De e minan s o
Repo Hai cu s and
Bank up cy
Jean-Ma c Bo azzi
Má io R. Páscoa
Guille mo Ramí ez
Wo king Pape
# 615
2017
De e minan s o Repo Hai cu s and
Bank up cy
By Jean-Ma c Bo azzi, M´
a io R. P´
ascoa and Guille mo Ram´
ı ez1
Pa is S. o Economics and Capula; Uni e si y o Su ey; No a S. o Business and Economics
Abs ac
Va ia ions in epo hai cu s play a c ucial ole in le e aging (o dele e aging) in secu i y
ma ke s, as obse ed in he wo majo economic e en s ha happened so a in his cen u y,
he US housing bubble ha bu s in o he g ea ecession and he Eu opean so e eign deb s
episode. Repo ades a e secu ed bu ecou se loans. De aul igge s insol ency. Colla e al
may be empo a ily exemp om au oma ic s ay bu c edi o s’ inal eimbu semen depends
on he bank up cy ou come. We show examples o bank up cy equilib ia. We in e how
hai cu s a e ela ed o asse o coun e pa y isks whene e a bank up cy equilib ium exis s.
1We acknowledge commen s om audiences a UECE 2016 Game Theo y Mee ings (Lisbon, No embe 2016)
and SAET 2017 (Fa o, June 2017). Guille mo Ram´ı ez was suppo ed by a doc o al g an gi en by FCT,
Po ugal (g an BD 74704/2010). P´ascoa and Ram´ı ez we e suppo ed by p ojec PTDC/IIM-ECO/5360/2012
(FCT, Po ugal).
1
1 INTRODUCTION 2
1 In oduc ion
In a epo ade, a secu i y is pledged as colla e al o a cash loan and can hen by eused by
he cash lende , ha is, pledged in a ano he con ac o sho -sold. The euse o he colla e al
makes epo ades qui e di e en om mo gage loans whe e he du able good colla e al s ays
pu . The esul ing le e age was s udied in de ail by Bo azzi, Luque and P´ascoa (2012), unde
he assump ion ha agen s always ul illed hei inancial obliga ions.
Le e age played a majo ole in he ecen inancial c isis o 2008. Leading o he c isis, i
was no only households ha we e highly indeb ed bu also la ge inancial ins i u ions. These
la ge ins i u ions u ned o he shadow banking sys em o inance hemsel es (see e.g. Go on
and Me ick (2010)). The epo ma ke is a c ucial pa o his sys em. The hai cu applied o he
loan gi en in a epo ade is in e sely ela ed o how much agen s can build up hei posi ions
in a secu i y by using he epo ma ke as a means o inancing secu i y posi ions. Secu i y
and epo ades can be combined in way ha allows secu i y posi ions o be inc eased, as he
secu i y ge s pledged as colla e al in epo, hen epledged o sho sold by he c edi o ( hen
again pledged by he coun e pa y o he sho selle and so on). How do colla e al euse and
hai cu s de e mine wha le e age is? The o me may be an ing edien bu does no de e mine
by i sel wha le e age is (and limi a ions on euse do no au oma ically ansla e in o a ge ed
educ ions in le e age). Fo long agen s o le e up o hai cu po en ial, he euse o he colla e al
only becomes necessa y when he weal h o hese agen s is high enough and he hai cu is low
enough ha he esul ing agg ega e le e aged long posi ions exceed agg ega e ini ial holdings
o he secu i y. I is ul ima ely he hai cu ha de e mines wha le e age is (and i could be
he sho s being le e aged ins ead). Mo e ecen ly, in he so e eign deb c isis o 2010-12, he e
was subs an ial dele e age (also o hose sho selling) caused by he consecu i e hikes in epo
ma gins on bonds issued by se e al Eu opean go e nmen s.
Gi en ha le e age and he hai cu a e in e sely ela ed, i is c ucial o unde s and how
he la e is de e mined. The hai cu is he di e ence be ween he alues o he colla e al and
he espec i e cash loan, a he ime when he epo ade s a s. I is usually exp essed as
1 INTRODUCTION 3
a pe cen age (less han o equal o 1) o he colla e al alue. Equi alen ly, he ini ial ma gin
cap u es ha di e ence by exp essing he colla e al alue as a pe cen age (g ea e han o equal
o 1) o he cash loan. A epo ade has a pu chase leg and hen a epu chase leg a a epu chase
p ice ha is locked in a he i s leg. The di e ence be ween he pu chase p ice ( he cash loan)
and he epu chase p ice is he epo in e es a e, ag eed upon in ad ance. Hence, in he absence
o de aul , he e would be no eason o cha ge a hai cu . The hai cu e lec s he cash lende ’s
pe cei ed isk o loss in he e en o he cash bo owe ’s de aul .
