Dissecting Fire Sales Externalities
Eduardo Dávila*
Harvard University
edavila@fas.harvard.edu
First Draft: February 15, 2011
This Draft: August 19 2011
This Compilation: August 19, 2011
Abstract This paper analyzes the sources of the
externalities generated by fire sales in the simplest
Walrasian borrower-lender problem. I identify three different channels and I brand them as the risk
sharing, collateral and margin channels. The risk sharing channel may create overor
underinvestment; the collateral and margin channels always create overinvestment. Unless
transfers are allowed, there are no feasible constrained Pareto improvements. When transfers
areallowed,constrainedParetoimprovementsaregenerallyfeasible. Thispaperrecognizesthat
government interventions are time inconsistent in this environment and shows how the welfare
implicationsoffiresalesmustbedecoupledfromamplificationmechanisms. Moreover,thispaper
clearlyidentifiestheeffectscausedbyexogenousandendogenousmarketincompletenessandits
interactions.
JELnumbers: D62,E44,G18. Keywords: fire sales, pecuniary externalities, collateral
constraints, amplification
mechanisms,risksharing,margins,timeinconsistency,financialregulation.
*IwouldliketothankPhilippeAghion,
MariosAngeletos, RobertBarro, JohnCampbell, EmmanuelFarhi, Jordi
Galí,OliverHart,GregMankiw,AndreiShleifer,AlpSimsek,JeremySteinandJeanTiroleforveryhelpfulcomments
anddiscussions,aswellasparticipantsinHarvardMacrolunch. FinancialsupportfromRafaeldelPinoFoundationis
gratefullyacknowledged. Remainingerrorsaremyown.
1 Intro duction
1Firesalesoffinancialorrealassetsarerecurringphenomenainsituationsoffinancialdistress.
We expect a
fire sale to arise when a shock hits the natural owners of a certain asset simultaneously,
forcingthemtoselltheirpositionstolowvaluationuserslessabletoholdtheasset. Thisdescription
offiresales,whichcanbetracedbacktotheseminalpaperbyShleiferandVishny(1992),leavesthe
followingquestionunresolved: iffiresalesaremerelyduetochangesinprices, whyshouldwecare
abouttheirrealeffects? Aren’tchangesinpriceasimpleredistributionofresourcesbetweenparties?
Ifwebelievethatmarketsarecomplete,anyobservedchangeinpricesismerelyavisibleexpressionof
Walrasianadjustment,whichinducestransfersbetweenagentsthatyieldaParetooptimalallocation and
make fire sales irrelevant from a welfare perspective. This is the content of the First Welfare
Theoremappliedtoafiresalecontext. Therefore,onlybycarefullyunderstandingthesourcesand
implicationsofmarketincompletenesswillwebeabletodeterminetherealeffectsgeneratedbyfire sales.
Thisisthegoalofthispaper.
This paper analyzes the different mechanisms by which pecuniary externalities induced by fire
saleshavefirst orderwelfare implications. Using thesimplest possibleWalrasianmodel, Iamable
toisolateanddistinguishamongthreedifferentchannels;Ibrandthemastherisksharing,collateral
andmarginchannels. Themainresultsofmyanalysis,whichcontrastwithpartsoftheprevious literature,are:
•Therisksharingchannelmaycreateover-orunderinvestmentwhilethecollateralandmargin
channelsalwayscreateoverinvestment.
2•When
ex-ante lump sum transfers between agents are not allowed, there are no feasible
constrainedParetoimprovingpolicies. Ifex-antelumpsumtransfersareallowed,whencredit
constraintsbind,therearefeasibleconstrainedParetoimprovingpolicies. Acorollaryofthese
resultsisthatsettingasimplecapitalrequirementisnotenoughingeneraltoachieveaPareto
improvement: ex-antecompensationisrequired.
•Amplification mechanisms are not the source of the pecuniary externalities described in this paper,
even though they can be present in this environment and be significant from a
quantitativeperspective. Normativeandpositiveimplicationsoffiresalesmustbedecoupled
andevaluatedseparately. Ishowhowwecanconstructconstrainedinefficientequilibriawithout
anyamplificationandefficientequilibriawithamplificationmechanisms. Thissharpdistinction
isnotpresentinpreviousliterature.
1ShleiferandVishny(2011)surveytheexistingliteratureonfiresales.
2This
apparently negative result does not necessarily imply that capital regulation is undesirable. It points out,
nonetheless,thatthepolicymakermusttakeintoaccountdistributionalconsiderations.
2
•A planner that seeks to implement constrained Pareto improving policies faces a time
inconsistency problem. All three channels are subject to this time inconsistency concern. To
myknowledge,thisisthefirstpaperthatcharacterizesthetimeinconsistencyproblemofthe
constrainedplannerinafiresaleenvironment.
•The amount of fire sold asset is the relevant variable to determine the magnitude of the risk
sharingexternality;however,thefullamountofcollateralizableassetheld,soldinthefiresale or not, is the
pertinent variable to determine the importance of both margin and collateral channels.
•Endogenousmarketincompletenesscreatesidenticalexternalitiestothoseimpliedbyexogenous
marketincompleteness. Endogenousmarketincompletenessmayalsobethesourceofadditional
externalitiesthatdependontheparticularspecificationofthecreditconstraints.
