogen ously match ed th rough comp
etitive markets .
6.1 Matching of Optimists and Mo derates in Debt and Short Markets
Und er ap propriate parametric res trictions the re e xists a quasi-e qu ilibrium in which trade rs
take the followin g p os itions . Firs t, op timists that can le verage , i.e., trade rs with typ e
T (1; ;1), inve st all of th eir wealth in th e asse t and th ey le ve rage as much as p ossib le
given th eir choice of contract ’. S econ d, op timists that can not leverage, i.e., trad ers with typ
e T21 (1; ;0), invest all of th eir we alth e ith er in th e asse t or th e sh ort contracts s old by mo
derates that can s hort se ll. Third, mo de rates that c an short se ll, i.e., trade rs with typ e
T3 (0;1; ), inve st all of the ir wealth in the b ond an d the y sh ort se ll as much as p oss ib le
give n their ch oic e of contrac t . Fourth , mo de rates th at can not s hort s ell, i.e ., trade rs
with typ e T (0;0 ; ), invest all of the ir we alth either in th e b ond or th e d ebt c ontrac ts s old
by typ e T14trad ers. In othe r words , typ e T1optimists b orrow f rom typ e Tmo de rates th at c
an not sh ort se ll, wh ile typ e T3mo de rates b orrow th e ass et Afrom typ e T24optimists that
cann ot leve rage. To se e the intu ition f or this matchin g, note th at typ e Tmo de rates re qu
ire a greater interes t rate than typ e T43mo de rates to part with th eir wealth (i.e., to le nd), b e
caus e, in e qu ilibrium, the y rece ive a gre ate r exp ec ted return on the ir wealth (s ince th ey h
ave the ability to s hort se ll). This implies that the d ebt contracts sold by typ e T1optimists are
pu rchase d by typ e Tmo de rates. A similar re as oning sh ows that the s hort c ontrac ts sold
by typ e T34mo de rates are b ou ght by typ e T2optimists. Given th is m atching, the charac
terization of the quasi-e qu ilibriu m follows closely the
analysis in Se ction 3 . In p articu lar, cons id er de bt c ontract p rice s given by (9), which c
orresp on ds to the valu ation of typ e Tmo de rates. Given thes e p ric es an d th e as set p
rice p2 E0[ v( s)] 1+ r;E1[ v( s)] 1+ r4 , The ore m 1 continue s to ap ply. That is, typ e Toptimists cho
ose to b orrow and le verage with a s in gle loan with riskin ess sle12 S that s olves p= popt).
Similarly, n ote that typ e T2( sleoptimists mus t b e indi/e re nt b etwe en inves tin g in th e as
set and th e sh ort contracts . Typ e TE1optimists exp ec te d return f rom inve sting in the ass
et is give n by[ v( s)] p2. Thus, c onsid er sh ort contract pric es:
qshort ( ) = 1E1[ v( s)] E1 [min ( ;1)] for e ach 2 R+. (25)
p
Given the p rice s in (25), typ e T2optimists ab sorb any p otential su pp ly of sh ort c ontracts f
rom typ e Tmo de rates. Hen ce, th e equilib rium in the sh ort contract market is d etermined by
typ e T33mo de rates op timal c ontract choice. I n ext ch arac te rize the optimal sh ort contrac t
and sh ow that a ve rsion of th e asymme tric d is ciplining res ult app lies als o in th is se tting.
29
6.
2
A
sy
m
m
et
ri
c
Di
sc
ip
li
ni
n
g
wi
th
S
h
or
t
S
el
li
n
g
m
o
d
e
ra
te
s
sh
or
t
se
ll
ac
c
or
di
n
g
to
a
u
n
it
sh
or
t
co
nt
ra
ct
=
v(
ss
h)
th
at
d
e
fa
ult
s
if
th
e
re
ali
ze
d
st
at
e
is
a
b
ov
e
s
o
m
e
th
re
s
h
ol
d
st
at
e
s
sh
2
S
.
T
hi
s
is
b
ec
a
u
se
,
fo
r
su
¢
c
ie
ntl
y
g
o
o
d
s
ta
te
s,
th
e
va
lu
e
of
th
e
pr
o
mi
se
d
as
s
et
ex
ce
e
ds
th
e
va
lu
e
of
th
e
p
os
te
d
c
oll
at
er
al,
a
n
d
th
e
s
h
or
ts
ell
e
r
n
d
s
it
o
p
ti
m
al
to
d
ef
a
ult
.
T
h
e
n
ex
t
re
su
lt,
w
h
ic
h
is
th
e
co
u
nt
er
p
ar
t
of
T
h
e
or
e
m
1
fo
r
s
h
or
t
co
nt
ra
ct
s,
ch
ar
ac
te
riz
es
th
e
th
re
s
h
ol
d
s
ta
te
ss
hf
or
th
e
o
pti
m
al
sh
or
t
co
nt
ra
c
t.
Give n the prices in (25) and the asse t price p2
s)] 1+ r;
The
or
em
6
(Asy
mm
etr
ic Di
sci
pl
inin
gw
ith
Sho
rt
Sell
ing).
Sup
pos
e
ass
ump
tion
(ML
R P)
hol
ds,
shor
t
cont
ract
pric
es
are
give
n by
(25)
and
the
ass
et
pric
e
sat
is e
sp
E0[ v(
E 1 [ v( s)]
1+ r
, typ e
T3
2 E0[
v( s)]
1+
r;E1[
v( s)]
1+
r .
In a
qua
si-e
quili
briu
m:(i)
The
re
exis
ts s
shsho
rt;
Tis
a
Dira
c
mea
sure
that
puts
wei
ght
only
at
the
cont
ract
=
v(
ssh2
S
suc
h
that
3v(
ssh),
i.e.,
mod
erat
es
(tha
t
are
abl
e)
shor
t sel
l on
ly
the
unit
shor
t
cont
ract
=) 1+
r. T
hes
e
mod
erat
es
inve
st al
l of
their
wea
l th
in
the
bon
d
and
shor
t sel
lt
he
ass
et
as
muc
h as
pos
sibl
e
subj
ect
to t
he
col l
ater
al
con
strai
n t.
(ii
)
T
h
e
th
re
sh
ol
d
st
at
e
s
sh
2
S
of
t
h
e
o
pti
m
al
sh
or
t
co
nt
ra
ct
is
ch
ar
ac
te
riz
e
d
as
th
e
u
n
iq
u
e
so
lut
io
n
to
:
p= pshort
R s s v( s)
h
dF1
( ssh) E[v(s)] =(1 + r) 1 +
F0( ssh1) R s s hs mins minv( s) dF0
s
minRs
min
R s sh dF1
s
sh
. (26)
dF0
Note that pshort ( s
sh
) de
scrib
es
the
price
for
whic
h th
es
hort
contr
this
cu
rve is
stric
tly
dec
re
asing
, with
p
sh
E
1
[
v
(
s
)
]
1
+
a
n
d
r
p
s
h
o
r
t
mE
a
[ v( s)] 1+ r
s
h
o
r
t
short
x
(
s
)
=
( ssh
mi n
( s)
1
f or any give n level of d ef ault th re shold sshf or sh ort contracts, th e ass et pric e is highe r
wh en optimism is c on ce ntrate d more on the re lative likeliho o d of go o d s tates . This illus
trates the asymme tric d is ciplining p rop e rty of the op tim al s hort contract.
The pro of of The orem 6 is relegated to App e ndix A.6. For an intu ition, n ote th at th e sh
ort contract d ef aults ab ove th e th re shold state ssh. Thus, they p ay th e s ame amou nt =
v( s) in th ese states. The n, u sing a s hort c ontract with thres hold sshsh, it is imp oss ib le for
mo de rates to b et on th eir p e ssimism ab out the relative like lih o o d of s tates ab ove s.
Con sequ ently, mo de rates p e ssimism ab ou t th e relative like lih o o d of go o d states is
not re ec te d in the as se t price, as sugges ted by (26). In contrast, m o d erate s c an b et on th
eir p es simism ab out the probab ility of states b elow sshshby se lling th e short contrac t. Thus
, this typ e of p ess im is m is re e cted in the ass et price.
Pu t d i/erently, it is easier for mo d erate s to b et on their p ess im is m ab out the p robab
ility of bad s tate s th an to b e t on the ir p es simism for th e relative likeliho o d of go o d states.
To b e t on the latte r typ es of p ess imism , mo derates ne ed to p ost a highe r level of
collateral (equivalently, the y ne ed to cho ose a short c ontrac t with a h igh d efau lt thresh
old ssh). Henc e, thes e typ es of sh ort sales are more d i¢ c ult to le verage , wh ich leads to
the asymme tric discip linin g resu lt with sh ort se lling. App e nd ix A.6 p rovide s a more
complete intuition th at p arallels the an alysis in S ec tion 3.1.
6.3 Equilibrium Asset Price with Short Selling
Th e equilib rium is characterized by typ optimists and typ e
e T1
T3
E [ v( s)]
le
. Note that typ e T
p + p 1) pE0[m in( v( s) ;v(
1+ r
r
(w1
sle))] 1+
E0[ v( s)] 1+
r;
mo de rates optimal contract choice, along with th e market c learing cond ition for the asse t,
wh ich I d erive ne xt. To s imp lif y the an alysis, sup p ose th e parameters are such that the
e qu ilibrium ass et p ric e satis e s p21
1 optimists s p en d a total
of
un its of the con sump tion go o d on the as set. Here , rec all that leis the fraction of investors
that are able to le verage , w1+ p 1is th e total we alth of optimists, an d th e sec ond term in
(28) is th e le verage ratio. Next note that typ e T2optimists (th at make an u nleveraged inve
stment in th e ass et) sp end a total of
(1 le) (w1 + p 1) W short
(29
)
un its of the cons ump tion go o d on th e ass et. He re , recall that typ e T2shortoptimists are
ind i/ere nt b e twee n bu ying the asse t an d bu ying the short contracts sold by m o d erate
s. Henc e, the y inve st in th e ass et all of th eir wealth ne t of W, which rep re sents th eir e
xp end itu re on sh ort c ontrac ts.
By marke t clearing in short contracts, W short is also equal to typ e T3 mo de
rates total
31
revenue f rom s ale s of sh ort contracts. The an alysis in the ap p en dix s hows th at this e xp
re ssion has a similar f orm to th e e xp re ssion in (28), and it is give n by:
1
W short = sh (w0 + p 0) pv(
Here, sh (w0 +
s) E1sh[m in( v( s)
;v( ssh))] E
p 0 ) de notes th e wealth
( v( s))
1+ r
p . (30)
of typ e T3shortmo de rates, and th e se cond te rm d enotes the short
leverage ratio, th at is, the total valu e of ass et short s old p er un it con sump tion go o d
sp end in g. M arket clearing for the as set implies th at th e total sp e nd in g on the as set,
th at is , the s um of the express ion s in (28) and (29), is equal to the total valu e of the ass
et, p. After su bs tituting f or Wfrom the express ion in (30) and re arranging terms, th e ass
et marke t clearing con dition can b e written as :
le
1
Th is expres sion s hows that s hort selling e /ec tively expan ds the su pply of the asse t, as
cap tured by th e s econ d te rm on the right hand sid e.
The equilib riu m tu ple
;s
sh
(p; s le
1( ssh
op le) = pshort
sh
le
t
sh
and ssh
le
le;
s
[v(s)] and[ ~ f1
v
(
s
)
]
~
F1
s
h
1
sh
sh
=
E
1
le
32
( v( s))
1+ r
sh))] E
w0 + p
) E1sh
lep
w +p1
1
p . (31)
+ (1
) w1 + p 1p = 1 +
0 v( s
shle[m in( v( s)
;v( s))] 1+ r
E0
) is charac terize d by th e op timality c on dition s p =
p( s), alon g with th e market clearin g con dition (31). Note that an increas e in th e frac tion of
short se lle rs, , d ec reas es the asse t pric e b ecau se it in crease s th e e /ec tive su pp ly of th
e as set. Converse ly, an in crease in th e fraction of leveraged inves tors, , inc re as es the as
set p rice b ecau se it increas es the dem an d f or the asse t, as cap tured by th e lef t h and s
ide of Eq. (31).
In ad dition , an in crease in the right- skewn ess of op timis m increase s th e ass et price.
To illu strate this e /ec t, c onsid er an equilib rium, (p; s sh), and sup p ose op timis ts op
timism is change d tothat s atis es ~ E(s) = f0(s) for e ach s2 [0; s]. Th at is, the distribu tion~
F1is equally optim is tic as the d istribu tion F, b ut its op timism is con centrated to th e right of
th e curre nt sh ort d ef au lt thresh old s. By Eqs. (10) and (26), this change in the typ e of
optimism le ads to an inc re ase in b oth def ault thres holds , sle, given the old equilib rium p
rice p. Note also that th e le ve rage ratio in (28) is in creasing in s, and th e sh ort le verage
ratio in (30) is de cre asing in s. He nc e, at th e old equilib rium p rice , this chan ge in crease s
optimists le verage ratio, wh ile it de crease s mo derates leverage ratio. Cons equently, th e
market clearing con dition (31) implies th at th e e qu ilibriu m p rice in crease s. That is , an inc
re ase in this typ e of right-skew nes s of optimism in crease s the ass et p rice also in the se
tting with short se lling.
Intu itively, whe n optimism is more right-s kewe d, op timists leve rage more by cho osing
larger and riskier loan s (cap tu re d by th e inc reas e in s), while s hort s elle rs leverage
less by p osting a gre ate r amount of c ollateral = v( ssh) for each u nit s hort contract (captu
red by th e in crease in s). This inc re as es th e de mand an d de creases th e e/ective su pp
ly for th e ass et (c f. (31)), which lead s to a h igh er equilibriu m p rice .
7 Dynamic Mo del: Financing Sp eculative Bubbles
Th e analysis s o far has con cerne d a two-p erio d ec onomy. However, the asymme tric d is
ciplining res ult als o has dynamic imp lic ations . This s ection c onsid ers a d yn amic exten
sion of the bas elin e setting to analyze the interaction of th e asymmetric d is ciplining
mechanism with the sp ecu lative c om p onent of asse t pric es id enti ed by Harrison and Krep
s (1978). T he an alysis in this sec tion sh ows that th e sp ec ulative bu bb le s are also as
ymme trically disc ip lin ed by en dogen ou s n ancial con straints. I rs t de scrib e the bas ic e
nvironm ent with out nan cial con straints and illustrate th at th e ass et pric e f eatures a sp ecu
lative comp on ent. I th en ch aracterize the dynamic equilib riu m with c ollateral con straints ,
an d analyze the e/ec t of b elief he teroge neity on th e s p ec ulative comp one nt.