In his a icle we model he limi ed commi men in ol ed in epo ades. In his espec , also,
he e is a key di e ence by compa ison wi h wha happens in many (bu no all) mo gages, as
cap u ed in he GE colla e al li e a u e. Repo ades a e ecou se loans, whe eas many (bu
no all) mo gages a e non- ecou se. I an household ha has signed a non- ecou se mo gage
decides o de aul , i would jus su ende he house and walk away wi hou su e ing any o he
penal ies. Tha is no he case in ecou se loans: in he e en o de aul , c edi o s can be epaid
abo e he colla e al liquida ion alue by o cing he bank up cy o he aul y bo owe and hen
becoming claiman s in he pa i ion o he bo owe ’s es a e. I may also happen ha c edi o s
end up eco e ing less han he colla e al liquida ion alue, when ha is he ou come om he
pa i ion o he es a e among all c edi o s. Repo colla e al is exemp ed om ce ain p o isions
o he US Bank up cy Code ha no mally apply o pledges, in pa icula , he au oma ic s ay
on en o cemen o colla e al in he e en o insol ency. Tha is, c edi o s can keep he colla e al
ha had been pledged o hem (and can sell i ) bu , when he bank up cy cou akes he inal
decisions, hey may ge mo e o less han wha hei claim was ( he p omised epaymen ) and
his may be di e en om he liquida ion alue o he colla e al.
I should be no ed ha when an agen goes bank up , i is no jus he epaymen o he cash
bo owed in epo ha is a s ake. I a secu i y happened o be pledged o his agen in epo,
hen his colla e al will no be gi en back o he cash bo owe s - a “ ail” occu s as a esul o
bank up cy - and he espec i e manu ac u ed di idends due o he bene icial owne will no be
paid also.
1 INTRODUCTION 4
De aul is a e y se ious e en and needs o be modeled by aking in o conside a ion he whole
bank up cy p ocess. I is no a decision ha can be aken asse by asse , compa ing p omised
paymen s and colla e al alues. Deb o s can’ be assumed o be epaying he minimum o hese
wo, con a y o wha happens in non- ecou se loans, as shown in a long s anding li e a u e
eme ging om he wo k by Geanakoplos and Zame in he nine ies (see Geanakoplos (1997),
Geanakoplos and Zame (1997) and Geanakoplos and Zame (2014)). Fo he same eason,
de aul can’ be a oided by designing con ac s so ha colla e al alues ne e all below p omised
paymen s, as was he case in a con empo aneous li e a u e da ing back o Kiyo aki and Moo e
(1997). Ga nishable es a es mus now be se agains o al deb s (ne o c edi s ha he de aul e
may be en i led o). This c ea es a non-con exi y in he bo owe ’s budge se ha seems o
ha e pu o p e ious esea ch e o s.
The e a e howe e in e es ing esul s ha can be es ablished, in spi e o he in insic non-
con exi y o indi idual decision p oblems. We conside binomial economies, whe e jus wo s a es
o na u e, Uo D, may occu a e he ini ial node (and each o hese s a es may be ollowed
wi hou unce ain y in o a hi d da e). Ou pape ocus on o e - he-coun e (OTC) epo, ha
is, ades ha a e no cen ally clea ed h ough an exchange (o cen al clea ing coun e pa y,
CCP), and bila e al (as opposed o i-pa y whe e colla e al selec ion, paymen , cus ody and
se lemen a e ou sou ced o a hi d-pa y agen ). Ou ini e-agen model does no le us explo e
he con exi ying e ec o la ge numbe s ha has been used in con inuum o agen s models in
se e al con ex s, including in consume bank up cy p oblems wi h unsecu ed loans (see A aujo
and Pascoa (2002) and Saba wal (2003)). Howe e , modeling he agen s se as a con inuum
is no app op ia e in a con ex o OTC epo whe e each ade should an icipa e coun e pa y
bank up cy isk and choose epo hai cu s acco dingly2.
Fo equilib ium o exis , le e age should be bounded. He e he e is ano he impo an dis-
inc ion be ween c edi backed by secu i ies and c edi backed by houses o p oduc i e esou ces.
2A bank up cy analysis migh be doable also o cen ally clea ed epo, bu agg ega e de aul isk should ake
he place o coun e pa y isk. The OTC case seems o be mo e in o ma i e and easie o ela e o he applied
li e a u e on he de e minan s o epo hai cu s.

1 INTRODUCTION 5
In he la e , he agg ega e supply o colla e al is ixed and, he e o e, unde exogenous colla -
e al ma gins, bo owing becomes bounded. In he o me , he colla e al supply is endogenous
since i includes sho -sales and, he e o e, he e a e no a p io i bounds on secu ed bo owing,
e en unde exogenous ma gins. Howe e , epo and secu i y posi ions mus be ela ed in ano he
way: he ne secu i y i le balance held by each agen mus be non nega i e. This is known as
he box cons ain and says ha in o o de o pledge he agen mus be long in he secu i y
and in o de o sho -sell he agen mus be long in epo ( he secu i y being pledged o him).
In he one-secu i y case, by combining he box and budge cons ain s we can bound secu ed
bo owing. This would be enough o bound all so s o le e age (long o sho ) in con ex ull
commi men economies. Howe e , in non-con ex economies allowing o bank up cy, we need o
bound secu ed lending as well, since an equilib ium o a unca ed economy (whose po olios
a e assumed o be ma ke easible) may ail o be an equilib ium. In he mul i-secu i y case, i
was al eady known ha , e en in he con ex ull commi men se ing, o he cons ain s should
be added wi h he pu pose o bounding epo and secu i y ades3.