3TherisksharingchannelisduetotheinteractionbetweentheWalrasianroleofpricesandthefact
thatmarketsareexogenouslyincomplete. Inthisparagraph,exogenousmeansthattheunderlying financial
friction or the particular missing market that originates the incompleteness need not be explicitly
determined in the economic environment. The fact that markets are incomplete and that intertemporal
marginal rates of substitution across agents are not equalized state by state
4makesitpossiblefortheplannertoinducewelfareimprovingpricedrivenredistributionbychanging
allocations. Intuitively, a planner can modify allocations to induce price changes that redistribute
wealth at each period/state towards those agents with higher marginal utility of wealth. The risk
sharing channel on its own can be understood as a particular case of the generic result shown by
Geanakoplos and Polemarchakis (1986) and Dreze et al. (1990), in which exogenously incomplete
markets are generically constrained inefficient. I want to make clear that only intertemporal risk
sharingplaysaroleinthispaper: evenwhenthereisnouncertainty,therisksharingchannelisfully active.
5Boththecollateralandmarginchannelsarecreatedbyendogenousmarketincompleteness;by
endogenousImeanthatfinancialconstraintsareexplicitlymicrofoundedandpricedependent. The
collateralchannelariseswhenpricetakingagentsdonotinternalizethattheirdecisionscandirectly
modifyprices,reducingtheborrowingcapacityofothers. Themarginchannelworksontopofthe
3Using
asset pricing intuition may be helpful here. The Fundamental Theorem of Asset Pricing (see for instance
Duffie(2001)orCochrane(2005))tellsusthat,whenmarketsarecomplete,thereexistsauniquestochasticdiscount factor (SDF) or,
equivalently, the marginal rate of substitution across any two states is equal for all agents. When markets are incomplete,
many SDF’s price assets, allowing different agents to have different marginal rates of substitution.
4Diamond(1967)andHart(1975)aretheseminalpapersinthisliterature.
5Aproperlymicrofoundedmodelinwhichpricesdonotaffectcreditconstraintsdoesnotgenerateanyexternality ofthiskind.
Thekeydistinctionforwelfareforthecollateralandmarginchannelsiswhetherpricesenterexplicitlyin thecreditconstraintsornot.
3
collateralchannelifborrowingconditions(margins)reactendogenouslytoprices,asisthecaseinany
modeldrivenbyvalue-at-riskconstraintsorinsituationsinwhichinformationalasymmetriesreact
toassetprices. ThecollateralchannelcanbemotivatedalongthelinesofHartandMoore(1994), where,
because of commitment problems, the market value of collateral at the time of repayment
arisesasthenaturalborrowingconstraint. ThemarginchannelhasthespiritofStiglitzandWeiss
(1981),inwhichpricesactasascreeningdevice. Itrelatesmoregenerallytomodelswhereasymmetric
informationplaysakeyroledeterminingborrowingconstraints. GreenwaldandStiglitz(1986)derive
generalresultsregardingconstrainedParetoinefficiencyintheseenvironments. Thebasicintuition
forwhytheselasttwochannelsaffectwelfareisdeceptivelysimple: wheninvestmentopportunities are
directly dependent on prices, leaving aside their effects on the budget constraints, price taking
behaviorneednotbeconducivetoParetooptimalallocations,sinceeachagentdoesnotinternalize
thathisdecisionsdirectlyaffectthechoicesetsofotheragentsinthemarket. Fromthisviewpoint, prices have
failed their full Walrasian duty of signaling scarcity through budget constraints, and begin to play a role
as drivers of market imperfections. It shouldn’t be surprising then that price
movementsinducerealexternaleffects. Inefficiencyofthecompetitiveequilibriumwithendogenous
borrowingconstraintsinWalrasianenvironmentsisalsodiscussedbyKehoeandLevine(1993)and
KilenthongandTownsend(2010).
Eventhoughbothexogenousandendogenousmarketincompletenesshavebeenstudiedbefore, it is not
clear in the literature how they relate to each other. This paper is the first to combine both approaches
and to tractably isolate how both frictions affect welfare in the fire sale context. I carry out my analysis
in the simplest Walrasian borrower-lender problem, the building block for any model in which financial
considerations play a relevant role. For instance, my results apply directly to financial intermediation
and housing market problems. A critical reading of the most recent literature that tries to apply these
concepts to financial environments suggests that many
authorsreferverylooselytofiresalesandpecuniaryexternalitiesassourcesofmarketimperfections, without
describing their precisenature. Ihope that the holistic framework provided by this paper
forcesfuturepolicydiscussionstobeexplicitaboutwhichparticularchannelsarebeinganalyzed.