7.1 Basic Dynamic Environment
Cons id er an in nite horizon ove rlapp in g gene rations e conomy in which the p erio ds and
ge neration s are d en ote d by n2 f0 ;1 ;:::g . Th ere is a continuu m of trade rs in e ach gen
eration n, who are b orn in p e rio d nand live in p erio ds nand n+ 1 . E ach trad er of ge
neration nhas an end owment of the cons ump tion go o d in p e rio d n, and con sume s on ly
in p erio d n+ 1 . The resou rce s can b e transf erre d b etwe en p e rio ds by inve sting e ith
er in the b on d Bor the as set A. Bon d Bis s up plie d e las tic ally at a normalize d p ric e 1 in
e very p erio d. E ach un it of the b ond yields 1 + run its of the c on su mption go o d in the n
ext p e rio d, and the n f ully de preciates (i.e ., th e b ond pays dividen d only once ). As set
Ais in xed su pply, which is norm alize d to 1 . The ass et yields anun its of divid en ds in e
ach p erio d n. Sup p ose that log d ividen d follows a ran dom walk, that is, the divid en d yield
f ollows the pro c ess
an+1 = ansn+1. (32)
is a ran dom variable with d is tribu tion Fwhich has a d ens ity f unc tion th at is continuou s
an d p os itive ove r S = smi n;sma x Rtrue. Su pp os e also th at 1 2 S and that the mean of
n+1++is normaliz ed to 1 . In other word s, th e ne xt p erio d d ivide nd yield u ctuates arou
Here, sn+1 s
nd the current divide nd yie ld an, with e xp e cted value equal to an. All youn g traders in p e
rio d nobs erve all past realiz ations of th e dividen d yie ld and the
cu rrent realiz ation an, bu t th ey h ave h eterogene ous priors ab out the n ext p erio d realiz
ation an+1. In each p erio d n, similar to the s tatic mo d el, th ere are two typ e s of you ng
traders, optimists and moderates, res p ec tively with priors F1and F0ab out th e next p e rio d
state s. Assumption (Od). Perio d nyoun g trade rs b elie f d is tribu tions F1and F0n+1for the
next p e rio d state sn+1have de nsity fu nc tions fthat are continuou s and p ositive over S .
The mo d erate b e lief distrib ution is given by F01;f= F0truewhile the op timistic d is tribu tion s
atis es F. In addition , trade rs b eliefs for the ran dom variables sn+ k1 OF0, for k 2 , are
identic al and given by th e true distrib ution Ftrue.
33
1 9One
way to interpret this as su mption is th at all trad ers know th e d ivide nd yield pro c
ess de scrib ed in (32), but in e very p erio d , some trad ers (op timists ) b ec ome optim is tic
regard in g the next p erio d re alization.Und er as su mption (Od), optimists exp e ctation f or
th e divid en d yie ld s in any f uture p erio d is given by
[an+ k] = En;1 [an+1] = E1 [an] (1 + ") .
En;1
Here, the parameter
[
] 1 >0
controls optimis ts level of optimis m (recallsth at the tru e distrib ution h as mean equal to
"
1). Cons equently, optimis ts p re sent
discnou nte d value of the fu tu re d ivide nd s can b
+
e c alc ulated as
1
E
n
;
1
[an+ k
= a (1 + ") .
r
k
1k=
1X
ppd (an)
v1
En;1 ] (1 + r)
n
Note that th e mo de rate prese nt discou nted value is given (an) = an
by ppdv 0
= !ian, whe re !i 2 R++. (33)
d
i;n)i2f 1 ;0 g
wi;n
20
n1
n
Lemma 1. Given any history (a0
=r. Thu s, op
timists overvaluation of the ass et is give n by "=r. Intu itively, optimis ts e xp ect th e next p
erio d re alization f or the dividen d yield to b e h igh er, an d th ey e xp ect futu re d ividen d yie
ld s to u ctuate arou nd this h igh er (exp ec te d) leve l. Th is leads to the valu ation di/e ren ce
"=r.
Similar to the bas elin e setting, sh ort sellin g the as set is ruled out by as su mption (S ).
Let (wde note typ e itrad ers en dowme nt of the con sump tion go o d , an d s up p ose
Th at is , young trad ers en dowme nts are p rop ortion al to the cu rrent d ividen d yield of
the asse t. Th is ass ump tion is n ot es sential f or the econ omic res ults , bu t it simp li e s th
e sub sequ ent analysis.Th is c ompletes the d esc ription of th e bas ic e lements of th e
dynamic e conomy. Note th at the ec onomy has a recu rs ive s truc ture. This is b ec au se
the d ividen d yie ld p ro ce ss f ollows a random walk (cf . Eq. (32)), an d youn g trade rs b e
lie fs are f orme d ind ep en de ntly of th e p ast divid end yie ld realiz ations (cf. ass ump tion
(O)). Th is obs ervation lead s to the f ollowing le mma, which p rovide s a su ¢ cient s tatistic
for the d yn amic e conomy and s imp li es the su bse qu ent notation .
;a
34
;:::;a
n
19
20
) of dividend y iel d realizat ions, the current
dividend y iel d ais a su¢ cient statistic for the determination of the equilibrium al l ocat ions in
this economy .
T h e r e c o u l d b e a nu mb e r o f e x p l a n a t i o n s f o r t h e s o u r c e o f t h i s ty p e o f o p t i m i s m . A
s i n S ch e i n k m a n a n d X i o n g ( 2 0 0 3 ) , o p t i m i s t s m ay b e ove r c o n d e nt a b o u t a s i g n a l t h e
y r e c e i ve a b o u t t h e n e x t p e r i o d s h o ck . A l t e r n a t i ve l y, o p t i m i s t s m ay b e s i m p l y o p t i m i
s t i c a b o u t t h e n e x t p e r i o d s h o ck , t h i n k i n g t h a t t h e c u r r e nt p e r i o d i s s p e c i a l . R e i n h a
r t a n d R o g o / ( 2 0 0 8 ) r e f e r t o t h i s ty p e o f o p t i m i s m a s t h i s t i m e i t i s d i /e r e nt s y n d r o m e
.
I t h a n k I va n We r n i n g f o r s u g g e s t i n g t h i s s i m p l i c a t i o n .
In view of th is lemma, le t a an 2 R++
p(a) = 11 + r
Sde
note th e c urrent d ividen d yield,
s sn+1
2S
de note th e n ext p erio d sh o ck, an d p(a) de note th e c urre nt ass et p rice .
7.2 Sp eculative Bubbles without Financial Constraints
As a b en ch mark, I rst con sider the ass et price in an e conomy in which in dividu als
can b orrow and len d free ly in a com p etitive loan market at the b enchmark rate r. In
oth er words , th ere exists no limite d liability or enf orc ement p rob le ms. In th is c ase,
optimists b orrow an d inves t in th e ass et an in nite amou nt wh ene ver th e as set p
rice is b elow th eir valu ation. Hen ce, the equilibriu m ass et price is equal to the
optimistic valuation :
a(1 + ") + Z p(as) dF1 , for all a2 R++. (34)
Th e rst term on th e right hand side is optimists exp ec ted d ividen d p ayo/ from th e
asse t, and the s econ d term is the ir exp e cted payo/ f rom th e sale of th e as set. E q.
(34) provides a recu rs ive ch aracte rization of the as se t pric e wh ich can b e s olved as
p(a) = a(1 + ")r " : (35)
= p(a) pNote that the asse t pric e p(a) is h igh er than op timists prese nt d is counted
valu ation, ppdv 1(a) =a(1+ ") rpdv 1. Th e c omp one nt of the ass et p rice in exces s of the p re
sent d is counted value of the holde r of th e as set, p(a) p(a), is what S ch einkman and
Xion g (2003) c all a sp e culative bu bb le . I als o de nepdv 1(a) p(a)= "r (36)
as the share of t he speculat ive component. The ass et pric e f eatures a sp ec ulative
comp on ent b e caus e optimists hold th e ass et n ot on ly for the higher e xp e cted divid
en d gains in the n ext p e rio d , but also s in ce they are p lan ning to se ll th e ass et to a
trad er who will b e e ve n m ore optimistic than the m in th e ne xt p e rio d . In vie w of th
es e exp ec te d sp ecu lative capital gains , optimists b id u p th e ass et p rice highe r th an
the pres ent discou nte d value of divide nd s.
The express ion in (36) also im plie s that the s p ec ulative c omp onent cou ld re prese
nt a large f raction of the ass et price, even for a relative ly s mall b elie f d isagre ement
"(esp ecially wh en the inte res t rate is low). The rationale for th is ob servation is relate d to
a p owerfu l ampli c ation e/ect: the dynamic mul tiplier. Note th at optimis ts in th e next p e
rio d also exp ect to make sp ecu lative cap ital gains by se lling the as set to yet more op
timistic trad ers in the sub sequ ent p e rio d, which increas es th e price in th e n ext p erio
d. Bu t th is fu rth er in creases the valu ation of c urre nt optimists wh o are plannin g to s
ell to f uture op timists, in creasing th e cu rrent asse t price f urther. In oth er word s, a high
as set price in th e ne xt p e rio d fe eds back into the as se t price to d ay, amp lif yin g the
e/ec t of h eterogene ous b eliefs and leadin g to a large s p ec ulative comp on ent.
I next in corp orate nan cial con straints into this econ omy. With nan cial cons traints ,
the
35
ass
et
pric
e do
es n
ot
ne
ces
saril
ys
atis
fy th
e re
curs
ion
in
(34)
.
Rat
h
er,
th e
ass
etp
rice
lies
be
twe
en
the
op
timi
stic
an d
the
mo
d
erat
e
valu
atio
ns ,
and
the
e
xact
recu
rs
ion
(an
d
the
sh
are
of th
es
p ec
ulati
ve
com
p on
ent)
is
dete
rmin
ed
by
the
typ
e
of
n
anci
al
con
strai
nts .
7.
3
Fi
n
a
n
ci
al
Fr
ict
io
n
s
a
n
d
D
y
n
a
m
ic
C
ol
la
te
ra
l
E
q
ui
li
br
iu
m
+I
m
od
el
nan
c ial
con
s
train
ts
us
in g
the
c
ollat
eral
equi
lib
rium
d
esc
rib
ed
in
Se
ctio
n
3.1.
In
parti
c
ular,
trad
e rs
in
eac
h ge
ner
atio
n
trad
e
coll
ate
raliz
ed
de
bt
cont
rac
ts
that
mat
ure
in
the
ne
xt p
e rio
d.
As
in
the
bas
e
line
setti
ng,
de
bt
cont
rac
ts
are
non
- rec
ours
e an
dn
on
-con
ting
ent.
For
mall
y, a
unit
debt
cont
ract
,
den
ote
d by
’2
R, is
a
pro
mis
e of
’un
its
of th
e
con
sum
p
tion
go o
d in
the
n
ext
pe
rio d
by
th e
b
orro
wer
coll
ate
raliz
ed
by 1
un it
of
the
as
set.
Giv
en
th e
curr
ent
divi
den
d
reali
zati
on
a, I
de
ne
the
valu
e
func
tion
as
th e
p
ayo/
of
the
ass
et
in
the
ne
xt p
erio
d,
T
h
e
v
(
a
;
s
)
a
s
+
p
(
a
s
)
f
o
r
e
a
c
h
s
2
S
.
(
3
7
)
d
e
b
t
co
nt
ra
ct
’d
e
fa
u
lts
if
a
n
d
o
nl
y
if
v(
a;
s)
<’,
a
n
d
th
us
it
p
ay
s
mi
n
(
v(
a;
s)
;’)
.
E
ac
h
d
e
bt
c
o
nt
ra
ct
’2
Ri
s
tr
a
d
e
d
in
a
n
a
n
o
ny
m
o
u
s
m
ar
ke
t
at
a
c
o
m
p
eti
tiv
e
pr
ic
e
q(
a;’
).
L
et
xA
i(a
)
;x
B
i+(
a)
d
e
n
ot
e
ty
p
e
itr
a
d
er
s
as
se
t
a
n
d
d
e
bt
h
ol
di
n
g,
a
n
d
+
i(a
)
; i
(a
)
d
e
n
ot
e
th
e
ir
lo
n
g
a
n
d
s
h
or
t
d
e
b
t
p
or
tf
oli
os
.
T
h
e
tr
a
d
er
s
p
ro
b
le
m
is
gi
ve
n
by
:
xA (a) Ei [v(a;s)] + xB Ei [min (v(a;s) ;’)] d i (a;’) ,(38)
i
i(a;’) RR+
Ei[min ( v(a;s) ;’)] d + i(a) (1 + r) + R
max
x( a) 0 ;
ii( a) ; i( a)
R+
+
(a) + xB iA i
(a) + ZR
(a;’) ZR+
Bi
q(a;’) d
+i
+ q(a;’)
i
+
d
(a;’)
(a) ;
R
p(a) ;[q(a;’)]’2 R+
+d iA i(a;’)
x(a) .a, Z
s
.
De ni tion
t 4 (Dynamic Col lateral E quil ibri um). Under assumptions (O) and (S), and
condition
. (A:50), a dynamic col lat eral equil ibrium is a col lection of prices a2 R++and al l
ocations such that, for each dividend real ization a 2 R xA i++(a) ;x(a) ; i, the al location of
each trader
i 2 f1;0g solves problem (38), and asset and unit debt markets cl ear.
p
x
!i
d
(a)
i
Note that, give n the value fu nction in the next p erio d (cf . Eq. (37)), th e e conomy in the
cu rrent p e rio d is very similar to the s tatic e conomy analyz ed earlier, with the main di/eren
ce that th e value f un ction , (37), also d ep e nds on the price fu nction. He nce , th e
dynamic equilib riu m is characterized with a xe d p oint argu ment. Th e line ar h omogene
ity of en dowm ents (cf . con dition (33)) en su re s th at th e pric e to d ivide nd ratio and the
loan riskin es s are in dep end ent of the cu rrent realization of a. The p ro of of the f ollowing
th eorem is relegated to Ap p en dix A.7.