In o de o gain in ui ion and allow o a ull cha ac e iza ion o equilib ia, we s a by
examining a one-secu i y and wo-agen case. In his simple case, he op imis is long in he
secu i y (sho in epo) and he pessimis is sho in he secu i y (long in epo). We ind equilib ia
whe e bo h o jus he o me go bank up ( he o me in he s a e whe e he secu i y has lowe
e u ns and he la e in he o he s a e). Then, we con empla e he mul i-secu i y and mul i-
agen case o see wha a e he de e minan s o hai cu s. On his issue, he e a e di e en iews
in he applied li e a u e. Go on and Me ick (2012) a gue ha hai cu s depend bo h on he
unde lying asse and on who is he coun e pa y in a epo ansac ion bu ha , pa icula ly in
imes o c isis, he la e gains impo ance. In con as , K ishnamu y e al. (2014) epo li le
a ia ion o hai cu s ac oss coun e pa ies and place much mo e weigh on he unde lying asse .
In an e (2015) a gues ha hese obse ed di e ences on hai cu s a ise because wo di e en
3See Bo azzi, Luque and P´ascoa (2012) on bounds ha esul om he seg ega ion o hai cu s o he dis inc ion
be ween deale s and non-deale s and Bo azzi, Luque and P´ascoa (2017) on bounds ha ollow om equi y
equi emen s in he spi i o he Basel egula ion o banks.
1 INTRODUCTION 6
ma ke s a e s udied: he bila e al and i-pa y epo.
The loss ha a lende may su e om coun e pa ies’ de aul may be ela ed o he colla e al
alling in alue (o being sold in a i e sale) bu , since he loan is ecou se, he loss canno be
associa ed o ha asse isk in such a simple way. I may happen ha he e is no asse isk bu
he coun e pa y isk will ne e heless go e n wha he lende ge s back, which does no ha e o
be equal o he colla e al liquida ion alue. Wha a secu ed c edi o eco e s in he bank up cy
p ocess depends on wha is he liquida ion alue o he whole es a e o he de aul e and how i
will be pa i ioned among all c edi o s, e en hough he exemp ion om au oma ic s ay allows
he c edi o o sell he colla e al while wai ing o he inal ou come o he bank up cy p ocess.
We cha ac e ize how hai cu s espond o asse and coun e pa y isks. Suppose he e a e
many ade s in he epo ma ke o each secu i y, epo a es a e secu i y-speci ic bu hai cu s
a e speci ic o each pai o ade s. In such compe i i e se ing, we should expec coun e pa y
bank up cy isk o a ec pai -speci ic hai cu s bu no he epo a e, as opposed o wha happens
in he wo-agen example. Say s a e Dis he s a e whe e bank up cy may occu . Suppose an
agen iis sol en in s a e Dand is, in e ms o he whole po olio, a ne c edi o o a coun e pa y
j(in s a e D) and he expec ed epaymen a e o his coun e pa y dec eases (an inc eased
coun e pa y isk). Then, agen iwould like o aise (lowe ) he hai cu cha ged o coun e pa y
jwhen accep ing colla e al om j, o secu i ies whose epo epaymen exceeds ( alls below) he
colla e al alue. Tha is, when he asse is isky om he c edi o ’s pe spec i e, hai cu s end
o mo e in he same di ec ion as he coun e pa y isk. Bu o he o he secu i ies ( isky om
he deb o s’ poin o iew, wa y o a epo ail), hai cu s mo e in he opposi e di ec ion.
Qui e di e en ly, in he wo-agen and one-secu i y example, coun e pa y isk a ec s he epo
a e and his e ec is s ong enough o make bank up cy a es dec ease as he hai cu inc eases.
In a small numbe s con ex , i is now he o he di ec ion ha may become mo e ele an : how
a e o e all sol ency a es a ec ed when he hai cu cha ged in one secu i y changes? Tha is
why in such ex eme non-compe i i e case, hai cu s and expec ed epaymen a es may mo e
oge he , con a y o ou esul s o he compe i i e case. To summa ize, he way coun e pa y
2 THE MODEL 7
isk may impac hai cu s depends on how compe i i e he epo ma ke is and o unde s and how
ha impac wo ks in he compe i i e case we need o couple his isk wi h asse isk. When
aced wi h a ise in coun e pa y isk, compe i i e c edi o s end o ask o highe hai cu s o
secu i ies ha exhibi an asse isk om he c edi o s’ pe spec i e, bu lowe hai cu s may a ise
i he secu i y in ol es he opposi e isk (a ail a he han a de aul isk).
2 The Model
2.1 Fundamen als
We conside a binomial economy wi h h ee da es. A an ini ial da e (da e 0) he e is only one
node in he e en ee, ollowed by nodes Uand Da he second da e. Each second da e no e
has a unique successo a he hi d da e: U+and D+a e he successo s o Uand D, espec i ely.
As we will see, he hi d da e jus se es o gua an ee ha secu i ies e ain alue a he second
da e, when bo owing and lending ansac ions a e se led (and we may wan o dispense wi h
he hi d da e in some cases, as discussed below).