6All my normative results use the notion of constrained Pareto efficiency. The relevant policy
questioninimperfecteconomiesisnotwhetheraplannercanavoidtheimperfectionsandachievean
unconstrained first best outcome, but whether, while being subject to the same constraints as the
agentsinthedecentralizedmarket,aplannercanengineeraParetoimprovement. Myanalysisshows
that,ingeneral,whenaplannerisnotallowedtouseex-antetransfers,therearenopossiblePareto
improvementsin a fire sale: the classic Walrasian mechanism, which enters in my model through
6ThereadermaywonderwhythegenericconstrainedinefficiencyresultsshownbyGeanakoplosandPolemarchakis
(1986),Drezeetal.(1990)andGreenwaldandStiglitz(1986)donotapplyhereunlessIallowforex-antetransfers. The
crucialissueisthat,inmyformulation,thefiresaleoccursinasinglestate,andpricesinotherstatesarefixed. With
morestates,itwouldbepossibletomodifyallocationsinsuchawaythatchangesinpricesindifferentstatescompensate
4
the risk sharing channel, is still powerful when fire sales are understood as a single event. Still
withoutusingtransfers,thesenegativeresultscanbeoverturnedifborrowersandlendersresources
aremerged,ashappensinrepresentativeagentmodels. Stein(2010)orWoodford(2011)arerecent
papersthatrelyontherepresentativeagentassumption,followingtheLucas(1990)approach. Those
modelsimplicitlyallowforanex-posttransferbetweenbuyersandsellersofassetsthatcompletely
eliminatestherisksharingchannel. Ifex-antetransfersareallowed,constrainedParetoimprovements
arefeasible.
Dependingontherelationbetweenex-anteandex-postinvestmentopportunities,therisksharing
channelbyitselfcangeneratebothoverorunderinvestment. Thecollateralchannelbyitselfalways induces
overinvestment and, as long as margins are countercyclical (the empirically relevant case),
themarginchannelalsopushestowardsoverinvestment. Theeffectthatultimatelyprevailsremains
anempiricalquestion.
Thispaperrelatestothegrowingliteratureonamplificationmechanismsandfinancialconstraints
increditmarkets,shapedbytheworkofBernankeandGertler(1989)andKiyotakiandMoore(1997).
Thisliteraturehasnoticeablybeenfocusedonfindingmechanismsabletogenerateenoughbusiness cycle
amplification, leaving aside the normative implications of financial constraints. Lorenzoni
(2008),theclosestpapertothisone,isanimportantexceptionthatfocusesonthewelfareimplications
oftheseconstraints. Hispapershowsthepossibilityofexcessiveborrowingduetotheexistenceof a pecuniary
externality; in my decomposition, the externality highlighted in Lorenzoni’s paper is
exclusivelyoftherisksharingtype. MycharacterizationoftheexternalitythroughanEulerequation
greatlyimprovestheexpositionoftheresults.
7Even though amplification mechanisms can be relevant for quantitative reasons, I show that
feedbackloops,cyclesorspiralsbetweenpricesandtheamountofassetssoldareneithernecessary
norsufficienttogeneratefiresalesexternalities;inotherwords,normativeandpositiveimplicationsof
firesalesmustbedecoupled. PreviousworkbyCaballeroandKrishnamurthy(2001)andGromband
Vayanos(2002)alsoanalyzesinefficienciesrelatedtopecuniaryexternalitiesinsimilarenvironments
butwithadifferentfocus. BrunnermeierandSannikov(2010)discusspecuniaryexternalitiesrelated to the
collateral and margin channels. The literature on financial intermediation and markets, for example,
Jacklin (1987), Allen and Gale (2004) and Farhi, Golosov and Tsyvinski (2009), is also related to this
paper through the risk sharing channel. In that line of work, the possibility of spot retrading in financial
markets, together with market incompleteness, reduces risk sharing
opportunities,makingregulationdesirable. AcemogluandZilibotti(1997)isanimportantpaperin
endogenousmarketincompletenessandGaleandGottardi(2010)isarecentrelatedpaperfocused
each other in the right way to induce Pareto improvements. Even though from a purely theoretical perspective my
problemcouldseemirrelevantandnongeneric,ifwethinkthatfiresalesaresituationsthathappenonlyinaparticular
state,thenmyformulationistheappropriateone.
7Krishnamurthy(2010)providesarecentsurveyoftheliteratureonamplificationmechanisms.
5
onbankruptcy. In order to understand the actual effects of fire sales externalities, the ultimate goal
must be
8to analyze the magnitudes of these externalities empirically and guide policy intervention. With
thatgoalinmind, anddespitethesimplicityofmyformulation, Ihavetriedtokeepthispaperas
closeaspossibletoacanonicalWalrasianframeworkusedinassetpricingandmacroeconomics. My hope is
that my model will provide a basis to document empirically or evaluate the effect of these externalities
in a full fledged quantitative framework. Some recent work by Bianchi and Mendoza
(2010),Bianchi(2010)andJeanneandKorinek(2010),amongothers,triestoexploretheimplications
ofcreditconstraintsquantitatively,butthatliteraturehassofaronlyfocusedonwhatIclassifyas
collateralexternalities. Thispapershowsthatthatlineofworkhasnoconnectionwhatsoeverwith
thegenericresultsofGeanakoplosandPolemarchakis(1986)andDrezeetal.(1990). Pulvino(1998)
andBenmelechandBergman(2011)areleadingempiricalcorporatefinancepapersthatalsoanalyze the
collateral channel. Consequently, I claim that deeper quantitative work integrating all three
mechanismsisstillpending.