36
i2f 1 ;0 g a2
E (pdTheor em 7 ( Existence and C har acteri zation of Dynamic E qui libr ium). Under
assumptions (Od), (S) and the parametric condition (A:50) in Appendix A.7, there exists a
recursive col lateral equilibrium in which p(a) = pda and s (a) = s dfor each a2 R. T he price to
dividend ratio, p pmi n d;pma x dd, is the unique xed point of the mapping P , w here Pd( ~pdd: h
pmi n d 1 r++;pma x d) is the col l ateral equil ibrium price of the st atic econ omy) = S ; v(sj pd) =
s(1 + pi) ; fFgi; f wi !igi; f 1d= 0; 0= 1g 1+ " r "i !! . (39)
Note that th is resu lt redu ce s the characteriz ation of the d yn amic equilib riu m to th e ch
arac teriz ation of the e qu ilibrium for th e static e conomy, E (p), alon g with a xed p oint
argu ment. Intuitively, Pd( ~pdd) is th e p rice to divid end ratio that wou ld obtain to day if the
futu re price to divid end ratio was given by ~pd. The u pp e r limit of the xed p oint interval,
pma x d=1+ " r ", is the pric e to divid end ratio th at would ob tain if op tim is ts always pric ed th e
ass et (i.e., it is the pric e in th e un con straine d ec onomy). Th e lowe r limit, pmi n d=1 r, is the
price to dividen d ratio that would obtain if mo derates always priced the ass et (i.e., it is the
mo de rate valu ation of the ass et). Th e equilib rium is in the inte rval pmi n d;pma x d . The n
ext example us es thischarac te rization to illus trate the e /ec t of n an cial c onstraints on th
e sp e culative comp on ent of th e as set p rice .
and F1 ;G [s] E0
1
Example 3. Consider the prior distributions F0
= 4. Figure 6 plots the price mapping, Pd
1 37
of Exampl e 1 in which the val uation
di/ eren ce for the next period shock is given by "= E[s] = 0 :1. C onsider the correspon ding dy
namic col lateral equil ibrium wit h int erest rate r= 0:15 and optimistic wealt h !( ), and shows t
hat it intersect s the 45 degree l ine exactly once, which corresponds to t he equil ibrium. The
equilibrium price is low er than the unconstrained l evel , how ever it is stil l higher than the
present discount ed val ue according to either the moderate or opt imistic priors (w hich are
close to each other). I n particular, in this exampl e, the price has a large speculative component
despite nancial constrain ts.
The gure al so il lust rates optimists balance sheet. O ptimist s downpay ment is about 1/4
of the asset price, and they borrow the remaining amount from moderates, col lateralized
against one unit of the asset. In particul ar, moderat e l enders, w ho correctl y know the
dividend yield process in (32), agree t o nance about 3/4 of the asset purchase despite the fact
that the present discounted val ue of the asset is l ess than half of it s price.
The las t featu re of this e xamp le provid es ins ights f or h ow the p rice can feature a large
sp ecu lative c omp one nt wh en optimists are nan cially con straine d. In th is examp le , len
ders have corre ct priors an d the y know that th e as set p rice is cons id erably greater than
th eir pres ent d is counted valuation. None theles s, they agree to extend large loans w hich
are in part collate ralized by the sp e culative comp onent of the price. This is b e caus e len
de rs valu ation of the ass et (th e lower green line in Figu re 6) also contain s a s p ecu
lative comp on ent, and thus it is highe r than their p re sent d is counted valuation. Intu itively,
len de rs agre e to exte nd large loans
20
18
16
14
12
10
8
6
4
2
8 10 12 14 16 18 20 22 0
Figu re 6: Th e xaxis is th e range of p oss ib le p rice to d ividen d ratios, pmi n;pma x . The
lowerdand higher gre en c urves res p ectively p lot the mo d erate and th e optimis tic valu
ations when the f uture price to d ividen d ratio is given by th e value at the xaxis . The re d cu
rve (inte rm ediate to th e two gree n cu rve s) p lots th e pric e mapp in g, P ~pdof th e red cu
rve with the 45 de gree lin e (d ashe d b lu e c urve). . Th e equilib rium is the intersec tion
38
20
18
16
14
12
10
8
6
4
2
8 10 12 14 16 18 20 0
Figu re 7: The lower (re sp . th e h igh er) re d line plots the cu rrent price to divide nd ratio as a f
unc tion of th e f uture p ric e to dividen d ratio for the b elie f distrib ution s in the rst cas e (resp
. se cond case ) of E xamp le 1. T he d ynamic equ ilibriu m is th e interse ction of th is cu rve
with the 45 de gre e line (d ashe d blue curve).
b e caus e th ey th in k that, sh ould the b orrower d efau lt, the y cou ld always se ll the collate
ral to anothe r op tim is t in the ne xt p erio d.
Pu t di/erently, a marke d characteris tic of this sp ec ulative episo de is that the bu bble
raises all b oats : b oth th e op timistic an d th e mo de rate valu ation s are greater than their
pres ent dis counted valuation s. Con seque ntly, optimists an d mo derates valu ation d i/e re
nce in any p e rio d (th e d i/erenc e b etween the two green line s in Figu re 6) is relatively s
mall. As in the un con straine d case, a large s p ec ulative bu bb le f orms f rom th e accu
mulation of s mall valu ation di/e re nce s th rou gh th e dynamic multiplie r. Th is is p erhap s unf
ortu nate, b ec ause a small valuation di/eren ce makes th e n an cing of the as set re latively e
as y, op ening the way f or large sp ecu lative b ub bles even whe n op timists are n an cially
con strain ed.
Natu rally, as th e p re vious se ction s s how, a s mall valu ation di/eren ce do es not
guarantee that n an cial c on straints are lax. Wh ethe r n anc in g will actu ally go throu gh,
and th e sh are of th e s p ec ulative c omp one nt, als o dep end s on a nu mb e r of other f
actors , su ch as op timists we alth le vel and the typ e of b e lief h eteroge neity. For example,
con sider the equ ilibriu m in Examp le 3 with th e only d i/e re nce th at the op timistic priors are
ch an ged to F1 ;B(de ne d in Examp le 1). Th is p rior leads to the same ass et valu ation, but it
is more lef t-skewed than F1 ;G. Figu re 7 s hows that, in res p onse to this ch ange, the s p ecu
lative comp one nt shrinks by ab ou t half .
The ne xt re su lt sh ows that this is a gen eral p rop e rty, th at is, an in cre ase in th e
rightske wnes s of op timis m un ambiguou sly inc re ases th e ass et price an d the sh are of the
s p ec ulative comp on ent. To s tate the res ult, I de ne the overvalu ation ratio d2 (0;1] as th e
u niqu e
39
( j
+ d E1 [ ( j
pd
v pd
solution to pd
. (40)
= (1
d)
E0
[v )] 1 + r
d)]
1+r
d
Intuitively,
d
cap tu re s th e f raction of the optimis m in prior b elief s th at is re ec ted in the asse t
price. I ge neraliz e the sp ec ulative comp one nt of th e ass et price (c f. Eq. (36) to
the nanc ially con straine d e conomy) as
= p(a) ppdv(a) , whe re
(a) = (1 d) ppdv 0 (a) + dppdv (a) . (41)
1
p(a)
ppdv
d
Unlike the un cons trained cas e, the margin al h old er of th e ass et is n ot n ec ess arily an
optimis t, he nc e the relevant prese nt d is counted value is de ne d as an ave rage of optimis
tic an d mo de rate pres ent disc ou nte d value s, weighte d by the overvalu ation ratio d2 1.The
f ollowing res ult es tab lis hes th at an in crease in th e right- skewn ess of b elief h eterogene
ity increas es the as se t price and th e s hare of the sp e culative comp on ent.
Theor em 8 (E /ect of Typ e of Heter ogeneity on the Sp ecul at ive Comp onent). Consider the
recursive col lateral equilibrium charact erized in Theorem 7 and let s ddenote the equilibrium
loan riskiness.
(i) I f optimists optimism becomes weakly more right-skewed, i.e., if their prior is changed
to~ F1that satis es ~ F1 RF1and ~ F1 OF0(so that assumption (O) continues to hold), then: the
price to dividen d rat io pd, the l oan riskiness s dd, and the share of the speculat ive
component dweakly increase. (ii) If moderates optimism becomes weakly more skew ed to the
left of s d, i.e., if their prior is changed to~ F0that satis es F0 R; s d~ F0and Fand the share of the
specul ative component d1 O~ F0, then: t he price to dividend ratio pweakly increase.d
Intu itively, if optimis ts optimis m b ec omes more right-s kewe d, the n f uture op timists will
p e rce ive lo ose r nanc ial con straints and they will b e able to bid up th e ass et pric e h igh
er. This implies that the resale option valu e to fu ture op tim is ts is h igh er, which le ads to a
gre ate r sp ecu lative comp on ent. Convers ely, if optimists optimism b e comes more le ft-s
ke wed , th en the sp e culative comp one nt b ecome s smaller b ec au se th e f uture op timists
will p erce ive tighte r n ancial cons traints . This res ult shows that bu bbles can c ome to an en
d b e caus e of a s hift in b e lief he teroge neity towards th e likeliho o d of bad events.
8 Conclusion
In this p ap er, I h ave the ore tically analyz ed the e/e ct of b elief he teroge neity on asse t
prices . The central f eature of the mo del is that, to take p osition s in line w ith th eir b e lie fs,
inves tors ne ed to b orrow f rom traders with di/erent b elief s us ing collate ralized contracts. Th
e len ders
2 1T
h e s h a r e o f t h e s p e c u l a t i ve p r e m i u m i s i n d e p e n d e nt o f t h e s t a t e a 2 Rb e c a u s e t h e f
u n c t i o n s p (a) ; ppd v 0(a), a n d ppd v 1(a) a r e l i n e a r l y h o m o g e n e o u s i n a.++
40
do not valu e th e c ollateral as much as th e b orrowe rs do, which rep re sents a con straint on
inve stors ab ility to b orrow an d leverage the ir inves tments . I have cons id ered th e e/ect of
this c onstraint on ass et pric es in a variety of s ettin gs that di/er in the typ es of collateraliz ed
contracts that are available f or trade . In the base line mo de l, I h ave res tricted attention to
non -contin gent loans and disallowed sh ort se lling, and I have relaxe d thes e re strictions in
two exten sions of the mo de l. In each of the se sce narios, my p ap er h as es tab lis hed that
optimism is asymmetric ally d is ciplined by e nd oge nous nan cial cons traints . In partic ular,
optimism ab out th e like lih o o d of bad states h as a sm aller e/e ct on as set p rice s than
optimism ab ou t the relative likelih o o d of go o d states. I h ave also cons id ered a dynamic
exten sion of the mo de l wh ich reve als that the sp e culative ass et price bub bles , identi ed by
Harrison and Krep s (1978), are also asymme trically d is ciplined by optim is ts n an cial cons
traints .
Take n togeth er, my res ults s ugges t th at certain ec on omic environme nts that gen erate
un ce rtainty (an d thu s b elief he teroge neity) ab ou t up side returns are con duc ive to as se t
price inc reas es an d sp ecu lative bu bb le s n ance d by c re dit. This p re diction is in line
with the obse rvations in Kin dleb e rger (1978), who has argu ed that sp e culative e piso de s
typ ic ally f ollow a n ovel event (which arguab ly gene rates u pside u nc ertainty), and th at the
eas y availab ility of cred it p lays an imp ortant role in th ese ep is o d es .
The as ym metric d is ciplining ch aracterization of as set prices also emp has iz es that what
inve stors d is agree ab ou t matters for ass et p rice s, to a gre ate r extent than the level of the
dis agree ment. In p articular, whe n op timists are n ancially c on strain ed, an inc re as e in the
level of b e lief h eterogene ity in gen eral h as ambiguous e/ects on as set pric es. Howeve r,
the e/ect can b e characteriz ed on ce the s ke wne ss of the in cre ase is take n into accou nt.
Ad dition al b e lief heterogen eity ten ds to d ec reas e asse t p rice s whe n it con cerns th e
likeliho o d of bad states , bu t it ten ds to inc re ase asse t p rices wh en it c on ce rn s th e
relative likelih o o d of go o d states . A growin g e mpiric al lite rature in n ance c onsid ers th e
e/e ct of the level of b e lief heterogen eity on ass et p rice s an d sub se qu ent ass et returns
(e.g., Chen , Hong an d Stein, 2001, Die th er, Malloy and Sche rb in a, 2002, and Of ek and
Richardson , 2003). My pap e r s uggests that a fruitfu l f uture re search d irec tion may b e to
emp irically investigate th e e/ec t of the s ke wne ss of the b e lief he teroge neity on as set pric
es.
41
A
A
p
p
e
n
di
c
e
s
A.
1
Pr
o
p
er
ti
e
s
of
O
pt
i
m
is
m
O
rd
er
Th
is
ap
p
en
di
x
es
ta
b
lis
he
s
th
e
pr
op
ert
ie
s
of
op
ti
mi
s
m
or
d
er
(c
f.
D
e
ni
tio
n
(1
)).
C
on
si
de
r
tw
o
pr
ob
ab
ilit
y
di
str
ib
uti
on
s
H;
~
H
ov
er
S
=
s
mi
n;
s1
~
H(
s)c
on
tin
uo
u
s
an
d
p
os
iti
ve
at
ea
ch
s2
S.
I
rst
sh
o
w
th
at
1
H(
s)
ma
x
R
wi
th
c
or
re
sp
on
di
ng
de
n
sit
y
fu
nc
tio
ns
h;
~
ht
ha
t
ar
e1
~
H(
s)i
s
str
ic
tly
in
cr
ea
si
ng
at
so
m
e
s2
Si
f
an
d
on
ly
if
th
e
ha
za
rd
rat
e
in
eq
ua
lit
y
in
(2
)
is
s
ati
s
ed
.
To
s
ee
th
is
,
co
ns
id
er
th
e
d
eri
va
tiv
e
of
1
H(
s)
1 ~ H(s)1
H(s) =
d ds
an
d
no
te
th
at
~ h(s) (1 H(s)) + h(s)
H(s))
2(1
1
~
H(s)
mi n,
x),
for e ach s2 [s;sma
th
is
ex
pr
es
si
on
is
p
os
iti
ve
if
an
d
on
ly
if
th
e
ha
za
rd
rat
e
in
e
qu
ali
ty
(2
)
ho
ld
s.
I
ne
xt
sh
o
w
th
at
th
e
op
ti
mi
s
m
or
d
er
is
w
ea
ke
r
th
an
th
e
m
on
ot
on
e
lik
e
lih
o
o
d
rat
io
pr
op
ert
y
(M
L
R
P)
,
that is, if ~ h( s)is stric tly in creasing over S, th en~ H O
ma x
h( s)
ma x,
H.
To
se
e
thi
s,
s
up
p
os
e
(M
L
R
P)
h
ol
d
s
an
d
no
te
th
at
th
is
im
pli
es
,f
or
~ h(s)h(s) h(
~s) <
ma x
e
ac
h
s<
s
~ h( ~s) for all ~s2
(s;s
)
.
Inte grate b oth side s of this e qu ation ove r
(s;s
~ h(s) (1 H(s)) <
h(s)
) to get
1 ~
H(s) ,
~HO
i
( ).
opt
per 1
Lem ma 2. Consider two probabil it y distributions F1
i
0
( s) >rper 1
rper 1
mi
n
mi n
op t
<0.
opt
which p rove s the hazard rate ine qu ality (2) and sh ows H. I n ext note th e followin g re sult, which d erives the implications of assu mpti
th at
key varid ( s)
. (iii) p( s) is continuousl y di/ erentiabl e an d strictl y decreasing,
p ds
i.e.,
ables use d in th e analys is , inc lu ding the exp ecte d payo/ of a loan with riskin es s s, E[min ( v(s)
= rfor each s>s per ( s) (cf
1
icular,
;v( s))], the p erceived interes t rate, r( s), and the op timality curve, p
s
and Fthat satisfy assumption (O ). (i) T he expected pay o/ of a loan with riskiness s, E[min ( v(s)
;v( s))], is strictly increasing in s. (ii) O ptimists perceived interest rat e r
Pro
of of
L
em
ma
2.