Figu e 1: E en s ee o he binomial economy.
Binomial models ha e been used o s udy he le e age cycle in economies wi h de aul on
non- ecou se loans (see e.g., Fos el and Geanakoplos (2012) and Fos el and Geanakoplos (2014)).
Gi en ha epo ades cons i u e ecou se loans, we will model de aul as a bank up cy p ocess.
The e is only one consump ion good. Ma ke s o his commodi y open a each e en . We
2 THE MODEL 8
deno e he p ice o his good a e en eby pe. The e is a ini e se o I≥2 agen s, indexed by i.
A bundle o commodi ies consumed by agen iis deno ed by xi= (xi
0, xi
U, xi
D, xi
U+, xi
D+). The e
a e also F eal secu i ies indexed by , each one being cha ac e ized by a ec o o non-nega i e
eal e u ns R = (R U , R D, R U+, R D+). Gi en spo p ices pe, he nominal e u n o secu i y
is peR e4.
T ading o secu i ies occu s a he i s and second da es. Each agen chooses a secu i ies
po olio φi∈
R
3Fconsis ing o posi ions in he Fsecu i ies a he ini ial nodes and nodes U
and D. Secu i y p ices a e deno ed by q≡(q
e)∈
R
3F. Agen s’ endowmen s o commodi ies a e
ωi∈
R
5
+, wi h ωi
s>0 in bo h s a es. Agen s ha e ini ial holdings, a da e 0, o each secu i y
,oi
>0. P e e ences a e desc ibed by u ili y unc ions Ui:
R
5
+→
R
. Fo each secu i y , we
no malize i s posi i e ne supply o be one: Pioi
= 1.
2.2 Repo ma ke s
Agen s can ha e nega i e posi ions in secu i ies, sho -sales a e pe mi ed. Sho -selling, howe e ,
is no he same as issuing (which we ake as gi en his model, ha ing occu ed p io o da e 0).
In o de o sho -sell a secu i y, an agen mus go i s in he epo ma ke and bo ow he desi ed
amoun o secu i ies. This is he way sho -selling is ac ually done in eali y.
Bo owing o secu i ies ac ually consis s in buying he secu i y and p omising o esell i o
he lende , a a u u e da e and a a p ede e mined p ice. The e is a di e ence be ween he
p ice a which a secu i y is bough , in he i s leg o he ansac ion, and he p ice a which i is
esold o i s o iginal owne , in he second leg o he ansac ion, a a u u e da e. This di e ence
is cap u ed by he epo a e. The highes epo a e wi hin i s class o secu i ies is e e ed o as
he gene al colla e al a e (GC).
The bo owe o a secu i y acqui es possession igh s associa ed wi h he secu i y. Howe e ,
any coupon o di idend paid o he bo owe du ing he e m o he ansac ion is passed h ough
o he o iginal owne ; his is called a manu ac u ed paymen o a manu ac u ed di idend.
4We could ha e conside ed nominal secu i ies ins ead.
3 A ONE-SECURITY AND TWO-AGENT MODEL 15
le e age:
zi≤oi
1−h
I we subs i u e xi
Uand xi
Din o agen i’s u ili y unc ion, we can w i e his p oblem as:
Maximize
Eiωi+aimax −βωi
U,(oi+ (1 −h)zi)RU+ηj
U[(h −RU)zi]+−[(h −RU)zi]−
+ (1 −ai) max −βωi
D,(oi+ (1 −h)zi)RD+ηj
D[(h −RD)zi]+−[(h −RD)zi]−
s. .
oi+ (1 −h)zi≥0
zi≤oi
1−h
The only decision a iable in he p oblem is ziand he agen only needs o decide whe he o
be long (zi>0) o sho (zi<0) in epo. Gi en ou assump ion on secu i y paymen s and
u ili ies, i is easonable9 o sea ch o equilib ia in which h ∈(EjR, EiR). Gi en he ela i e
weigh s o each s a e in agen s iand ju ili y unc ion i is also easonable o s a o equilib ia
by assuming agen i o be epo sho (zi<0), and j o be epo long (zj>0).
Being sho in epo, agen ican po en ially ans e consump ion om s a e D o s a e U,
which gi es him compa a i ely mo e u ili y. In o he wo ds, agen iis an op imis wi h ega d
o his secu i y (as he pu s mo e weigh in he s a e whe e he secu i y pays mo e) and his
sugges s ha he should be long in he secu i y and le e age his long posi ion by being sho in
epo. Howe e , aking a sho epo posi ion is no gua an eed o inc ease his consump ion in
s a e U, o o yield an inc ease in o e all u ili y, since his depends on he agen ’s coun e pa y
e ec i e epaymen a e in s a e U(ηj
U).
We canno comple ely ule ou agen i aking a long epo posi ion e en hough his would
ans e consump ion om a high u ili y s a e o a s a e wi h low u ili y. The eason o his is
9In ac , i is easy o see ha i h > EiRo h < EjRbo h agen s will wan o ake ei he long o sho epo
posi ions, and he e canno be ma ke clea ing.