Section2laysoutthemodelanddiscussestheconvenienceofthevarioussimplifyingassumptions.
Section 3 solves the decentralized problem and section 4 analyzes the solution to the planner’s
problem, both in the case when ex-ante transfers are allowed and in the case when they are not.
Section 5 treats the time inconsistency problem faced by the planner. Section 6 discusses some
results and extensions and section 7 concludes. Appendix A contains proofs omitted in the main text
and appendix B describes the case with a single noncontingent asset. The online appendix
presentsadditionalmaterial.
Areadereagertoseethemainequationofthepapershouldgodirectlytoequation(27).
2 The mo del
9Themodelhastwoperiodst=
0,1 andtwostatesatt= 1,denotedasG(goodstate)andB(bad state).
TheprobabilityofthegoodstateispGandtheprobabilityofthebadstateispB= 1 -pG.
Therearetwogroupsofagentsofunitmass. Irefertothemasentrepreneursandoutsideconsumers,
respectively. The entrepreneurs represent the natural holders of capital or high valuation users
andaresubjecttocreditconstraints. Theoutsideconsumersareunconstrainedinequilibriumand represent
low valuation users or unsophisticated holders of capital. There are two types of goods:
consumptionandcapitalgoods.
8Partsofthatliteratureusetheassumptionthattheborrowingcapacityofanassetdependsonitscurrentpriceand
Thatformulationdoesnotruleoutdefaultandishardtorationalizewithanymicrofounded model.
9Aversionofthemodelwithoutuncertainty(i.e.,withasinglestate)isenoughtoshowthemostbasicresults. See
theonlineappendixforthatformulation.
6
notonitsfutureprice.
2.1 Entrepreneurs
Entrepreneursareriskaverseexpectedutilitymaximizerswithdiscountfactor1 (forsimplicity),so
theymaximizeU(C0) + pGU(C1G) + pBU(C1B). U(·) isincreasingandconcave: U >0,U<0. C0, C1Gand
C1B denote the consumption of the entrepreneurs at t= 0 and at the good and bad state att= 1
respectively. Entrepreneurs haveendowments of theconsumption good,e0, eand e1B. These agents
own a technology that transforms capital at t= 0 into AGor AB1Gunits of the consumption good at t= 1
depending on the state. k0and p100respectively denote the amount of capital chosen by the
entrepreneurs and its market price at t= 0 . The consumption good can be
transformedintocapitalonaone-to-onebasisandviceversaatt= 0,whichimpliesthatp= 1. p1Gand
p1B0denote the price of capital in each state at t= 1. Since t= 1 is the last period, the capitalheldfromt=
0 tot= 1 becomesuselessfortheentrepreneurs,sotheyareforcedtosellall theircapitalatt= 1
totheoutsideconsumersinthemarket. Theoutsideconsumersfullyabsorb
thecapitalfromtheentrepreneurs,generatingafiresale;theyaregladtoreceiveitbecausetheyare
abletotransformcapitalintoconsumptiongoodsinthesameperiod,rightbeforetheworldends.
Entrepreneurscantradeatt= 0 intwostatecontingentassets,oneforeachstate. Theseassets havet= 0
pricesq0Gandqanddeliveroneunitoftheconsumptiongoodateachrespectivestate. a0Gand a0B0Bdenote the
net position in each asset, i.e., a>0 means that the entrepreneurs are holding the statesasset (saving)
anda0s0s<0 means that they are selling it (borrowing). Outside consumers, described below, take the
opposite side of these trades. The time subscript for each endogenousvariables,0
or1,denotestheperiodinwhichthevariableisdetermined.
Entrepreneursfaceborrowingandlendingconstraintsintheamountsthattheycanbuyorsell
ofeachcontingentasset. Foreachstates= {G,B},theirassetholdingsmustsatisfythefollowing constraints:
-[ 1 -ξ(p1s)] p1sk0 =a0s = Ls
(1)
=a
0s
1sk0
Theborrowingconstraint-[ 1 -ξ(p1s)] p1sk0
1s
7
1s
1s
1sk0
10
0
is pledgeable would not modify
impliesthatthemaximumamountborrowed by the
entrepreneurs depends on the value of the capital in the future, p, corrected by a pledgeability
coefficient 1 -ξ(p), which I make explicitly price dependent. We can think of the functionξ(p) ∈[ 0,1]
asthemarginrequiredwhentheentrepreneursborrowagainstcollateralwith futurevaluep.
Withoutthemargincorrection,i.e.,assumingξ(p1s) = 0,∀p,thisconstraintcan
berobustlymicrofoundedthroughacommitmentproblemàlaHartandMoore(1994). Theoutside consumers
are only willing to lend up to the market value of collateral p1sk, since entrepreneurs
cannotcommittokeeptheirpromisesandtheoutsideconsumerscanonlyrecovertheirinvestmentby
seizingthecapital. Thistypeofborrowingconstraintrulesoutdefault. Analternativeformulation in which a
portion of the returns in the consumption good Ask0
Thisistheassumptionmade,forinstance,inthebasicversionoftheneoclassicalgrowthmodel.
anyofmyresults. NotethatnotallofAsk0couldbepledged, sincethatwouldmakethemarkets
effectivelycomplete.