Par t
(i ).
Note
that
the
de
rivati
ve of
Ei s
smin[
min
(
v(s)
;v(
s))]
=
RdE
v(s)
dFii(
s) +
v(
s) (1
F[mi
n(
v(s)
;v(
s))]
d si (
s))
is
give
n
by=
v(
s)
f i( s
)+
v0(
s) (1
Fi(
s))
v(
s)
f( s
)=
v0(
s) (1
Fi(
s))
>0,
(A.1
)
w
hi
ch
c
o
m
pl
et
es
th
e
p
ro
of.
4
2
Part (ii) . The de rivative of
1+ rp er
1( s) 1+ r
=
E1[m in( v( s)
;v( s))] E0[m in( v(
s) ;v( s))]
can b e calc ulated
as
2
(E0 [min ( v(s)
;v( s))])
dE1[m in( v( s)
d 1 + rper
=
E0 [min (v(s) ;v( s))] E1 [min ( v(s)
;v( s))] d s
d 1( s) 1 + r
;v( s))]
s
0
( s))
[min ( v(s) ;v( s))] (1
, (A.2)
E1
F02
= E0 [min ( v(s) ;v( s))] (1
( s)) (E[min ( v(s) ;v( s))])
F1
1 F10( s) 1 F( s)
smi n;sma x , (A.3)
whe re the las t line us es Eq. (A:1).
for e ach s2
I n ext claim th at
E1[min (v(s) ;v( s))] E0[min (v(s)
;v( s))] <
1
F1
w
he
re
th
e
rs
t
in
e
qu
ali
ty
us
es
1
F
1
0
(
s
)
1
F
(
s
)
,
dE1[m in( v( s)
;v( s))] d s
th
e
ha
za
rd
rat
e
in
e
qu
ali
ty
(2
)
an
d
th
e
se
co
nd
in
eq
ua
lit
y
us
e
s
th
ef
ac
t
th
at0
(
s)(
s)
1
Fo
pt(
s)
in
E
q.
(1
0)
,
no
te
th
ati
s
str
ict
ly
in
c
re
as
in
g.
Th
is
pr
ov
es
th
e
c
lai
m
in
(A
:3
)
an
d
c
o
m
pl
et
es
th
e
p
ro
of
of
thi
s
pa
rt.
P
art
(iii
).
U
si
ng
th
e
d
e
n
iti
on
of
p
dpopt( s)
ds
f0 ( s) ( s) + f1( s)1
11 + r 0 v( s) f0( s) +
@
F01( s)( s
)1 F
1
s
v(
s)
dF1
1
F1( s)
1
A
1
0
0
( s)
R smax
1
which, in view of Eq. (A:2), proves that the p erceived interest rate 1 + r per 1( s) is s trictly inc re asing. To prove the claim, note that f or
F( s)01( s))+ v(
1 F0 ( s) 1 v(
F
s) (1 F s sminv(s) dF0+ v( s) (1 F01( s)) R( s)) =
s) (
f
s)
1
=1
1+
f( s) 1
f( s)
( s)) ( [v(s) j s s] v( s)) ,(A.4)
(1 F0
r
F( s)
1 F
E1
whe re the rs t lin e app lies the ch ain ru le and the s econ d line sub stitu te s E[v(s) j s s] and re ( ) and pmc ( ).
arranges te rms . The te rm, f00( s) 1 F( s)f11( s) 1 F( s) 1op t, in Eq. (A:4) is p ositive from th e hazard
rate in equality (2). Sinc e the term s, (1 Fdp( s) d s0( s)) and (E1[v(s) j s s] v( s)), are als o p
ositive, it follows that<0, comp le tin g th e p ro of of the lemma. I next present th e n al resu lt of this
app e ndix, wh ich u ses as sump tion (O) to d erive th e e/ec ts of
an in crease in optimis ts (mo d erate s ) optim is m on th e cu rve s popt
and F0
1
O
Through ou t th e app en dice s, th e notation
~ E1
22
43
Lem ma 3. Consider two probabil it y distributions that satisfy assumption (O ). (i) Suppose optimists become w eakl y more opt imistic,
consider their beliefs are changed to
F1
~
mi n;sma F1. T hen: (i.1) Conditional expectations
~ [v(s) j s s] E
x
increase, that is,
F1
E1
2
2).[v
(s) j
s s
] for
each
s2
[s
[sj
s
s]
co
rr
e
sp
on
d
s
to
th
e
co
nd
iti
on
al
e
xp
ec
tat
io
n
(i.2) T he optimal it y curve
( s) shifts up pointwise, that
popt
is,
s; ~ F1
popt ( s; F1) if pmc ( s; F1
.mi n;s
1
pmc
ma x(
s; F1) for
each s2 s
o
p
t
m
(
(
s
;
s
;
F
F
1
1
1
[ v(
(ii) Suppose moderates become weakly more
c optimistic, i.e., consider their bel iefs are changed 1+ r
to
~
F
)
i
f
p
m
c
s
( = pmc
)<
;
O
o
p
t
(
w
h
i
c
h
E [ v( s)]
1+ r
~ F0
( s) shifts up pointwise, that
is,
a
l
s
o
s
a
t
i
s
e
s
F
1
pop
t
p
(i.3) T he market clearing curve changes as fol lows:
;
s ~
F0
popt
mi n;sma
x
. (A.5)
~
F0
F0 O
so t hat assumption (O) continues to hol d). Then: (ii.1)T he optimal ity curve p
(ii.2) The market cl earing curve mc ) for each s2 s( s) shifts ( s;
p
up pointwise, that is,
F0
pm
c
s ~
;
F1
pmc
( s; F1) for each s2
x .
.
smi n;sma
Pro of of Lemma 3. Par t (i .1). De n e th e f unc tion g: S! Rwith
g( s) = ~ [v(s) j s
E1
s] E1
;s [v(s) j s
s] . (A.6)
th at g(s
) = 0, an d note als o th at the s tatem ent in the lemma is equivalent to th e followin g claim:
mi ng( s)
0 for each s2 ma x). (A.7)
[s
I will rst nd an upp er b oun d for the de rivative of g( s) which I will th en use to p rove the claim in
(A:7). To put an u pp e r b ou nd on the d erivative of g( s), c onside r rs t th e de rivative of the
ma xNote
cond itional
exp ec tation
mi n[v(s) j s s] at some s2 [s;sma x
E1
). With s om e re arrangeme nt, th is derivative c an b e written as
d
[v(s) j s s] = f11( s) 1
[v(s) j s s] v( s)) .
d sE F( s) (E1
1
Using th is express ion , th e d erivative of g( s) can b e writte n as
f( s) = ~( s) 1 = 1~ ( s) 1 F1( s)1f[v(s) j f11( s) 1 F( s)[v(s) j s s] v( s))
F1~ f1( s) ~ F1( s) 1 ~s s] v( s) ! ~ E1 (E1
1( s) 1 F
g0E1( s)
f
( s) g( s) ,
[v(s) j s s] v( s) +
1
(A.8)
whe re th e sec ond lin e follows by rearranging te rms an d su bstituting th e de nition of g( s)
from Eq.
~ E1
~ F1.
44
(A:6). Note that [v(s) j s
s] v( s) >g( s) and the rs t term in Eq. (A:8) is always non of op timists according to
the b elief distrib ution
to p rove th e c laim in (A:7), s upp os e the contrary, th at is , su pp ose th ere
exists ~s<s
su ch that g( ~s) <0 . Cons id er ne xt
ma xNext,
ma x^s=
sup f s2 [ ~s;s) j
g( ~s)g .
ma xma xsinc e g(s) = 0 . Then , E q. (A:9) app lies for ^sand imp lie s g0Note th at ^sexists
and that g( ^s) = g( ~s) <0 by the c ontinu ity of th e fu nction g( ). This fu rthe r implies
that ^s6 = s( ^s) ~ f( ^s) 1 ma x1~ F1( ^s)g( ^s) <0. This f urther imp lie s that th ere exis ts s2
( ^s;sopt( s) can b e writte n as) su ch th at g( s) <g( ^s) = g( ~s), wh ich contrad ic ts
the de nition of ^s. This proves th e c laim in (A:7) and comp le te s th e pro of of the rs t
part. Part (i.2). Note , by E q. (18), that the optimality curve p
pop ( s) = 11 + r [v(s)] + (1 F0 ( s)) ( [v(s) j s s] E0 [v(s) j s s])) .
t
(E0
E1
p ositive (sinc ~ O F1), which provid es the f ollowing upp e r b ound on th e d erivative (
e
F1
of g0
s):
mi ( s) g( s) for e
1
g f( s) ~1~
n
0 F( s) 1
ach s2 [s
g( s)
;sma x). (A.9)
: S! Rwith
The n, u sing part (i.1) sh ows th at poptma x( s) in (13) and note th at wma x( s) sh if ts u p p ointwis
e, comp le tin g th e p ro of. Part (i.3). Cons id er the d e nition of w( s) do es not d ep en d on
F1mc, as it d ep en ds on mo derate s valu ation of deb t contracts. E q. (A:5) then follows by the
de nition of p( s) in (14). Intu itively, th e change,~ F1 F1, on ly a/ec ts pmcmc( s) by in cre asing
optimists valuation . Thu s, it only shif ts th e p( s) cu rve in c ase (ii) region of Eq. (14), while it
leaves it con stant in oth er case s.
Part (ii. 1). Similar to p art (i.1) of th e lem ma, d e ne th e f un ction gmix
( s) =
gmix popt
;
s ~
F0
( s; F0)
popt .
g
I will prove a s tronger claim, that
Note that the statement in th e lemma is equivale nt to the claim:
mix
( s)
0 for each s2 smi n;sma x . (A.10)
To p rove the claim in (A:11), n ote that using Eq. (A:4) and rearranging term s, the d
erivative of
gmix
d
g
mix
( s)
ds
0 f or e ach s2 smi nmi n;sma x , (A.11)
= 0.
which imp lies the claim in (A:10) sinc e gmixs
( s) can b e writte n as
f11( s) 1 (
F( s)
s) ~
( s # (E1 [v(s) j s
fF ~00( s) )
F0
dgmix ( s)
"f
= 11 +
ds
r
O ~ F0O F0 implies
Next note that F1
f
0
F
0
s] v( s)) . (A.12)
0
( s)
f00( s) 1
F( s)
0
0
0
4
5
. Af ter re arran gin g term s, th is f urth er im plie s
( s) ~ f( s)
~( s) F( s)
~ f~ F( s) 1 f( s) 1
( s)
F
0~ f0~
F0( s) 1
( s)
1
f11( s) 1
F( s)
1
( s) .
Using th is inequality in E q. (A:12) and noting th at
[v(s) j s
s] v( s)
0 proves the claim in
E1
O
F0 implies
~ E0
[min ( v(s)
;v( s))] E0
[min ( v(s) ;v( s))] for each s2 S
:
~
F0
(A:11), comp le tin g th e p ro of of the lemma. Part (ii .2). First note that, applying th e argu ment in
part (iii) of Lemma 2 f or the d is tribu tions
By Eq. (13), this f urther implies w ma
s ~ F0
wma ( s;
F0
). Us in g th is inequality an d the fact that~ E0[v(s)] E0[v(s)], Eq. (14) implies th at p( s) sh if ts u p
x1
x 1mc
;
p ointwis e, comp le tin g th e p ro of.
A.2 Characterization of Quasi-equilibrium
This se ction c omp letes the analys is of the quasi-e qu ilibriu m, by p roviding th e pro ofs f or Theorem 1
and Eq. (14).
Pro of of Theor em 1. I prove th e theorem in two step s. I rst sh ow that ’= v( s) 2 supp
1 only
if smaximize s the leveraged return e xp res sion in (11). I then s how that the p roblem h as a unique
solution ch aracterized as the solution to Eq. (10). This es tab lish es that 1is a Dirac measu re at the
contract v( s), comp le ting th e s ketch pro of provid ed after the th eorem s tateme nt.
To prove th e rst ste p, rs t note that optimists d ebt contract choice can b e re stricted to’ 2 v mi
ns ;v(sma x) without loss of gene rality, i.e., su pp os e 1(C) = 0 for e ach C
R+n v mi ns ;v(sma
x) 2 3.Cons id er the chan ge of n otation ~s = v 1(’) and let de note th e pu sh forward measu re
of 1over S= s ~ S = mi n v;s ~ma xS , i.e .,for each B orel s et~ S S. (A.13)
Using th is n otation , and after s ub stitu ting the deb t p rices f rom Eq. (9), optimis ts p roblem in a
quasi-equ ilibriu m can b e writte n as :
x
A
max
1
[s
xA 1 0 ; 2 M([
smin;smax])
[ s E0
Z[ smin;s
;smax]
s.t. pxA 1
E1Optim
[v(s
E1 [min (v(s) ;v( ~s))] d (
)] min;smax]
~s) ,
Z
E
1
[min ( v(s) ;v( ~s))]d (
1+r
~s)
max]
A 1.
mind ( ~s)
x, Z
+p1
w1
is ts solve a line ar optimization prob le m. At the op timu m, the b udge t c onstraint bind s. Th e
collate ral cons traint also bind s b e caus e (sinc e p>[ v( s)] 1+ r) optimis ts always pref er b orrowing and
inve stin g in th e ass et to n ot b orrowin g. Then , letting de note the Lagrange mu ltiplier for th e bu dget
con straint and the Lagran ge multip lier for the c ollateral cons traint, the rst ord er con dition s are give n
by:
v s2 3mi nThis is b ec ause any s af e contract with ’<v s mi n can b e rep licated by th e alternative safe
contractm ax (which h as the additional b en e t of u sing less collate ral), an d any contract ’>v(s) that de
faults in all s tates can b e rep lic ate d by th e c ontrac t v(s). m ax
46
[min ( v(s) ;v( ~s))] A 1 E1
1+r
1 [v(s)] .
E0
[min (v(s) ;v( ~s))] + (A.14) with e qu ality if ~s2 supp( ) .
Moreover, the rst ord er con dition with re sp ec t to x
= pE
leads
to
Plu gging th is express ion for into (A:14) yie ld s the following rst orde r c ondition :
[v(s)] E1
To this e nd, c onside r th e d erivative of 0 [min ( v(s) ;v( ~s))] =(1 + r)
; (A.15)
RL 1
L1
d
0( ~s) = 1[m in( v( s) ;v( ~s))]
RL 1( (1 F0 ( ~s)) (1
( ~s)) . (A.16)
d~sR 1+ r
~s) 1 +
F1
L1
r
p E
E1 d
mi n[v(s)] v smi n ) =(1 + r) >0
0 1[v(s)] =(1 + r) = 0.
d~sR
and RL 1
L
1
~s= smax
smi n =
~s= s
(
L
[min
R ( v(s) ;v( ~s))] p Ewith s trict inequ ality
only if ~s2 supp( ) .