3 A ONE-SECURITY AND TWO-AGENT MODEL 16
ha bank up cy limi s he u ili y loss in s a e Uand, depending on how la ge his long epo posi-
ion is allowed o be, he inc ease o u ili y expe ienced in s a e Dcould mo e han compensa e
his loss.
Ul ima ely, whe he agen i akes a long o sho epo posi ion will depend on his endowmen s
on each s a e and in how much he can le e age his posi ion. I le e age is low enough as o ule
ou bank up cy, hen inecessa ily akes a sho posi ion (zi<0).
S a ing wi h he assump ion ha iis sho in epo, his posi ion will be de e mined by he
box cons ain . In ac , as he u ili y unc ion is linea , agen iwill pick he la ges possible sho
epo posi ion. Since we know ha xi
s≥(1 −β)ωi
s>0 no ma e wha he po olio migh be,
agen iis no cons ained in his choice by non nega i i y o xi
sin any s a e. I is jus he box
ha de e mines wha ha la ges sho epo posi ion is. This is zi=−oi
1−h=−1
1−h.
Ma ke clea ing equi es ha i agen iis sho in epo, agen jmus be long. Again, he
linea i y o u ili ies equi es ha j akes he maximum posi ion ha he can in he epo ma ke .
We know his posi ion o be zj=oj
1−h=1
1−h.
Wi h hese epo posi ions, agen s iand ja e sol en in s a es Uand D, espec i ely. In ac ,
in s a e Uagen ihas a non-nega i e inancial income: (oi+ (1 −h)zi)RU+ηj
U[(h −RU)zi]+−
[(h −RU)zi]−= (oi+ (1 −h)zi)RU+ηj
U(RU−h )|zi| ≥ 0>−βωi
U. The e o e, agen idoes
no become insol en , ac ually makes xi
U≥ωi
U(and analogously o agen jin s a e D).
Howe e , agen iis dec easing consump ion in s a e Dand we canno be su e o his sol ency
in ha s a e. The same applies o agen jin s a e U. I we le αi
s= 1 i agen iis sol en in
s a e sand αi
s= 0 when he decla es bank up cy, we ha e ou possible cases o conside :
Case αi
Uαi
Dαj
Uαj
D
1 1 0 0 1
2 1 0 1 1
3 1 1 0 1
4 1 1 1 1
I will be use ul o deno e by zs he posi ion o he sho epo agen and zl he posi ion o he
3 A ONE-SECURITY AND TWO-AGENT MODEL 17
long epo agen . We ha e a gued ha he sho agen (whoe e he is) will be sol en in s a e U
while he long agen will be sol en in s a e D.
Nex we no e ha whe he an agen goes bank up o no in a ce ain s a e depends en i ely
on how he agen ’s obliga ion in ha s a e compa es wi h he ga nishable po ion o his income.
Fo a gi en βand hwe can compu e he (g oss) epo a e ha equalizes he wo and ha we
deno e s o he sho agen and l o he long agen . In he case o he sho agen we ha e:
s=RD
h+1−h
h·βωs
D
os
I < swe ha e αs
D= 1 and, i > s,αs
D= 0.
Fo he long agen we ha e ha :
l=RU
h−1−h
h·βωl
U+ 2olRU
ol
I < lwe ha e αl
U= 0 and, i > l,αl
U= 1.
Suppose ha he pa ame e s o he model a e such ha RD/h < s< l< RU/h. I we
conside a gi en hai cu and o a ixed βwe ha e ha depending on he epo a e, bank up cy
coe icien s a e necessa ily as ollow:
Case αs
Uαs
Dαl
Uαl
D
< s1 1 0 1
s< < l1 0 0 1
l< 1 0 1 1
Now, o each , we can compu e he consump ion o bo h sho and long agen s. Fo he sho
agen we ha e:
xs
U=ωs
U+αl
U(h −RU)zs+ (1 −αl
U)[βωl
U+ 2olRU](8)
xs
D=ωs
D+ max{−βωs
D,(h −RD)zs}(9)
In (8) we ha e w i en αl
U(h −RU)zs+(1−αl
U)[βωl
U+2olRU] ins ead o ηl
U[(h −RU)zs]+. The
wo e ms coincide because ηl
U=αl
Uwhen he long agen is sol en in s a e Uand, when he
3 A ONE-SECURITY AND TWO-AGENT MODEL 18
long agen is insol en , we ha e ha βωl
U+ 2olRU=−ηl
U(h −RU)zl
U. F om ma ke clea ing
we ha e ha zl=−zs, so ha :
ηl
U[(h −RU)zs]+=ηl
U(h −RU)zs=−βωl
U+ 2olRU
(h −RU)zl
U
(h −RU)zs
=[βωl
U+ 2olRU]zs
zs=βωl
U+ 2olRU
Analogously, we can w i e he consump ion o he long agen as:
xl
U=ωl
U+ max{−βωl
U,2olRU+ (h −RU)zl}(10)
xl
D=ωl
D+ 2olRD+αs
D(h −RD)zl+ (1 −αs
D)βωs
D
(11)
F om his consump ion o he sho and long agen , we can compu e hei espec i e u ili ies o
a gi en alue o . The inal s ep o con i m ha consump ion plans and po olios co espond
o an equilib ium is o check o op imali y. This is done by compa ing agen s’ u ili ies wi h he
le els o u ili y hey would a ain by aking he opposi e ac ion (e.g. a sho agen deciding o
ake a long posi ion ins ead) while conside ing he choice o he o he agen as gi en. Tha is,
he sho agen mus compa e his u ili y wi h he u ili y he would ge i he chose he po olio
zsl >0 ins ead. The consump ion implied by his po olio would be gi en by:
xsl
U=ωs
U+ max{−βωs
U,2osRU+ (h −RU)zsl}(12)
xsl
D=ωs
D+ 2osRD+ (h −RD)zsl
(13)
No e ha in (12), e en hough he (long) coun e pa y migh be insol en in s a e U, his does
no a ec consump ion o he sho agen because now, when he is also aking a long posi ion,
he e m (h −RU)zsl cons i u es an obliga ion o he agen and he epaymen a e o his
coun e pa y is i ele an .