A constant margin, ξ(p1s) = ξ, ∀p1s, can be easily microfounded as a fixed transaction cost that an
entrepreneur pays if he defaults. There are different theories that justify the existence of price
dependent margins ξ(p). By appealing to those, I leave the underlying contractual frictions that
determine margins open to interpretation. My formulation is meant to incorporate the ideas
developed in models with endogenous margins, such as Brunnermeier and Pedersen
(2008),Geanakoplos(2009)andShin(2010),amongothers. Value-at-riskconstraintsorinformation
asymmetries,particularlyinadverseselectionproblems,arethenaturalmicrofoundationsforthese
endogenous margins. Contagion effects or search related disruptions could also be considered as
the source of ξ(p1s1s). I assume that ξ(0) = 1, ξ(1) = 0, which implies that there are no margins when
borrowing in the good state, and that margins are countercyclical, i.e., ξ(·) <0 , as shown
empiricallybyAdrianandShin(2010). Ihaveoptedforincludingthischannelbecauseofitsrealworld
importance. A reader who feels that the margin mechanism lacks an appropriate microfoundation
cansimplyignorethischannel,i.e. setξ(p1s) = ξ,∀p1s ,andomitallmyreferencestoitthroughout
therestofthepaper. Theconceptualdistinctionbetweenrisksharing/Walrasianexternalitiesand
collateral/frictionalexternalitiesremainsunchanged,aswellastherestoftheresults. Inthatcase,
thecreditconstraintswouldread:
p
=
=Ls
1
s
1
a
s
0
Thelendingconstrain
k
s
tas
=
ξ
0
L
s
s
and-[ 1 -ξ(p
s
)] p1sk0
isanaturalmeasureofmarketincompletenessinthismodel.
1 s
s
s
8
,whichpreventsentrepreneursfromchannelingtoomuchwealth towards
a particular state, reflects commitment problems on the part of outside consumers. If we
takeliterallytheassumptionthatonlythecapitalgoodcanbeseized,weshouldexpectL= 0. A
positivevalueforLcanberationalizedifitdenotesthestatecontingentamountofstoresofvalue intheeconomy.
ThisconstraintcanbeinterpretedalongthelinesofHolmstromandTirole(1998)
orHolmstromandTirole(2011),whoemphasizetheimportanceofoutsideliquidity/storesofvalue.
Therefore,IassumethatL=0 withoutlossofgenerality.
Observethat,inagivenequilibrium,atmostonecreditconstraintforeachstatebinds. Which
constraintactuallybindsiscrucialforthepolicyimplicationsofthemodel. Note that missing markets in this
environment can be thought of extremely tight credit
constraints: if we assume for a state sthat L
= 0 and ξ(p1s) = 1 ,∀p1s
, the model represents
asituationinwhichthereisnocontingentassetforthatstate. Thisisaverynaturalwaytocombine
theinsightsoftheliteraturesinbothendogenousandexogenousincompletemarkets.
Agraphicalrepresentationofthecreditconstraintscanbeseeninfigure(1). Thedistancebetween L
Borrowing Lending
0 Ls
- [1 - ξ(p1s)] p1sk0
Figure1: Creditconstraints.
2.2 Outside
consumers
Theoutsideconsumers,whoarecrucialtosimplifytheanalysisandkeepittractable,areri
skneutral, haveadiscountfactorof1
andhavelargeendowmentsoftheconsumptiongoodineachperiodand state.
Because they trade frictionlessly in state contingent securities, their prices are
pinned down at q0s= ps. In the good state, they run a linear technology that
transforms one unit of capital
intotheconsumptiongoodandviceversa,whichimpliesthatthepriceofcapitalinthegoo
dstate is pinned down, p1G= 1 . In the bad state, the outside consumers use an
increasing and concave production technology F kC 1B , with F > 0 and F< 0.
Thus, the outside consumers solve maxkC 1BF kC 1B -p1BkC
1B ,implyinganequilibriumpricep1B= F kC 1B ,wherekistheamount
ofcapitalthattheoutsideconsumersneedtoholdinequilibrium.
Sincetheyabsorballthecapital
soldbytheentrepreneursinthebadstate,inequilibriumkC 1B= k0,sop1B= F (kC 1B).
Iassumethat p1B= F 0(·) =1 .
Theoutsideconsumerssimplyrepresenttheunsophisticatedagentsthatmusthold
theassetduringafiresale,providingadownwardslopingdemandcurve.
Figure(2)showsasimplifiedrepresentationoftheproblem.
pG
p1G = 1
=F
=1
p
0
p1B
pB
Fire Sale
(k0) = 1
Figure2: Eventtree.