~
This equation1 implies th at any s2 supp( ) maximize s R( ~s), comp le tin g th e rs t step of th e p ro
s
of. As the se cond step , I show that problem (11) has a un ique solution , and I ch aracterize th e s
)
olution.
( ~s), which c an b e written as
=
E
1
p v(s
Thus, the de rivative in (A:16) satis e s the b oun dary
conditions
Note that
(
~s)
j
R
L
1
min
>0 and dd~s
RL 1
(
~s)
j
<0 . (A.17)
Eq. (A:16) also leads to the rs t ord er cond ition
[v(s)] p E
[v(s)] E (sma x) = E1
d
mi n( ~s) = 0 for ~s2 [s;sma x) i/ RL 1( = 1 F10( ~s) 1 F( ~s) : (A.18)
d~sR ~s) 1 + r
L1
Plu gging this rst order cond ition into (11) and rearranging term s yields p= poptopt( ~s). By Le mma 2, p
p= p ( ~s) is strictly de cre asing, which imp lies that th ere e xists exactly one s2 S(the solution
toopt( s)) that satis es th e rs t order con dition in (A:18). By the b ound ary con ditions in
(A:17)
d d~sR(L ~s),
1
it follows that RL 1
47
and the c ontinu ity
of
( ~s) has a un ique maximu m charac te rize d as th e solution to E q. (10). This establish es th e s
econd ste p, and complete s the pro of of Theorem 1.
Pr
o
of
of
E
q.
(1
4)
.
C
on
s
id
er
op
ti
mi
s
ts
bu
dg
et
c
on
str
ai
nt
(5
)
an
d
no
te
th
at:
= p xA 1= 1 + xB 1B 1+ Z+
q(’) d
xA 1= p
w1 p
xA 1
1 + xB 1
q( s) Z
B
1
p x
(’)
1
d
+ q( s) x,
1 + E0 [min (v(s)
;v( s))] 1 + r
A 1.
’
whe re th e se cond line use s the f act th
at
A
1
is
a
Di
ra
(A.19)
1
’
A1
1
+x
=w
x
+x
1
(’)
c
m
ea
su
re
at
s,
an
d
th
e
la
st
lin
e
us
es
th
e
fa
ct
th
at
op
ti
mi
st
s
c
oll
at
er
al
c
on
str
ai
nt
(6
)
bi
nd
s.
S
ub
s
tit
uti
ng
co
ntr
ac
t
pri
ce
sf
ro
m
E
q.
(1
5)
,
th
e
pr
ev
io
u
s
di
sp
la
ye
d
e
qu
ati
on
im
p
lie
s
th
e
p
eri
o
d
1
o
w
of
f
un
ds
co
ns
tra
int
:
Next n ote th at op timists cho ose B = 0 exce pt for th e corn er case
x
.
In
vi
e
w
of
thi
s
ob
s
er
va
tio
n,
E
q.
1
(A
1
p=
E [ v( s)]
1+ r
:1
9)
ch
ar
ac
te
riz
e
s
op
ti
mi
s
ts
d
e
m
an
d
for
th
e
as
se
t.
R
e
ca
ll
al
so
th
at
m
o
de
rat
es
cho os e
xA 1
p= 1 0
1
= 0 exce pt for the corn er case
p=
w + E0 [min (v(s)
;v( s))] 1 + r
E 0,
wh ich characteriz es m o d erates deman d f or the ass et. Finally, re call th at th e as s
clearin g con dition is give n by xA 1A 0= 1. The re are th re e c as es to c onside r. First c on
as e (ii), i.e ., s upp ose th e marke t clearin g pric eis given by some p2 E0[ v( s)] 1+ r;E1[ v( s)
In th is cas e, m o d erates de mand xA 0, which implies th at optimists d emand must b e g
xA 1= 1. Using th is in Eq. (A:19) (alon g with xB 1= 0), th e marke t clearin g pric e is solve
= wma x
1( s)
. (A.20)
0
Characteri zation of Equil ibrium I next p rovide an analytical charac te rization of th e mi
n
quasi-e qu ilibriu m de scrib ed by p= p( s) = pmc( s), wh ich will b e us efu l f or s om e of the sub
sequ ent pro of s. Note that if optimis ts wealth is not to o large , in partic ular, if
wmax
E 0 [ v( s)] ; E 1 [ v( s)]wmax 1 , pis ind eed the market clearin g pric e, provin g c as e (ii). If ( s) 0 E1[ v
As long as th e expres
lie s ins
1( s) 0
1+ r
1+ r
sion
id e
arket clearing price is p=E[ v( s)] 1+ r(alon g with allo cation s xA 1= 1 and xB 1
Fin ally, ifwmax 1( s) 0 E11[ v( s)] 1+ rE0, the n th e marke t clearin g p rice is p=[ v(
ations xA 1<1 and xB 1= 0), provin g c as e (i). This comp le tes th e p ro of o
optAnalytical
p= pmc ( s ) =
1
0
4
8
<
[
v
1 + r , (A.21)
the n th e two c urve s intersec t in
( th e case (ii) re gion of Eq. (14) and the e qu ilibriu m p air (p; s )
0
is charac te rize d as follow
E s: s)
1
1
w + 11 + r
E0
w
1
[min (v(s) ;v( s ))]
]
v
s
, (A.22)
whe
re s
is the un iqu e solution
to
( s( ssmax
)) dF
)Z
G( ) 0 1 F0 ) 1 F
s
1E0
1
))] = w
1 (1 + r) .
[min ( v(s)
;v( s
(A.23)
In this case , op timists take loans with riskine ss s 2 smi n;sma xE0[ v( s)] 1+ r;E1[ v( s)] 1+ r and th e p rice s atis es p 2 .
in Eq. (A:23) is di/e rentiab le and s trictly de creasing, which imp lie
to E q. (A:23).
s
(v(s) v( s
1
If th e op p osite of cond ition
(A:21) holds , then the two cu rve s
inte rs ect in th e c ase (i) re gion of
Eq. (14). In this c ase, optimists n
ancial cons traints are n ot b in
ding, the y b orrow with a safe loan
(with riskines
mi n= s) and th ey bid u p the as set price to the op timistic valuation , i.e., E [ v( s)]
1+ r
s s
p=
1. This an alysis als o ve ri e s that
th e two curves n ever inters ect in
case (iii) region of Eq. (14), which
implies that the equilib rium price satis es p> E 0. This completes the analytical characteriz ation of equilibrium.
A.3 Characterization of
Collateral Equilibrium
This s ection comp le te s the
characteriz ation of the collateral
equilibriu m by provid in g th e pro of
of The ore m 2.
Pro of of Theor em 2. As the rst step
, I sh ow th at the p rice s and allo
cations in The orem 2 cons titute a
collate ral equilib riu m. I ne xt p rove
the esse ntial un ique nes s of the
collateral e qu ilibrium.
EExi
stence of the C ollater al Equili br
ium. I claim th at the allo cation in Th
eorem 2 cons titutes a c ollate ral
equilibriu m. The analys is f or the
corner p rice p=[ v( s)] 1+ ris s
traightforward . The re fore , su pp os
e that the asse t p rice satis e s p2 E0[
v( s)] 1+ r;E1[ v( s)] 1+ r 1. Eq. (14) en
sures that the as set market clearin g
con dition is satis e d. Henc e, all that
remains to che ck is that loan market
is in equ ilibrium.
This amou nts to ch ecking that de
bt contract ch oic es are optimal f or
trad ers after re laxing the restric
tion s
0 = 0 and + 1= 0, and that de bt contract p rice s c le ar the marke t. I ne xt es tab lish an eas
y-to-che ck cond ition f or equilibriu m in the loan market. Rec all that mo derates
rate of return on cap ital is give n by
1 + r, wh ile optim is ts rate of re
tu rn on c apital is given by
L1
( s
( s ) >1 + r(cf . E q. (11)), wh ere the in equality f ollows since p<
RL
1
qbi
(’) = E0
L1
+,
d0
(’) = E1
[min ( v(s) ;’)]
R( s
Note that thes e are the p rice s that
wou ld make mo derates (resp . optim
is ts ) in di/erent b etwe en hold ing a
d ebt c ontrac t ’and holding th eir e
qu ilibrium p ortfolio.
Similarly, c onside r the ask prices
f or a deb t c ontrac t ’that would make
the trad ers in di/erent b e tween se
lling the deb t contract ’and holding
the ir equilibriu m p ortf olio. Th ere is
a s light c om plication b ec au se , to
b e able to s hort s ell the contract ’, th
e trade r mu st als o h old 1 un it of th
e asse t. Hen ce, consid er the cros s
investment s trategy of s hort sellin g
one un it of c ontrac t ’and bu yin g
one un it of th e asse t. Let qask i(’) de
note the p rice th at makes typ e itrad
ers in di/erent b etwee n pu rs uing
this strategy and h old in g their
equilibriu m p ortfolio. Note that qask
0(’) and qask 1(’) are resp ectively
49
de ne d
) . (A.24)
. Given th e rate s of return 1 +
rand R 1), con sider trad ers bid
pric es f or each de bt contract ’2 R
[min ( v(s) ;’)] and
1+r
qbid 1
E [ v( s)]
1+ r
de n ed as the solution s to:
E0 [v(s)] E0ask 0[min (v(s) ;’)] p q(’) = 1 + E1 [v(s)] E1ask 1[min ( v(s) ;’)] p
( s) .
rand
q(’) = RL 1
(A.25)
The b id and as k pric es in (A:24) and (A:25) can also b e u sed to d e n e th e aggregate b id an d as k price for th e c
ontrac t ’, give n by:
(’) = m ax qbid (i ’) and qask (’) = m in qas (’) .
i
ki
qbid
i
Note th at, if the pric e of a contract ’is b elow qbidask(’), a trader would d emand in nite un its of the contract, wh ich wou
ld violate marke t cle aring. S im ilarly, if th e pric e is ab ove qbid(’), a trade r would se ll in nite un its, which wou ld again
violate marke t clearing. M ore over, non -ze ro trad e in a contract require s at le ast on e typ e of trad er to b uy th e
contract an d an other typ e of trad er to s ell, which c an hap p en only if q(’) = qask(’). I t follows th at the loan market is
in equilib rium if and only if d ebt contract p rice s an d allo cations satis fy the followin g cond ition:
bid
qbi (’)
(’) q(’) qaskask(’) wh ene
for s om e
ve r ’2 supp
i.
qask (’) with e qu ality i/ ’= v( s ) . (A.27)
(A.26)
d
(’) , and q(’) = q(’) = q
qbid
i
I ne xt s how th at the loan marke t allo c ation of The ore m 2 satis e s the loan market e qu ilibriu m con dition (A:26).
In particu lar, I claim:
Note that the claim in (A:27) is true for all ’2 R+Note th at th e de bt contract pric es of T heorem 2 (cf. E q. (15)) are
chose n su ch th at q(’) = qbid(’). Moreover, th e allo cation s are such that the re is trade only for contract ’= v( s ). Hen
ce, th e claim in (A:27) implies (A:26), which e nsu re s that the loan marke t is ind eed in equ ilibriu m., if it is tru e f or all
’2 v smi n ;v(sma x) . To prove th e claim for the re le vant s et of deb t contracts, ’= v( ~s) for some ~s2 S, rst n ote
that
qbi (v( ~s))
d i <qask i
(v( ~s)) for each ~s2 S and i. (A.28)
which is straightforward to che ck. The re is a wed ge b etwe en e ach typ e trade rs b id and as k p rice s, intu itively b
e caus e b uying the d ebt contract h as n o collate ral require ments wh ile se lling th e d ebt contract require s the trade r
to p ledge collateral (an d thus, th e trad ers ask price to s ell a contract is h igh er).
Se cond , note
is the un iqu e solution to p roblem (7) by de nition , an d
that s
thus
T
h
i
r
d
,
r
e
c
a
(
))] p E
01[min
(v(s) ;v( ~s))] p E
[min ( v(s) ;v( s
R
L
1
1
= qbid 0
0
q
is equal to 1 for ~s=
s
E1
a
s
k
1
ask 1
s
)
Using
th is inequality an d th e de n ition of q
=
E
0
[min ( v(s) ;v( ~s))]
1+r
[m in( v( s) ;v( ~s))] E[m in(
v( s) ;v( ~s))]
ll th at
(v( ~s)) with e qu ality i/
~s= s
. (A.29)
(
~s)) E
v
mi n, and is strictly in cre asing in ~s. By Eq.
(
1+ r
qbid 1( v(
(A:24), it f ollows that qbid 1( v ( smin))= RL( s) <1, an d th at ~s)) qbid
qbid 0( v( smin))
qbid 1( v( ~s))
qbid 0( v( ~s))0( v( ~s))
50
is strictly increasing in ~s. Then , the re are two cas es to c onside r. As th e rs t case ,may n ever excee d 1 , that is, it
may b e the
1.1
1.1
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6 0.8 1 1 .2 1.4 0 .5
0.6
0.6 0.8 1 1 .2 1.4 0 .5
Figu re 8: Th e left p an el disp lays th e bid an d ask pric es f or the c ase in
wh ich th e in equality in (A:30) holds , and th e right p an el d is plays the
case in wh ich th e ine qu ality in (A:30) fails. The sh aded areas d isplay the
se t of all p oss ib le equilib riu m de bt c ontrac t pric es in each case.
(v( ~s))
(v( ~s)) for e ach ~s2 S . (A.30)
<qbid 0
cas e that
qbid 1
In th is cas e, combining E qs . (A:28),(A:29) and (A:30) proves the c laim in (A:27).
The le ft p anel of Figure 8 plots the bid an d as k p rice s in this rst case . Th e gu
re illus trates th at, in this c as e, th e quasi-e qu ilibriu m de bt p rices in (9) and the
collateral equ ilibrium d ebt price s in (9) are identic al.
As th e s econ d c
ase,
qbid 1( v( ~s))(v(
qbid 0( v( ~s))
qbid
1bid 1
cross
~s)) <qbid (v(
0
~s)) f or all ~s ~scrosscross(v( ~s)) f or all
(v(
~s<~s. (A.31)
~s)) qbid 0
cross
su ch thatmay e xce ed 1 for s u¢ ciently large ~s. That is , it may b e the case that the
re exis ts ~s
,q
1
))] = is u niqu ely de ne d as th e s olu tion
Note that, in this case,
to
~s
)1+r
E [min (v(s) ;v(
~s
[m in( v( s) ;v( s
in( v( s)
;v( s ))]
0 [m
<
cross ))]
E0cross[min (v(s) ;v(
~s
RL
,
cross
RL (
~
s
. It can also
( s) 2 which im plie s
Moreover, it can b e che cke d E1 ))] E
1+ r 4
th at
~s
2
To
see
this,
c
on
side
r
th
e
leveraged
retu
rn
express
ion
(11),
which
c an b e re
4
written as
L (
1
s
p E0 [min (v(s) ;v( s ))] = E1[s] RL
E1 [min (v(s) ;v( s ))]
1+r
1( s )
R) .