The long agen mus also compa e his u ili y wi h wha he would ge i he chose he sho
posi ion zls <0 ins ead. In his case his consump ion would be gi en by:
xls
U=ωl
U+ (h −RU)zls
(14)
xls
D=ωl
D+ max{−βωl
D,(h −RD)zls}(15)
3 A ONE-SECURITY AND TWO-AGENT MODEL 19
We ha e ha he o iginal consump ion plans a e op imal (and we ha e an equilib ium) i i is
ue ha Us(xs
U, xs
D)≥Us(xsl
U, xsl
D)and Ul(xl
U, xl
D)≥Ul(xls
U, xls
D).
We can o example, s udy an economy wi h ini ial pa ame e s:
β= 0.35 RU= 1.4ωi
U= 4 ωj
U= 6 ai= 0.9
h= 0.9RD= 0.1ωi
D= 2 ωj
D= 4 aj= 0.2
The ollowing a e equilib ia in which iis epo sho and jis epo long o his economy:
αi
Dαj
Uxi
Uxi
Dxj
Uxj
DUiUj
0.7755 0 0 8.9 1.3 3.9 4.9 8.14 4.7
0.8044 0 0 8.9 1.3 3.9 4.9 8.14 4.7
0.8333 0 0 8.9 1.3 3.9 4.9 8.14 4.7
0.8622 0 0 8.9 1.3 3.9 4.9 8.14 4.7
0.8911 0 0 8.9 1.3 3.9 4.9 8.14 4.7
0.9200 0 0 8.9 1.3 3.9 4.9 8.14 4.7
0.9488 0 0 8.9 1.3 3.9 4.9 8.14 4.7
0.9777 0 0 8.9 1.3 3.9 4.9 8.14 4.7
1.0066 0 0 8.9 1.3 3.9 4.9 8.14 4.7
1.0355 0 1 8.68 1.3 4.12 4.9 7.942 4.744
1.0644 0 1 8.42 1.3 4.38 4.9 7.708 4.796
1.0933 0 1 8.16 1.3 4.64 4.9 7.474 4.848
1.1222 0 1 7.9 1.3 4.9 4.9 7.24 4.9
1.1511 0 1 7.64 1.3 5.16 4.9 7.006 4.952
1.1800 0 1 7.38 1.3 5.42 4.9 6.772 5.004
1.2088 0 1 7.12 1.3 5.68 4.9 6.538 5.056
1.2377 0 1 6.86 1.3 5.94 4.9 6.304 5.108
1.2666 0 1 6.6 1.3 6.2 4.9 6.07 5.16
1.2955 0 1 6.34 1.3 6.46 4.9 5.836 5.212
1.3244 0 1 6.08 1.3 6.72 4.9 5.602 5.264
No ably, he e a e no equilib ia co esponding o cases 3 o 4 in his economy. We can compu e
he exac alues o sand l:
s=RD
h+1−h
h·βωs
D
os=0.1
0.9+0.1
0.9·0.35 ·2
1= 0.1888
l=RU
h−1−h
h·βωl
U+ 2olRU
ol=1.4
0.9−0.1
0.9·0.35 ·6+2·1.4
1= 1.0111
Figu e 3 shows agen s iand j’s p oblems when he (g oss) epo a e is 1.18 and clea ly show
ha i is op imal o i o be sho in epo (as much as he box cons ain allows him) and
3 A ONE-SECURITY AND TWO-AGENT MODEL 20
o agen ji is op imal o ake he highes long posi ion ha he can. The Figu e shows
consump ion in s a es Uand D o each agen . Some imes, as wi h xi
D, a kink occu s in he
agen s consump ion a he poin whe e ziis such ha he agen s obliga ions equal his ga nishable
income and he agen is indi e en be ween being sol en o decla ing bank up cy. In o he cases,
as o xi
U, no kink is obse ed. This is because he epo posi ion ha equalizes obliga ions and
ga nishable income occu s ou side he in e al ha cons ain s epo posi ions. In his case, zi
a which he kink would occu is zi= 14.14, which is he po olio ha sa is ies he condi ion
−βωi
U= (oi+ (1 −h)zi)RU+ (h −RU)zi. Simila ly, o xj
U he kink whe e jis ma ginally
sol en occu s o zj= 15.9, also beyond he uppe bound on zj. In he case o xj
D, wo kinks a e
obse ed. The one o he le co esponds o he epo posi ion ha makes he agen indi e en
be ween being sol en o no . The one a zj= 0 occu s because when zj<0, he agen is a
deb o in s a e D(meaning ha (h −RD)zj<0) and so he is no a ec ed by i’s epaymen a e
ηi
D<1. When zj>0, he is a ne c edi o , is a ec ed by i’s epaymen a e (meaning ha his
income is ηi
D(h −RD)zjins ead o (h −RD)zj) and his educes he slope o xj
Das a un ion
o zj.