2.3 Remarks ab out mo deling choices
Manyoftheassumptionsmadesofarmayseemsomewhatarbitraryandunrealistic;Iwa
nttoargue
nonethelessthattheyrepresentthesimplestformulationthatencompassestherelevant
channelsin
9
as
myanalysis:
11•Numberofperiods:
theuseofonlytwoperiodshastwoimportantimplications. First,allthe
capitalaccumulatedatt= 0 mustbesoldatt= 1 ,sinceithasnootheruselookingforward. Second, it
prevents borrowing from the future as a way to reduce a possible fire sale. Both
dimensionscaneasilybereintroducedintothemodelatthecostofcomplicatingtheanalysis, so I leave the
extension to three periods for the time inconsistency analysis in section 5. A version of the model
with three periods must include at least the two following assumptions:
a)theentrepreneursarenetsellersofcapitalinthebadstateandb)theamountofresources
thatcanberaisedinthefiresalestateislimited(i.e.,someformofslowmovingcapital;see Duffie (2010) for
a state-of-the-art review of the literature). Assumption b) guarantees that
thefiresalecannotbeavoidedbypledgingfuturereturnsandassumptiona)iscrucialtohave
therightdistributionaleffectsoffiresales. Inasimilarformulation,Lorenzoni(2008)andGai
etal.(2008)allowformultipleequilibriainthefiresalestateduetoamplificationmechanisms.
Iwanttoremarkthatalltheinefficienciestreatedinthispaperaremarginalinefficienciesand do not rely on
Pareto ranked equilibria. My formulation will make clear that those feedback mechanisms, although
realistic and capable of generating additional amplification, are not at all necessary to analyze fire
sales. The normative and positive implications of fire sales have
differentsourcesandthispapershowshowtheycanbedecoupled.
12•Numberofstates:
asimplermodelwithasinglestateandnouncertaintyisenoughtogenerate most of
the results in the paper. I present that simple version in the online appendix. The main drawback of
the single state formulation is that, in what I denominate belowas case 3 in equilibrium,
entrepreneurs would have to be net savers in equilibrium, which may be counterfactual. The
current formulation allows entrepreneurs to borrow at t= 0, by having q0Ga0G+
q0B<0,whileatthesametimearranginginsurancetowardsthefiresalestate, thatis,a0Ba0B>0.
•Riskaverseversusriskneutralagents: myformulationwithriskaverseentrepreneursmakesthe
problemnaturallyexportabletomorequantitativeenvironments. Analternativedescription, with risk
neutral entrepreneurs but endogenous hedging motives due to concave production and imperfect
financial markets, along the lines of Froot, Scharfstein and Stein (1993) or
Krishnamurthy(2003)isaplausiblealternative. Underthoseassumptions,theentrepreneurs
wouldbecomeeffectivelyriskaverse. Thispapermakesclearthat,eventhoughmarginalutility of
wealth/investment opportunities can be explicitly dependent on prices, as in, for instance,
11Forinstance,ShleiferandVishny(1992)uselongtermdebttoinducedebtoverhangandLorenzoni(2008)imposes
directlyaborrowingconstrainttosatisfythisrequirement.
12That is thesituation in which the entrepreneurs want to arrange insurance for the firesale state. In aone state
model,theentrepreneurswouldbenetsavers.
10
Gromb and Vayanos (2002) or Lorenzoni (2008), that dependence is not the direct source of
thepecuniaryexternalities. Thefactthattheoutsideconsumersareriskneutralprovidesalot
oftractabilitybypinningdownthetimezeropriceofthestatecontingentsecurities. Relaxing
thisriskneutralityassumptionwouldinvolveaccountingforriskandinsuranceconsiderations,
unnecessarilycomplicatingmyanalysis.
•Walrasian versus contracting formulation: this paper is purposely written in a Walrasian setting,
since it is the most natural environment in macroeconomics and for quantitative analysis. Unlike
Lorenzoni (2008), all trading is carried out in state contingent securities instead of discretionary
contracting. Moreover, the Walrasian formulation makes the role of ex-antetransfers,
whichcanbehiddenintothetermsofcontractingproblems, explicit. This
restrictsthespaceoftransfers,sincetheagentscannotredistributeresourcesbymodifyingthe
termsoftradeinthestatecontingentassets,anditmakesthemodeleasiertoscaleuptomore
complexenvironments.
•Specification of borrowing constraints: there is a clear asymmetry in the way the constraints
in(1)arewritten,sincetheamountthattheentrepreneurscanborrowdependsonthecapital they keep, its
price, and a margin while, the maximum feasible amount of savings is capped by an exogenous
value. The frictions discussed above that support this specification can be rationalized by assuming
that entrepreneurs carry out collateralized borrowing against the
valueoftheirassets,whileoutsideconsumershaveadiversifiedpoolofresourcesthatseparates their
commitment problem from their specific capital holdings. A crucial difference between
thispaperandthemajorityoftherecentliteratureisthattheborrowingconstraintdoesnot
dependonthecurrentpriceofcapitalp0butonitsfuturepricep1. Thealternativeassumption
thattheborrowingconstraintdependsonthecurrentpriceofcapitalisexploitedbyBianchi
andMendoza(2010),Bianchi(2010)andJeanneandKorinek(2010). Despiteitsconvenience in terms of
generating amplification, their formulation lacks an explicit microfoundation and
opensthedoortothepossibilityofdefault,whichisimplicitlyneglectedinallthosepapers.
•Linear versus concave technology: I assume a linear technology in the good state for the outside
consumers simply to shut down any price fluctuation of capital in the good state. This assumption
proxies for the fact that markets for capital are deep in good states, and
sellingassetsdoesnotgenerateanypriceimpact. Inanycase,aslongascreditconstraintsdo
notbindinthegoodstate,anykindofdemandforcapitalworkswithoutchanginganyresult.