( s) >
Note th at RL 1
>~
s
.
1
E
[m in( v ( s ) ;v ( s)) ] 1+ r
<
1
0
[
v
s
)
]
p
E
bid and ask prices
bid and ask prices
(
E0 [m
in(
v(
s)
;v
(
s))
]R
1
)
)
]
51
b ec ause op timists always h ave th e option of b uyin g the as set withou t b
orrowing. He nc e, the p re vious in equality impliesE[m in( v ( s ) ;v ( s, which can b e
rewritten as)) ] E[m in( v ( s ) ;v ( s<1+ r RL 1( s ).L 1( s )
b e see n th at2
5
(v(
~s))
qask 0
, with equality i/ ~s= s
cross~s ~s
bid.
M
oreo
ve r,
u
sing
Eqs.
(A:2
8),
(A:2
9),
(A:3
3)
and
(A:3
1), it
als o
f
ollo
ws
that
q(
~s)
<qas
kcros
s(
~s)
for
each
~s
~s.
Th is
com
plete
s the
p ro
of of
clai
m
(A:2
7),
and
es
tab
lish
es th
at th
e
cross(v(
~s)) for e ach ~s ~s. (A.33)
qask 1
(v( ~s))
qask The n, usin g E qs . (A:28), (A:29) and (A:31), it follo
allo
catio
n ch
arac
teriz
ed in
The
ore
m2
is in
dee
da
colla
teral
equil
ibriu
m.
The
right
pan
el of
Figu
re 8
plots
the
b id
an d
ask
pric
es in
this
s
econ
d
case
.
This
gu
re
illu
strat
es th
at, in
th is
cas
e,
the
qu
asi-e
quili
b riu
m
de
bt
price
s in
(9)
and
the
colla
teral
equil
ibriu
md
eb t
pric
es in
(9)
are
not
the
sam
e, b
ut
the
di/er
en
ce in
price
s do
es
not
over
turn
the
opti
malit
y of
th e
d
ebt c
ontr
ac
t s.
Figu
re 8
als o
illus
trate
s
that
the
deb
t
cont
ract
p
rice
s
are
n ot
un
ique
ly d
eter
min
ed in
e qu
ilibri
um
(exc
ep t
f or
the
p
rice
of th
e
opti
m al
cont
ract
v(
s ),
wh
ich
is un
iqu
ely
de
te
rmin
ed).
In
parti
cu
lar,
any
p
rice
fu
nctio
n
q( )
su
ch
that
q(v(
~s))
2 q
bid(v
(
~s))
;qask
(v(
~s))
ca
n su
pp
ort
th e
e qu
ilibri
um
allo
catio
n in
equil
ib
rium
.
How
ever
, the
e qu
ilibri
um
allo
catio
ns in
the
loan
mar
ket
an d
th e
e qu
ilibri
um
ass
et
price
pis u
niqu
ely
dete
rmin
e d,
as I
next
prov
e.
Es
se
nti
al
U
ni
qu
en
es
s
of
C
ol
lat
er
al
E
qu
il
ibr
iu
m.
I
rst
pr
ov
e
th
at
th
e
e
qu
ili
bri
u
m
as
se
tp
ric
e
p
is unique ly d etermined . Let
and R1
R0
de note trade rs equilib rium rate s of retu rn on cap ital (in th e ab ove equilibrium, R= 1 + rand
R( s)). Sinc e trad ers always h ave th e option to inve st in th e b ond , Rand R1are always weakly
greater than 1 + r. M ore over, in e qu ilibriu m s ome investors must agre e to h old th e b on d, wh
ich implies that eithe r Rmu st b e equ al to 1 + r. Sin ce optimists have a gre ate r valu ation of the
asse t, the equilibriu m rates of retu rn satisfy R[ v( s)] 1+ r= 1 + r. I n ext claim th at, for any given pric
e p2, op timists rate of re tu rn is u niqu ely de term in ed as R= RL 1( s), an d th e loan marke t
allo cation s are un ique ly d etermined by The orem 1. To p rove this c laim, c on side r optimis
ts b id an d ask p rice s in (A:24) and (A:25) for an arbitrary p rice
E1 [v(s)] E1bid 0[min ( v(s) ;v( s))] p
q(v( s)) = R1E1
0
1
= E1 [v(s)] E1ask 1[min ( v(s) ;v( ~s))] p
q1bid 0[min ( v(s) ;v( ~s))] p q(v( ~s)) ,
(A.34)
1 R0
;or R1
= RL 1
0
1
0
E1[ v( s)]
1+ r
E0
level p2 E0[ v( s)]
1+ r
and an arbitrary required rate of retu rn R1 [ v( s)]1E ;
1+ r
[v
(s
)]
E(
v(
~s
))
2
5
) in the se expres sion s with R1). Eq
ired rate of retu rn (and Eq. (A:24) sh
It f ollows th at the loan market is at eq
1(v( ~s))
qbid 0(v( ~s)) f or all ~swit
(8)). The n, this rate of re tu rn R1satis
Note that f rom the de n ition of qask i(v( ~s)) in (A:25), the in equality in (A:33) is e qu ivalent to
E0[v(s)] E0[min (v(s) ;v( ~s))] 1 + r E1[v(s)] E1[min (v(s) ;v( ~s))] RL 1( s cr oss) for each ~s ~s.
(i.e ., r
RL 1
( A.32)
Rec all that
E1[ v ( s )]
E0[ v ( s )]
E1
E1[m in( v ( s ) ;v ( ~s,))which implies that
] E0[m in( v ( s ) ;v ( ~s
[min (v(s) ;v( ~s))] E0
)) ]
RL
;
1
[s] E1[s] E0
[min (v(s) ;v( ~s))] E[min (v(s) E1 [min (v(s) ;v( ~s))]
( s)
;v( ~s))]
1+r
in view of the d e n ition of ~scr oss. Th is p rove s the inequality in (A:33), which in tu rn sh ows
whe re the last inequality holds for each ~s ~s
qu ality in (A:33).
0
52
whe re th e rs t e qu ality use s the de nition of
qask 1
ask 1
qas (v(
k 1 ~s))
(v
(
s)
),
th
e
se
c
on
d
e
qu
ali
ty
us
e
s
th
e
de
ni
tio
n
of
q(
v(
~s
)),
an
d
th
e
la
st
in
eq
ua
lit
yf
oll
o
w
s
fro
m
th
e
in
e
qbid 0
(v( s)) and th e f act that
qask 1
; E [ v( s)]
1+ r
(v( s)) =
qbid 0
qu
ali
ty
mi n
1
E0
[
v(
s)]
1+
r
1
=
0
(which c orresp on ds to th e op timal contract s
s
i
n
c
1
E 0[ v
r;
mi n=
e
mi n
with
Note that g
s2 Ssolves problem (11). In particu lar, for any pric e p2 , the un ique loan m arket allo cation is th (v( ~s)). The c omparis on b etwe en
e same as the quasi-equilibriu m loan market allo cation characteriz ed by Th eore m 1, and
that
=R
E1[ v( s)] 1+
r
0[
v( s)]
1+ r
E1
mi ns
s
v s
g [v [v
( (s (s
)j)j
s s
s
) s] s]
E .
=1
ma x)
= 0 . I c laim
that
~
E
1
~
E
g( s)
[v(s)] = E the un ique rate of return th at equEilibrates th e loan market is given by R
( s). I next prove th e un
, optimists loan market
investme nt on the ass e
Comb in in g this with the
un iqu ely d etermined .
The ab ove analysis also
riu m allo cation s are un
price of the optimal contr
allo c ations are not n ec
e c ontrac ts ’ v. Th is e
comp le te s th e pro of o
r
A.4 Comparative Sta
Pro of of Theor em 3. Pa
0 for all s2 smi n;sma , (A.35)which implie s Eq. (19) in the main te xt. Mos t of the p ro of then follows by th e argume nt
x
L= pp (p 0
Si
n
ce
p
w
e
ak
ly
in
c
re
as
es
,
th
e
le
ve
ra
ge
rat
io
al
provid ed after The ore m 3. For th e c omparative s tatics of the leverage ratio, sub stitu te E q.
(A:22) into (16) to get:
w1) =
01
1
+
w1
p
.
so
w
ea
kl
y
in
c
re
as
es
,
co
m
p
le
tin
g
th
e
pr
o
of
co
nd
iti
on
al
on
th
e
cl
ai
m
in
(A
:3
5)
.
,
the
sam
e
argu
m
ent
use
d in
th e
p ro
of of
p art
(i) of
Lem
ma
3 sh
ows
th at
R
mi
nSe
cond
, su
pp
ose ,
to
reac
hac
ontr
ad ic
tion ,
th at
th
ere
exist
s
~s2
[sg
mi
ns
=0,
this f
urth
er
impli
es th
at th
ere
exist
s
^s2
[smi
n;sR
)
with
g
(
s
)
0
f
o
r
a
l
l
s
2
[
s
;
s
m
a
x
)
.
(
A
.
3
6
)
g(
~s)
<0.
Sinc
e;~s
) su
ch
that
g(
^s)
=0
and
g0mi
n(
^s)
<0 ,
sinc
e
oth
erwi
s e,
th e
di/er
entia
ble f
unc
tion
g( )
cou
ld n
ot b
e
com
e ne
gativ
e
over
th e
ran
ge
[s;~
s).
5
3
f11( s) 1
F( s)
over the ran ge s2 sR;sma
x
1~
~ To prove
F1( s) 1 f
ince
( s)
the claim in (A:35), rst note t
Cons id erin g Eq. (A:8) for s= ^sand us in g g( ^s) = 0 implies
g ( ^s) = ~ f1~ F1(
0
^s) 1 ( ^s)
f11( ^s) 1 ! ~
F( ^s)
E1
~ f 1 ~ F1 (
^s) 1 ( ^s)
Note also th at F0
0(
s)
0
F0
[v(s) j s ^s] v( ^s)
( s)
f11( ^s)
1 F(
^s)
0
0,
R
0R(as
^s<s). Sin ce
g0
0
0
0(
s) ( s) we akly in crease s f or e
1 F ach s2 s
0
.
~
F0,
~
F0
mi n
popt
whe re th e ine qu ality f ollows ( ^s) <0 by the ch oic e of ^s, the previou s d is played in equality yields a contradiction , comple ting
since
th e p ro of.
Part (ii) . App lying the pro of of part (i) for distributions F
~ sh ows th
F at
0
~ E [v(s) j s s] f or e ach s2 S . (A.37)
, which f urther implies( s) over this ran ge. In view of th is ob servation and E q. (A:37), Eq. (18) implies that
f
( s) for e
;
~ f( s)
0
ach s2
s
s
1
1
~
F(
implie
~
R; s
~
is weakly inc re asing f or s2 smi ; s
~s) 1
n
F
F0 s
F( ~s)
; s . (A.38)( s). Note that sin ce FR; s
E0 [v(s) j s
s]
mi n
Next cons id er the e/ec t on th e marke t clearin g curve pmc
The n, the same step s as in the pro of of p art (ii.2) of Le mma 3 app lies in th is case and sh ows
pm ( s) we akly in crease s f or e
c
ach s2 smi n
1 F0 ( smin)
26
; s : (A.39)
=
1,
0;
~
F1
tha
t is
,
0
opt
;
~
F
smi n
opt
(10) implies
s;~
F1
mc
0
1
popt
( s) for. Th
each s
2 s en, E q.
s; s .
0
m
c
( s) is a d ecreasin g relation and pmc
mi
~
= popt
s; F
1 for e
n
F
ach s2
Using Eqs. (A:38) and (A:39) alon g with the f act th at
popt
( s) is a we akly increasin g relation, it follows th at pis weakly greater at th e ne
oint, comp le tin g the pro of .
1 ~
Pro of of Theorem 4. Part (i). The fact that
= 1 ~ F0 (
F0( s) 1
F( s)
smin)
0~
F0
and F0 are equally optimis
over
implies that
0
). Using th is along with th e previou s
disp laye d equation s hows E q. (A:38), that is , p s2 mi ns; s . Note also that2 6 s2 mi ns; s ~ E0[v(s)] Eimplies that
pmc( s) we akly inc reas es for each[v(s)]. Comb in ing th is with the fac t that( s) cu rve remains cons tant overOn th e
oth er han d, the e/e ct on sis ambiguous . Th e e /ec t on s; s ~ F( s) = Fexce pt for the f act that itsde p en ds on how
mu ch the p( s) cu rve sh ifts up. If th e e/ec t on the p( s) cu rve is stron g, p e rh aps b ec ause 0is small, the n smay de
crease. To s ee the intuition f or this re sult, su pp ose many units of the ass et are alre ad y e ndowed to optimists, i.e . 1is
high. The n the inc re ase in lend ers valu ation of d ebt contracts acts s im ilar to a p ositive wealth s ho ck to optim is ts (cf
. Eq. (13)), b ecaus e optim is ts c an b orrow more agains t the u nits the y alre ady own. This e/ec t te nd s to lower the le
ve rage ratio, and if su¢ ciently stron g, it can ove rc om e the e/ect from the s hift of th e p( s) cu rve which (s imilar to p art
(i)) ten ds to inc reas e th e le ve rage ratio.
54
By p art (i) of
Lemm a 3,
F imp
O
1
lies
p
s;
F0; ~
F
o
p
t
popt
( s; F0;F1
( s) f or
0
each
lower b ou nd exte nds , th at is ,
E0
p
m
c
(
(
s
)
s
)
i
f
=
E 0[ v( s)]
1+ r
(
E [ v( s)]
1+ r
opt
s
)
mc
>
, wh ich further imp
p lies
pand s
m
c
1
and
~
(
;sma x),
s) for e ach s2 [ s;s). (A.40)
ma x(
m
a
F
1
1
s
)
1
f
o
r
1
x
1 ~F
e
1
a
c
h
s
2
[
pmc s; ~ F0; ~
F1 (
= pmc
s
;
s
( s) if pmc ( s) >
[ v( s)]
1+ r
fn or e ach s2 smi; s
0
.
mc
";s
Note also that the pro of of The orem 2 es tab lish es that p= pmc . Us in g th is in th e previous d isplayed equation implie
ombining th is with E q. (A:38) and us ing the f ac ts tha
o d s2 ( s( ) is a d ecreasing curve an d p( ) is an inc
tion p oint of thes e cu rves is weakly to th e right of s w
or th e leverage ratio f ollows by the same argum ent as
laim th at Fhave the same d is tribu tion cond itional on
with s> s . That is, f or any s2 ( s
this, n ote that by as sump tion
f(s) 1 F( s) To se e
=
(s) 1 F(s) =
~ f1~
F1(s) 1
1 ~
F11( s) 1
F
ma x).