(a) Agen i’s p oblem. (b) Agen j’s p oblem.
Figu e 3: How consump ion and u ili y a he second da e ela e o epo posi ions, when = 1.18.
As is clea om he p e ious discussion, o a gi en se o pa ame e s, he e a e mul iple
equilib ia. Rega dless o his inde e minacy, we ha e wo condi ions ha mus be sa is ied in

3 A ONE-SECURITY AND TWO-AGENT MODEL 21
any equilib ium:
ηi
D=βωi
D(1 −h)
( h −RD), ηj
U=−[βωj
U+ 2RU](1 −h)
( h −RU)
(16)
These equa ions sugges ha ∂ηi
D
∂h =βωi
D(RD− )
( h−RD)2. Fo he equilib ia we ha e p esen ed he e we
ha e ∂h
∂ηi
D
<0, and ∂h
∂ηj
U
<0.
Figu e 4 shows how his ela ed o he equilib ium alues o ηi
Dand ηj
U, o h anging om
0.85 o 0.99.
Figu e 4: How ηi
Dand ηj
U ela e o hwhen = 1.18.
Obse e ha haicu s mo e oge he wi h he coun e pa y’s epaymen a e ( his mus always
happen in his 2-agen and 1-secu i y economy. As we will see in sec ion 4, in a compe i i e
se ing, whe e many agen s ade many secu i ies, he impac o he hai cu in one secu i y on
he insol ency o an agen becomes less no iceable. I is he o he di ec ion ha becomes mo e
ele an : hai cu s ise in esponse o lowe epaymen a es o he coun e pa y, o secu i ies
ha in ol e a isk om he c edi o ’s poin o iew (ha e a colla e al liquida ion alue below
he p omised epo loan se lemen ). Tha is, in a compe i i e se ing, c edi o s end o ocus
on how o p o ec hei indi idual c edi s a he han ying o in luence he sol ency o he
coun e pa y.
4 HAIRCUTS 22
4 Hai cu s
Hai cu s in pai wise epo ades a e endogenously de e mined in he equilib ium ha we de ined.
Exis ence was es ablished and cha ac e ized o he 2-agen and 1-secu i y case o a se o gi en
pa ame e s. We discuss now wha may go e n hai cu s, ha is, how should we expec hai cu s
o be se in equilib ium, depending on wha a e he pa ame e s and o he equilib ium a iables
o he ele an pai o epo ade s.
Suppose agen ihas a possession alue o secu i y a he ini ial node, ha is, a binding
box cons ain o secu i y a he ini ial node - mo e p ecisely, he shadow alue µi
0o his
cons ain is posi i e. Deno ing by λi
eagen i’s mul iplie o he budge cons ain a each
node eand νij
he mul iplie o he lowe bound on epo posi ions alue, om he i s o de
condi ions o agen i’s p oblem, we ge he ollowing exp ession o hij
hij
=τij
(p, q) + νij
λi
0
1− Ps
λi
s
λi
0
αi
sκij
s
(17)
whe e
τij
(p, q) = 






Ps
λi
s
λi
0
αi
s(1 −κij
s)(q s+psR s)
q 0,i µi
0= 0
1−Ps
λi
s
λi
0
αi
sκij
s
(q s+psR s)
q 0,i µi
0>0
and κij
s=γij
sηj
s+ (1 −γij
s), γij
s= 1 i Iij
s>0, γij
s= 0 i Iij
s<0, αi
s= 1 i agen iis sol en in
s a e s,αi
s= 0 i igoes bank up in s a e s, and ηi
ssa is ies (5).
I is wo h ecalling ha ηj
sis he e ec i e pe cen age o his deb ha agen jpays o all o
his coun e pa ies, so i can be used as a measu e o coun e pa y isk: he lowe ηj
sis, he iskie
(o less sol en ) agen jis in s a e s, and his mus be aking in o accoun by agen s deciding
ha ing jand coun e pa y and, in pa icula , in se ing he e ms o epo con ac s (hij
).
Equa ion (17) is ue in any equilib ium and can be used o s udy he incen i es ha coun-
e pa ies iand jha e o ei he inc ease o dec ease he hai cu (1 −hij
) associa ed o hei
epo ansac ions in esponse o an inc ease in he isk o one o he coun e pa ies. Le ’s look
a he de i a i e o hij
wi h espec o ηj
D, unde he assump ion ha agen s’ ma ginal a es o
income subs i u ion emain unchanged. To be mo e p ecise,
4 HAIRCUTS 23
Assump ion (Λ): agen i’s ma ginal a es o subs i u ion o income ac oss he i s wo da es,
λi
s/λi
0, a e no a ec ed by a change in he coun e pa y j’s e ec i e epaymen a e ηj
D.