Thefactthatthepriceforsoldcapitalisdownwardslopinginthebadstateis,however,crucial for my results.
The specific formulation with a concave technology is not necessary and it is used here only for
simplicity. An alternative but equivalent formulation could assume that a
newgroupofriskaverseagentswholacksdiversificationopportunitiesisforcedtoholdthese
11
firesoldassets,demandingalargerriskpremiumwhentheyareforcedtoholdmorecapital.
3 Decentralized problem
I first set up the full entrepreneur’s problem and subsequently analyze the properties of the
competitiveequilibriumbyincorporatingthepricesderivedfromtheoutsideconsumers’optimality
conditions. It is important to note that in the decentralized problem, each entrepreneur follows
price-taking behavior, unlike the planner in the following section. A representative entrepreneur
solves:
U(C0) + pGU(C1G) + pBU(C1B) (2)
max
k0,a0G,a0B,C0,C1G,C1B
Subjecttothefollowingconstraints13
= e0 -q0Ga0G+ G
[ -q0Ba0B] (λ0 (pGλ1G
0G
p1G
k0+ a
1G
=e
:U
0
= pBλ1B
1B0G
=
e
0G
0G
0
0B
+[
p1B
B]
k0 + a0B
(pG
)] p1Gk0
0B
)] p1Bk
(
p
(pBλ1B
0G
0B)
(9)
B
1G
G
B
0G
:q0B
0B
0
13
:p0λ 0 = p λ1B (p1B + AB) + pGλ1G (p1G + AG) + pGν0G [ 1 -ξ(p1G)] p1G + pBν0B [ 1 -ξ(p1B)] p1B
1B,C1G
1B 0B
0G
12
0,a0G
0B,C0
: theinitialbudgetconstraint(3);thestate-by-statebudget
constraints(4)and(5);andthedoublesetofcreditconstraintsforeachstatecontingentsecurity(6) to(9).
+
) (3) C1G+ A) (4) C1B+ A) (5) a= LG(pGη0G) (6) a=-[ 1 -ξ(p1G) (7)
a=LB(pBη0Bν) (8) a=-[ 1 -ξ(p1B0ν
Theoptimalityconditionsfortheentrepreneursproblemarecharacterizedby:
p
C
0
0
k
0
C1GC0
:U
+ p (C1B) = λ1B
C1B : U (C
) = λ1G
0G
(C0) = λ0
Gη
Bη
(10)
a:qλ= pGλ1G-p0Gν(11) aλ-p0B+ p(12) k0Bν0B
,a,C(13) Acompetitiveequilibriuminthiscontextisasetofallocationskand
pricesp,p1G,p,q,qsuchthatbothentrepreneursandoutsideconsumersbehaveoptimally,
givenprices,andmarketsclear.
The parentheses represent the Lagrange multipliers used for each constraint. The multipliers are defined to be
nonnegative.
Given the assumptions made about the behavior of outside consumers and the available
technologies, the equilibrium prices are pinned down to p0= 1 , p1G= 1 , q0B= pB, q0G= pG
1G
0G(k0). Thefirstorderconditionswithrespecttoa0G,a0B
andp1B = F as:
14canthenbewrittenin
equilibrium
= λ λ1s -η
1s+
ν0G s+ ν0sλ[0 1 -ξ(p1s)] p1s
+ ν0B
0
andk0
(16
)
λ0 -η
+ As)
0
0λ0
1s
ν0s
λ0
14
λ0 1 = E(14) λ= λ1B
0B(pPresentValue
CollateralValue
Condition(16)isastandardEulerequationforthecapitalgoo
d. Avariationalargumentallows ustoderiveit. λrepresentsthemarginalcostatt= 0
ofanadditionalunitofcapital(ifp = 1,
thiswouldbep),andinequilibriumitmustbeequaltotheexpectedmarginalbenefit,(p1s0)
unit
sidetheexpectation,λ0(p1s+ A),representstheclassicalassetpricingcondition and the additional one,[ 1
sof-ξ(p1s)] p1s, represents the collateral value of capital when relaxing the borrowing constraint.
wea
Conditions (14) and (15) are equivalent Euler equations for the state contingentassets.
lthv Equation (16) makes clear that constrained agents in decentralized markets value more assets
aluthat relax their borrowing constraints. Note how the dividend generated by the investment, A,
eda
doesnotcarryanextrapremiumbecauseitisnotcollateralizable. Asimilarideahasrecentlybeen
tλ1s
emphasizedbyGarleanuandPedersen(2010)inaCAPMenvironment. Moreover,ifwehaddifferent
)] tradedassets,andtherewereanex-anteproductionstage,thoseassetswithhighercollateralvalue
p1swouldbecreatedtothedetrimentofthosethatdonotincreaseborrowingcapacity.
new I now proceed to characterize the 3 different cases that can arise in equilibrium depending on
unit
which credit constraints bind. Note that the key variables that determine which borrowing
s of
constraints actually bindarethemarginalutilityofwealth in eachperiod/state,λ0andλ1s; these
borr
areendogenousobjectsthatdependonendowments,borrowingcapacity,availabletechnologiesand
owi
assetprices. Rememberthat,atthemargin,anyentrepreneurmustbeindifferentbetweenconsuming
ngandinvesting.