(A.41)
Moreover, taking th e d erivative of th is equation with res p
ect to simplies
ma ). (A.42)
~ f1~ (s) for e ach s2
x
F1(s) 1 [ s;s
ma xUsing E qs . (A:41) and (A:42), it follows that, for each
s2 ( s;s),
f11(s) 1
F(s) =
~ F1
~
F1
( s)
=
~ f1 ( s
~ ),
F1
mi n
f or s= s
1
0
opt
( s) and ( s) cu rves . First cons id er th e optim ality cu
pmc
rve popt
popt s ~ F0; ~ F1 =
s ~ F0;F1 for e ach s2 ( s ;sma
;
popt
;
x) .
f
1
1
f
1
1
(
s
)
(
s
)
1
1
F
(
F
(
s
)
s
)
1
=
F
1
1
(
s
)
1
F
(
s
)
=
~ f1
1 (s
=
(s) 1
. Us in g Eq. (A:40)
s
+ "and taking "! 0 sh ows th at, in this c ase,
F1and~ F0
an ~
d F
are the same distrib ution. By the s am e argume nt, Fare
also the same d is tribu tion. It f ollows that th e asse t p rice
premains con stant f or th is case .
Cons id er next the c as e in wh ich c on dition (A:21) holds so
that s
. Con sider re sp e ctive ly the s hift in th e p( s) and n ote th
at Eq. (A:40) in E q. (10) implies
). Us in g th is in the previou s d isplayed equation sh ows
By part (ii) of Le mma 3, F0 O ~ implies
s ~
pop
mi n>s
( s; F0;F1
F0 popt
;
F0;F1 popt
( s) we akly de cre ases for each s2 ( s;sma x) . (A.43)
t
Next consid er th e marke t c learing c
urve p
55
mc
( s). Note
that s
>smi n implies
popt
( s) =
pmc
( s)
<
un ch anged in a n eighb orho o d ;
( s
+ "). Note also th at the de cre ase in op timism of F0
s
mc
E 1 [ v( s)]
1+ r
leaves
, thus by Eq. (A:5) (cf .
pmc
we akly de creases p( s) downwards p ointwise . It follows that
pm ( s) we akly dec re ases for
; s + ") . (A.44)
c
each s2 ( s
Comb in in g Eqs. (A:43) and (A:44) and us in g the fact th opt ( s) is a dec reas in g cu rve
at p
while pmc
is an in creasing curve, the ass et p rice pis weakly lower at th e new interse ction p oint. This
complete s the pro of of Theorem 4 .
(
s)
A.5 Collateral Equilibrium with Contingent Contracts
W ith continge nt loans, trad ers s olve th e f ollowing an alogue of p rob le m (7):
[v(s)] + xB i+ Ei [min ( v(s) ;’ (s))] d
i(’ ) RD
i
(’ )
D
D
s.t. pxA i
d
i
D
xA iEi
max
xi 0;
+ i; i2
Ei[min ( v(s) ;’ (s))] d (1 + r) + R+ xB i+ ZDq(’ ) d + i(’ ) Zq(’ )
d iA i(’ ) x.(’ ) wi+ p i. Z
M( D )
The same an alysis as in the p ro of of Theorem 1 ap plie s and sh ows that optimis ts cho os e a
de bt contract,
[’ (s) 2 [0;v(s)]]s2S, that maximize s th e leveraged retu rn
(21).
opt;contPro of of Theor em 5. I rst c laim that th e le verage d re tu rn in (21) is maximize d by th
e solution to equation p= p( s). S econ d, I c laim that the op timal leverage d return, RL;cont
1ma x(’ sj p), is gre ate r than 1 + rif and on ly if p<pma x. Th is imp lies th at optimists make a le
ve raged investme nt in the asse t if and only if p<p, which c omp letes th e p ro of of the
theorem. To p rove th e rst claim, n ote that th e de bt contracts c an b e restric ted to the s et
such that ’(s) 2[0;v(s)] for e ach s2 S. Th e s ame ste ps in the pro of of Theorem 1 show that ’
2supp 1 if an d only if ’solves Prob le m (21). To charac te rize th e solution to this p roblem
, n ote th at th e de rivative ofRL;general 1(’ ) with resp ect to ’(s) is given by:
@RL;cont 1(’ = f1 1+ r(s) pRss0maxmin’ (s) f1(s)
) @’ (s)
dF0
f0(s)
RL;cont 1(’ ) ! .
1+r
From this express ion an d ass ump tion (MLRP), it f ollows that the d
erivative
@RL;cont 1( ’ ) @’ ( s)satis
e s a cuto/ prop erty. In pa
that
@RL;cont 1(’ >0 f or e ach s< s<0
) @’ (s)
f or e ach s> s .
Cons equently, th e optimal level of promise for each s tate s2 Sn f sg has a corne r solu tion .
Henc e, the solution ’to Problem (21) has the form in Eq. (22), exce pt p otentially a Leb e sgue
meas ure z ero of s tate s. In partic ular, th e contract s p eci ed in Eq. (22) is one optimal s olu
tion to P roblem (21).
f 1( s )
Moreover, the thresh old state s2 Sis characteriz ed as the solution to)
56
f 0( s
=
RL;cont 1( ’ ) 1+ r
, wh ich after
us in g the form in E q. (22) can b e writte n as
1 1+ rRsmax
popt;cont
v(s) dF1
0
sR
= f1( s)
f( s) .
1( s(p)) f
p= popt;cont
RL;cont
1
min
m
i
n
1
1+ r
p
s
smin
v(s) dF
0
Rearran ging this exp re ssion sh ows th at the
thresh old state is charac terize d as th e u niqu e
solution to
( s), comp le tin g th e p ro of of
the rst claim. To p rove th e se cond
claim, x a pric e le vel p, consid er th e
corres p ond in g op timal thresh old s(p),
and note th at th e op timal leverage d return is
give n by:
j
p
’s
= E1
[v(s)]
R( s(p))
v(s) dF1
= (1 + r) f
( s(p)) . (A.46)
0
s( p) s
( p)
ma x
R s( p)
1 1+ r
v(s) dF
sopt;cont
0
ma x
opt;cont
’
crosss
L;cont 1
cross(s
m
a
x
j
p
( s(p))H(sinc e s(p) is op timal), an d th e secon d equality us es E q. (2
ere, the rst e qu ality us es the fact th at p= p E qs . (23) and (24)), an d thu s
is th e op timal thres hold c orresp on ding to pric e leve l p. Using Eq. (A:
that R(’scr o ss j pma x) = 1 + r. Since RL;cont 1 s( p)is d ecreasing in p, it f ollo
raged retu rn is greater than 1 + rif and only if p<pma x, but they are in di/e
in the ass et and the b ond if. Hen ce, optimists b orrow an d inve st in th e
p
>
p
m
a
x
.
T
h
i
s
c
o
m
p
l
e
t
e
s
t
h
e
p
r
o
o
f
o
f
t
h
e
t
h
e
o
r
e
m
.
A.6 Collateral Equilibrium with Short
Selling
This app e ndix p rovides a p ro of of Theorem 6
in S ection 6. It the n u ses the p ro of to provide
an intuition for the asymmetric disciplining p rop
e rty (which is more comp lete than th e intu ition
p rovide d in the main text). Th e app e ndix e
nds by d eriving th e total e xp end itu re on sh
ort sales, Eq. (30), w hich is us ed in the main
text.
Similar s te ps as in the derivation of Th eorem
1 show that the de fault thresh old sshfor s
hort contracts maximizes :
[min ( v(s) ;v( ~s))] E1[ v(
s)] p
=
Rshort ( ~s) = v( ~s) Ev( [min ( v(s) ;v(
0
~s) 1+ r E10
~s))]
. (A.47)
Eq. (26) corre sp ond s to th e rs t ord er cond
ition for this m aximization problem. Un der as
sump tion (MLRP), the unique solu tion to th is
equation maximize s the re tu rn in (A:47), c om
pletin g the s ketch pro of of The orem 6.
To interpret problem (A:47), note th at Rshort
0( ~s) is the re tu rn of s hort s elle rs from se
lling one un it of th e s hort contract = v( ~s).
More s p ec i cally, s hort sellers rece ive
qshort (v( ~s)) =
E1
[min ( v(s) ;v( ~s))] = [v(s)]
E1
p
f rom the sale of th is contract, an d the y u se th
is amount toward s mee tin g th e collate ral re
qu ireme nt. Howeve r, the y n eed to p ost a
total ofv( ~s) 1+ run its of th e consu mption go o d
as collate ral. Thus, th ey pay the di/eren ce (the
de nominator of (A:47)) ou t of their wealth . In th
e ne xt p erio d, s hort se llers rece ive v( ~s)
from the c ollateral that th ey have p os te d, an d
they exp ec t to pay E0[min ( v(s) ;v( ~s))]
on the promises the y have made . This is b e
caus e, short se llers re turn th e as set if th e
realized s tate is b e low ~s, bu t the y d ef ault
on th e s hort contract if th e realize d s tate is ab
ove ~s. In the latte r s cen ario,
57
(A.48)
sh
ort
se
llers
lose
on ly
the c
ollat
eral
that
they
have
p
oste
d,
wh
ich
is
wort
h v(
~s).
Hen
ce, s
hort
s
elle
rs e
xp
ec
ted
pay
men
t is
give
n by
E0[
min
(
v(s)
;v(
~s))
].
Prob
lem
(A:4
7)
cap
tu re
s the
es
senti
al
trad
e -o/
that
s
hort
se
lle rs
are
facin
g.
Note
th at
mo
dera
tes
p
e
rc
e
iv
e
d
int
e
re
st
rat
e
on
a
s
ho
rt
c
on
tra
ct
is
gi
ve
n
by
:
1 + rper ( ~s) E[min ( v(s) ;v( ~s))]
0
E10[min ( v(s) ;v( ~s))] =E1[ v( s)] p
(
~s
)
<r
for
th
e
eq
u
ili
bri
u
m
s
ho
rt
co
= E0[min ( v(s) ;v( ~s))]
E1[min ( v(s) ;v( ~s))] E1
[v(s)] . (A.49)
p
ntr
ac
t
~s
=
s
sh
per (
0 sh
s
This express ion f urth er implies th at . Intu itively, sh ort sellers exp ect to make a net p ositive return, r r), by sellin g the s hort contract
rper 0
and b uyin g the b on d with the p ro cee ds. Moreover, un der assu mption (MLRP), th is retu rn is
inc re as in g in th e sh ort thresh old ~s. This is b e cause , the highe r ~s, the les s of te n th e shor
contract def aults , and the greater p ortion of the asse t th e s hort s elle rs e/e ctive ly se ll. On the
oth er hand , problem (A:47) sh ows th at a
v(
highe r thre sh old ~srequire s s hort sellers to p ost a greater amount of
~s)
collate ral,
1+ r
sh
sh
( ssh) is dec re as in g in the de fault
I next p rovide the intuition
thres hold
f or why th e fun ction
pshort
sh
sh
Cons id er ne xt th e intuition f or th e as ymm etric discip lining prop
erty of p
per
0
( ssh
( ss
h
ssh
sh,
then Eq. (A:49)
( ssh) p erce ive d by mo de rates
is highe r. Thus ,th e as set price do
es not n eed to b e to o high to e
ntice mo de rates to ch o os e th e
sh ort contract with th re shold s.
Cons equently, with th is typ e of b
elie f h eteroge neity, th e ass et pric
e is close r to the m o d erate valu
ation. In contrast, su pp ose the b
elief hete rogen eity is conc entrated
more on the relative likelih o o d of s
tate s ab ove s. In this c ase, Eq.
(A:49) implies that the p e rce ive d
retu rn we dge r rsh) is lower. The n,
mo d erate s are e nticed to ch o os
e the thres hold le vel sonly if p
rice s are s u¢ ciently highe r than
the mo d erate valuation . He nce ,
optimism ab out the p robab ility of
state s b elow sis disc ip lin ed
more than optim is m ab out the re
lative like liho o d of states ab
ove s, as sugges te d by the f orm
of pshortsh). Fin ally, con sider the
de rivation of the total exp en diture
on short sales, d enoted by W . Note
thatis th e amou nt of wealth mo de
rates ne ed to allo cate to sell one u
sh
s
This
restr
ic ts
sh
ort
se
llers
ab
ility
to
lever
age
the
ne t
re
turn
r r,
shor
t se
llers
trad
e o/
grea
ter
le
vera
ge
agai
nst a
lowe
rn
et
retu
rn.
This
trad
e-o/
is
resol
ved
by p
rob
le m
(A:4
7),
and
lead
s to
the
op
timal
s
hort
c
ontr
reveals th at the return we dge r rper 0
per 0(
). It follows that, when cho
s osing s
.
ac t
ch
arac
teriz
ed
by
(26)
.
,
an d
why
it h
as
the
as
ymm
e tric
dis
cipli
n in
g
prop
erty.
Con
side
r rs
t the
form
er s
tate
men
t,
whic
h is
equ
al to
sayi
ng
that
the
d
efau
lt th
re
shol
d sf
or th
e op
timal
s
hort
cont
ract
is
dec
re
as in
g in
the
asse
t
price
.
Note
th
at,
by E
q.
(A:4
9), a
high
er p
rice
pinc
reas
es
th e
wed
ge r
r)
that
s
hort
selle
rs
exp
ec t
to
mak
e.
This
in
centi
vize
ss
hort
selle
rs to
leve
rage
mor
e, by
cho
os in
ga
lowe
r def
ault
thre
sh
old
.
Intuit
ively
, as
p
rice
s
are
high
e r,
s
hort
se
llers
se e
a
gre
ate r
barg
ain
in
shor
ts
ellin
g
and
the y
lever
age
the
ir sh
ort s
ales
mor
e.
sh
ort
).
To
un
d
er
st
an
d
thi
s
p
ro
p
ert
y,
su
pp
os
e
th
e
eq
uil
ibr
iu
m
de
f
au
lt
th
re
s
ho
ld
is
gi
ve
n
by
s,
an
d
c
on
si
de
r
ho
w
h
ig
h
th
e
as
s
et
pri
ce
sh
ou
ld
b
e
(r
e
lat
iv
e
to
th
e
m
o
d
er
at
e
va
lu
ati
on
)
to
en
tic
e
m
o
d
er
at
es
to
ch
o
os
e
a
s
ho
rt
co
ntr
ac
t
wi
th
thi
s
de
fa
ult
thr
es
h
ol
d.
If
th
e
b
eli
ef
h
et
er
og
en
e
ity
is
co
nc
e
ntr
at
e
d
on
st
at
es
b
e
lo
w
s
h
s
per 0
sh
sh
sh
s
h
(
sh
short
(
s
v( ssh
) 1+ r
(w0 + p 0). Thu s, the total nu mb er of s hort c ontrac
ts
v( ssh . Typ e T3
) 1+ r
(w0
v( ss
h) 1+ r
sold by mo d erate s is give n
by
.
v(
~s)
1+ r
qshort
qsho r t
v(
~s )
1+ r
v ( s sh)
1+ r
W
sh(
short
= v(
1+ r
w0+ p 0 )
sshsh)
s
. The total e xp e nditure on sh ort sales is the n given
by:
qshort
v( ~s)1 +qshort + p 0)v(
r
~s) 1+ r
58
v(
from Eq. (A:48), an d rearranging terms yields th e express ion (30)
for
Su bs tituting f or qshort ~s)
1+ r
Wshort.