Al hough we migh no wan o ake his assump ion li e ally, i is use ul o ge a sense o
how hai cu s mo e wi h coun e pa y isk in a con ex whe e agen iis ading in many secu i ies
and has many coun e pa ies, so ha a small a ia ion in he de aul a e o one o hem in some
s a e won’ a ec he op imal in e -nodes de la o s o agen i.
This assump ion holds o linea u ili ies ( ecall ha he bank up cy s uc u e ensu es he
posi i i y o consump ion in each s a e, which implies ha DUi
s(xi) = λi
sps) in he case o epo
o ma u i y (dispensing wi h he hi d da e and allowing o ps= 1) and p o ided ha agen i
is consuming a he ini ial da e (so ha DUi
0(xi) = λi
0p0) and ha he equilib ium commodi y
p ice p0is no a ec ed by a small change in he e ec i e epaymen a e ηj
Do coun e pa y j
in s a e D.
∂hij
∂ηj
D
=
−αi
Dγij
D·1
q 0·λi
D
λi
0
·[(q D +pDR D)−hij
q 0 ] + hij
∂
∂ηj
DP
s
λi
s
λi
0
αi
sκij
s
1− P
s
λi
s
λi
0
αi
sκij
s
(18)
When mos o he esponse o a a ia ion in coun e pa y isk is channeled in o a change in
hai cu s, a he han a change in he epo a e, we can be mo e speci ic abou he di ec ion o
change. We say ha epo a es a e compe i i e i ac ions by a pai o agen s iand j, in pa icula
ac ions ha change hei sol ency a es (ηi
sand ηj
s) do no a ec equilib ium epo a es. This is
a easonable assump ion i he e a e many agen s (and he e o e, many pai s o coun e pa ies)
in he economy, bu no o be expec ed in an economy wi h only wo (o e y ew) agen s, as in
he example o sec ion 3. We ha e,
P oposi ion 1. Suppose epo a es a e compe i i e. Le us e alua e he impac o ηj
Don hij
,
unde a scena io whe e agen s’ ma ginal a es o subs i u ion a e no a ec ed. Say Iij
D>0(iis
a ne c edi o in he epo ma ke wi h espec o j) and iis sol en in s a e D(αi
D=ηi
2= 1). I
agen j’s expec ed epaymen a e ηj
Ddec eases, agen iwill wan o:
•Inc ease he hai cu (1−hij
)he cha ges (pays) in his epo long (sho ) posi ions wi h agen
j, o secu i ies such ha hij
q 0 > q D +pDR D.
4 HAIRCUTS 24
•Dec ease hai cu s paid o (cha ged o) agen j o his sho (long) epo posi ions in secu i ies
gsuch ha hij
gqg0 g< qgD +pDRgD.
I Iij
D<0(iis a ne deb o in he epo ma ke wi h espec o j), o i iis insol en in s a e D,
he has no incen i es o inc ease o dec ease he hai cu (1 −hij
)in esponse o expec ed changes
in ηj
D.
Rema k 1. The las pa o he p oposi ion e lec s he ac ha i iwe e a ne deb o o agen
jins ead, he would no be en i led o any sha e in he liquida ion o agen j’s es a e in he e en
o agen j’s bank up cy. No e ha as long as agen s iand j ade in he epo ma ke , one o
hem mus be a ne c edi o and p oposi ion 1 applies o ei he io j, as long as he agen is
sol en in s a e D.
P oo . See he appendix.
Suppose ha iis a ne c edi o wi h coun e pa y j, and ha zij
>0, and ha hij
q 0 >
q D +pDR D. I agen ian icipa es a dec ease in j’s expec ed epaymen a e, ηj
D, hen iwould
like o cha ge ja highe hai cu (by lowe ing hij
). To unde s and why his is so, no e ha he
magni ude o j’s ne deb o i, is gi en by zij
[q 0 hij
−(q D +pDR D)] and lowe ing hij
would
educe his deb and, he e o e, he loss esul ing om agen j’s bank up cy.
Now suppose hij
q 0 < q D +pDR D. I e e y hing else is as in he p e ious pa ag aph,
a dec ease in ηj
Dwill be an incen i e o i o collec a lowe hai cu om j(by aising hij
).
E en hough agen iis a ne c edi o o agen jwhen adding up all o his epo ansac ions
wi h j, he has now a deb o jassocia e o his posi ion on secu i y wi h absolu e alue
zij
[(q D +pDR D)−q 0 hij
]. Tha is, he colla e al kep by iwhen lending cash o jhas now a
highe ma ke alue han wha jowes o i. I hij
inc eases, he ge s o keep mo e o he colla e al
in he e en o j’s bank up cy.
In bo h cases, ihas incen i es o espond in a way ha coun e ac s he loss in income when
jbecomes mo e insol en in s a e D. The app op ia e esponse depends on he ela ionship o
he alue o j’s deb (q 0 hij
) wi h he ma ke alue o he colla e al (q D +pDR D). This