A.7 Characterization of Dynamic Equilibrium
This s ection completes th e characteriz ation of the d yn amic e qu ilibriu m an
alyze d in Sec tion 7, by provid in g a p ro of of The orem 7. I rst note a p re
liminary lemma which is nec ess ary for the pro of of th e the orem.
Note that th e p roblem of trade rs in th e d yn amic ec onomy (cf . (38)) is
similar to their p roblem in the static econ omy (c f. (7)), with the on ly d i/ere nce
that the asse t is not en dowe d to the cu rrent you ng gen eration. The n ext lem
ma u ses th is ob se rvation to sh ow that a recu rs ive c ollateral equilib rium c
an b e cons truc ted b ased on th e an alysis in Sec tion 3. The res ult re qu ires
the cond ition
!0
1 + "r " , (A.50)
which ens ures that youn g trad ers e ndowment is su¢ c ie nt to pu rchase th e
e ntire ass et s up ply. Rec all that a loan with riskin ess sis a d ebt c ontrac t ’=
v(a; s) that de fau lts if and only if the ne xt p e rio d state is b elow the thresh
old level s2 S.+; s Lem ma 4. Consider a dynamic economy with con dition
(A:50), and suppose there exist s a col lection of price and loan riskiness pairs,
(p(a) 2 R(a) 2 S )a2 R++, such that for each a2 R, the pair (p(a) ; s ++(a))
corresponds to the col lateral equil ibrium characterized in Theorem 2 for t he
static economy
(S ; v(a; ) ; f Figi
; fwi
!iagi
;f
1
= 0; 0 = 1g ) . (A.51)
Then, there exists a recursive col l ateral equilibrium in which, for each a 2 R++,
opt imists make l everaged investments in the asset by borrowing through a
single loan with riskiness s (a) and the asset price is p(a).
p(a) ;[q(a;’)]’2 R
; xA (a)
i
;xB i (a) ;
+i
(a) ;
+
Pro of of L emma 4. Let the tuple
B0
prices
b e s uch that, th e pric es and allo c ations f or each acorre sp ond to the c ollate
ral equilib rium of the s tatic ec onomy in (A:51). I claim th at this tup le corres p
ond s to a dyn amic e qu ilibrium with a mo d i ed b on d allo cation f or mo de
rates , ~x(a). Note that optimis ts prob lem (38) is equivalent to th eir p rob le
m (7) in this static e conomy, give n
p= p(a) and q(’) = q(a;’) f or e ach ’2 R
: (A.52)
Hen ce optimists allo cation s are als o op timal in the dynamic ec onomy. Mo
de rates problem is s lightly di/e rent sinc e 0= 1 in the static econ omy
whereas 0= 0 in the dynamic e conomy (as th e ass et is he ld by the old gene
ration). Becau se of th is di/erenc e, the allo cation xA 0(a) ;xB 0(a) ; + 0(a)
; 0(a) violates the b ud ge t c onstraint of mo d erate s in the dynamic e
conomy by an amount p(a). Consid er inste ad the mo di ed b ond allo cation
~x (a)
B0
x B (a) p(a)
0
0 for e ach a2 R++. (A.53)
i
(a)
i2f 1 ;0 g a2 R++
27
xA 0(a) ;~xB(a)
0 ; + 0 (a) ;
0
(a)
27
B0
59
Note that th e allo cation satis e s mo d erate s b udge t cons traint. Wh en this is the cas e, it can also b e s een that this allo
cation solves Problem (38) give n th e p rice s in (A:52).
To see this, note that E q. (A:53) implies x
(a) >p(a) >0 for the static ec on omy in (A:51),
Hen ce, if the ine qu ality in (A:53) is satis ed , the n th e conj ectured tup le with the mo
(a)
di ed ~xB 0
con stitu te s an equilib rium of the d yn am ic e conomy. To verify th e in equality in (A:53), c onside r
mo derates b udge t constraint in th e static ec onomy
xB (a) + ZR+ q(a;’) d + 0 (a;’) = !0a+ p(a) , (A.54)
0
whe re the equality h olds sin
ce 0
= 1 and xA (a) = 0 . Next note th at
0
(a) = p(a) ;
whe re the rst line follows f rom th e ine qu ality q(a;’) p(a) (which f ollows from Eq. (9)), th e s econ
d lin e us es th e de bt market c le aring cond ition (8), the third lin e u ses th e collateral constraint (6),
and the last line us es the asse t marke t clearing c ondition xA 1(a) = 1. Using th e last disp laye d in
equality, the bu dget con straint (A:54) implies
xB (a)
!0a 1 + "r " a p(a) ;
0
q(a;’) d + 0
ZR+
p(a) x(a;’) p(a) d + 0 (a;’)
ZR= p(a) ZR++A 1
(a;’)
d 1
whe re the se cond ine qu ality f ollows from c on dition (A:50), and the third ine qu ality follows from the f
ac t th at p(a) is we akly le ss th an the unc onstrain ed le ve l1+ " r "a. It follows that ~xB 0(a) in (A:53) is p
ositive , completing th e p ro of of Lemma 4.
Lemma 4 red uce s the ch aracteriz ation of the d ynamic equilib riu m to th e static case , along with
a xed p oint argu ment (since the valu e fu nc tion v(a; ) de p en ds on the price fu nc tion). I ne xt u se
this charac te rization to prove Th eorem 7.
aand s (a) = s d(sj pd) a, whe
d
Pro of of Theor em 7. Plugging the con je cture, p(a) = pd
re
v (sj pd) = s(1 + pd) . (A.55)
d
2 S, into (37) implies that the valu e f un ction is also linearly h omoge ne ou s. In particu lar, v(a;s) = v
Next n ote th at u sing th e con jec tu re in th e ch aracterization of th e static e qu ilibrium (cf. E qs. (10)
; s d; s d
28
d
d
d d)) , (A.56)
p
d
opt
d
xA 02 8(a) ;~xB 0(a) ; + 0(a) ; 0The notations popttrac ts . As their b ud gets and b on d h old in gs are red uce
d by the same amount p(a), the allo cation (a) is op timal for mo de rates in the dynamic e conomy.( ; v)
;pmc( ; v) resp e ctive ly de note the fu nctions popt( ) ;pmc( ) evaluated with the partic ular value fu nc tion
v( ).
60
and (14)) an d us in g linear h omoge neity in a, the con stants ) in (39). In partic ular, (pare characteriz ed as the collate
pd
(p) is th e un ique s olution to the following equations:
=p
( ; v ( j pd)) = pmc ( ; v ( j
sd
sd
p
( ; v). Given a pair, whe re th e notation
( j p dd)) de notes the f un ction ( ) evaluated with the value f un
popt
ction
(pdd
popt
v ( j ; s), that solves (A:56), Lemma 4 implies th at th e c on je ctured allo c ation which f urth er implies
d pd
that mo de rate traders are ind i/erent b etwee n h old in g b ond s and de bt con -
is an equilib rium. The re main in g step is to characteriz e the s olu tion to the xed p oint equation
(A:56) . To th is en d,
d
d
let (Pd ( ~pd)
i ( ~pd)) de note the s olu tion to (A:56) whe n the value
;Sd
given by vd
) (i.e ., whe n the fu tu re pric e to divide nd ratio is given by ~p). Th en, th e s olu tion to (A:56) is
a xed p oint of the mapping P( ) over th e intervalh pmi n d=1 r;pma x d=1+ " r ". I next claim that Pd( ) is
stric tly increas in g over this inte rval, and it s atis es the b ound ary con ditions
d
pmi n d >pmi nand Pd
Pd
d
d(pma x d)
pma x d. (A.57)
d
1
d
d)
(cf . Eq. (A:55)), 0
!1
1 F0
First sup p ose
d)
2 smi n;s
d(
1
~
p
~pd
mi
n
is con stant. S econd , s up p ose c on
dition
d
( ) has a un ique xe d p oint 2 (pmi n
pd
d;pma x d
This claim imp lies th at that ], wh ich charac te rize s th e dynamic equilib rium.
Pd
To p rove the c laim, I rst show that th e loan riskin ess ( mi n) 2 [s;sma x
S
~
p
mi
= 1, and E
) 1 + " s1 + r . (A.58)
d (sj
~pd
n
( ~p
<(1 + ~p
( s) Z
(A:58) is satis ed , and thus
S
d)
1
=s
=w 1+r .
1+
~p
ma x
) is weakly increasin g in
~p. There are two cases de p end ing on c on dition (A:21). Us in g the valu e fu nc tion v) = s(1 +
~p[v(s)] = 1 + ", cond ition (A:21) can b e writte n as
is su¢ c ie ntly large that this con dition is violated . In th is c ase, by the charac terization
in the pro of of The orem 2, the loan ris kine ss S
is determine d as th e unique solution to Eq. (A:23). This equation can b e s impli ed to
smax
( s) 1 F
s
(v(
s)
v( s)) dF
d
1
d
d
(
~
p
d
d d(
~pd) is weakly increasing in
~pd
d) is s trictly inc re asing in ~pd, note
th at
The p ro of of Th eorem 2 sh ows th at th e le ft h and side of this expression is a strictly de creasing fun
ction of s. Since th e right h an d side is d ecreasin g in ~p, it f ollows that in th is case S) is
increasing. Comb in in g the two cas es, S. Next, to s how th at P( ~p
d
d
(Sd ( ) ; v
~p
( j ~pd
d
[min
( s;S
d
0
,
d
d
d
Pd
)
1
+
"
1
d(
d
( ~pd) = pmc
d
[
m
1 +s (i) and
whe re the sec
E (ii) ofi E q. (1
~ on d e qu ality comb ines case
0
the valu e fu nc
) and
n us
p tion v(sj ~p) =+ s(1 + ~p) (cf . E q. (A:55)
layed e qu ation can b e written as
r
d))] 1 + r
~
(
p
;
!
v
) = m in ))P= min E[v(sj ~p)] 1 + r
dd
1 (
+ (1 + ) E0
.
d
~p
~
p
1
(
;
!
1
1
( ~pd
d)
is we akly increasing in ~pd, this e qu ation imp lie s that( ~pd) is s trictly inc re as in g
Pd
in ~pd
d
Sin ce . Fin ally, to sh ow that P( ~p) satis e s the b ound ary cond itions in (A:57), n ote th at E q. (10)
Sd
implies
P ( ~pd) =
d popt
(S ( ~pd) ;
d
vd
Using the de nition of popt
( j
~pd)) .
( ) f rom Eq. (10) and sub stitu tin g vd (sj ~pd) = s(1 + ~pd) (cf . Eq.
61
+
(A:55)), th e p re vious disp laye d e qu ation c an b e written as
(Sd d( ~p( sd max sdF1! .
1
~pd)) Z
(Sd
S d(
P ( ~pd) = 1 + ZsSd(
sdF + 1 F0 )) 1 F
~p
d ~pd1 + r
~pmind)
0
Next, c onsid er th is expres sion f = 1 and note th
at
or ~pd
r
smin
)
Sd
>1
+
P
d
1
S F1
1
+
r
d
s
d
F
+1
F0
s
d
F
0
d
1
r
d0
s
1d F11 F
sF1mi nmi n(s
ma x in
s
1
1
m
a
x
r
r
0
the rs t line with s
Z
d1
1 r Ssmax
r>Sd
1A
r",
d)
0
d
Sds
1
P
s
)
s
d
F
1r "
d
d)
d
d
d(
and Sd
~pd) we akly in crease for each ~pd. Sin
ce pd
( ) sh if ts u p, it follows that pd
Here, th e sec ond line replace . Similarly,, an d the ine qu ality f ollows sinc e th e expres sion in th e rs t line is a d ecreasin g fu
sS
nction of S
r"
=1+
1+ " r "
1+ " r "
0
@ZssSd(m
d
E(
~p
inmin
d)
0
1+ " r "
1+ " r "
Sd+ 1 F
( ~p
1
+1 F
Z
min
1+ " r "
(
E1
d
d
s
d
d
sdF0
d
d
Z
smax
1r"
max
!=
sd
F
1
d
(
~
p
[
1+"
1+r
s] =
Z mi
1+"
(sma xma
x(s) 1 F) Z
n
1+r
d
whe re th e s
econd lin e
replaces S
1
+
smax
sdF1
)Z
= 1 +1 +
rE
r
.
S . It f ollows th at Psatis e s th e b ou ndary cond itions in (A:57), com
The ore m 7.
in the rst lin e with s
"
r
"
Pro of of Theor em 8. Part (i). S ince (P
( ) ;Sd ( )) is the static e qu ilibrium f or the econ
~
~ omy
p
p
) and s in ce optimis ts optimism b ecome s weakly more right-skewed, The orem 3 ap plie s and sh ows that P(
~pis th e xe d p oint of th e s trictly in cre asing mapp in g P) and sinc e Pwe akly in creases . Next note that S( )
is a weakly in creasing fun ction (by the pro of of The ore m 7) and th at th e equilib riu m price to divid end ratio, p,
weakly in creases . S in ce S( ~p) also we akly increase s f or each
= Sd (pd d. P lu ggin g in the value fu nction,
0
( j pd) = s(1 + pcan b e
rewritte n asd
d
1
d
= ppdv(a) =
p(a)
1pd
(1
d)
1r +
p
pd
1+" .
r
, it follows
= (1
~ that the equilib rium loan riskines
s, s d
d
1+r+
), weakly in creases . Next cons
onent,
1 + " 1 + r.
vd
) (cf . Eq. (A:55)), an d us in g E
)1
d
1 + pd
[s] = 1 and E1 [s] = 1 + ", E q. (40)
d
Note also that Eq. (41) can b e writte n
as
Comb in in g the las t two disp laye d e qu alities, the sh are of the sp e culative comp on ent is give n by:
d
1 + p= 1 1 +
1 =rd
we akly in creases
Sin ce pd, d
also we akly in creases , completing th e p ro of f or th e rst p
art.
62
(
d
a
n
d
~
p
S
)
d(
~pd)) is th e s tatic e qu
conomy E ( ~p
d
d
;
S
d
Pd ( ~pd) and sinc e ( ) sh if ts u p, it follows that pd
Pd
63
d
Part (ii). S in ce
) and sinc e op timists optimism b ec omes weakly more s kewed to th e left of s
(Pd
app lies and s hows th at P
. S in ce is th e xed p oint of the s trictly in creasing mapp
( ~pd) we akly increas e for
in g
pd
each ~p
also we akly in creases , completing th e p ro of of The orem 8.we akly in creases . The same step s f
or p art (i) sh ow that
dd,
The orem 3
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