cap
acit IhavepurposelyassignedaleadingroletotheLagrangemultipliers(λ,η,ν)inthepaper,since
y theymakemyresultswidelyapplicable. Ifwehadchosenamoregeneralmodel,e.g.,withEpsteinByassumingasufficientlywell-behavedF (·),auniquesolutionisguaranteed. Evenifthereweremultipleequilibria,
valu
edallofthemmustsatisfy(14)to(16)andmyanalysiswouldbelocallyvalid.
at 13
ν,
ina
dditi
ont
oth
eval
uet
hatt
hen
ewu
nito
fca
pital
gen
erat
esw
hen
rela
xing
the
borr
owi
ng
con
stra
int,
[1
-ξ(p
λ1s.
The
first
ter
min
Zinpreferencesorhabitformation,wewouldonlyneedtolookatthecorrespondingvaluesforthe
marginalutilityofwealthtoextrapolatetheresultsofthispaper.
Inordertosimplifytheanalysis,Imakethefollowingassumption.
Assumption1. a)Inequilibrium,themarginalutilityinthebadstateisstrictlylargerthaninthe good state:
λ1B>λ1G. b) In equilibrium, the marginal utility at time t= 0 is strictly larger than
themarginalutilityinthegoodstate: λ150>λ1G. Dependingontherestofparameters,adjustingthe
endowmentsisenoughtosatisfythisassumption.
Assumption 1.a) is innocuous and it just makes the bad state worse than the good state.
Assumption1.b)forcestheentrepreneurtoalwayshittheirborrowingconstraintstowardsthegood
stateandsimplyreducesthenumberofuninterestingcasestoanalyzeinequilibrium.
Combining the first order conditions and making use of assumption 1, an equilibrium can be
characterizedinthefollowingway:
Proposition 1. Depending on parameter values, there are three different cases that arise in
equilibrium:
•Case 1: Entrepreneurs hit the borrowing constraint towards both good and bad states: λ0> λ1B>
λ1G. Their asset positions satisfy: a0G= -[ 1 -ξ(p1G)] p1Gk0and a1B)] p1Bk0.0B= -[ 1 -ξ(p
•Case2: Entrepreneurshittheborrowingconstraintonlytowardsthegoodstatebutnotthe
savinglimitortheborrowingconstrainttowardsthebadstate: λ1B= λ. Theirasset positionssatisfy: a0G= -[
1 -ξ(p1G)] p1Gk0anda0B∈ -[ 1 -ξ(p1B)] p0>λ1Bk01G, LB .•Case 3: Entrepreneurs hit the borrowing
constraint towards the good state and the saving limit towards the bad state: λ1B> λ0> λ1G. Their
asset positions satisfy: a1G)] p1Gk0anda0B= LB.0G= -[ 1 -ξ(p
Controllingforriskaversionanddependingonhowattractiveinvestmentopportunitiesareatt= 0, the
entrepreneurs follow a pecking order in borrowing: if the marginal utility of wealth is really
largeex-ante,theywillborrowasmuchastheycan,pledgingincomeinbothgoodandbadstates. If investment
opportunities are not so attractive ex-ante, i.e., λ0<λ1B, they will start shifting
resourcestowardsthebadstate,inwhichmarginalutilityofwealthishigher. Theintuitionfortwo
statesextendsnaturallytoamodelwithseveralstates: inthatcase,wewouldfindcutoffssuchthat
anentrepreneurborrowsthemaximumpossibleamountagainststateswithlowestmarginalutility,
15Dependingontherestoftheparametersofthemodel,itmaybenecessarytosetanegativeendowmenttoachieve
thedesiredvalueforthemarginalutilityofwealthineachstate. Thiscanbeunderstoodasahabitorasubsistence level.
14
stays inside his credit constraints for intermediate states and sends all the savings towards those
stateswiththelargestmarginalutilities.
1 andλ1B
T
6 0
h
e
r
e
l
a
t
i
o
n
b
e
t
w
e
e
n
λ
0
G
=0
ν
ν
λ0
andλB,thesetofbindingconstraintsdiffers.
= 0 ν0B>0 η0G0B
ν
willbecruciallaterforthewelfareimplicationsoffiresales. If
entrepreneursborrowbecausetheyreallyvaluewealthatt= 0,reducingthesizeofthefiresaleand
compensatingoutsideconsumersex-antewillnotbeoptimalfromaplanner’sperspective. Notealso that in
case 2 entrepreneurs could be either borrowing or saving in equilibrium, but that will not
haveanywelfareimplications.
Figure (3) provides graphical intuition for proposition 1. Depending on which regionλ
falls
withrespecttoλ
Case 2
>0 η0G
ν0G = 0 η0B
0B
=0ν
λ1G
0G >0 η0G
0B
= 0 η0B
Case 3
=0
>0
0G
>0 η
Case 1
λ1B
=0
Figure3: Threetypesofequilibriumcasesdependingthesetofbindingconstraints.
Without assumption 1, three more cases could arise in equilibrium: λ0 = λ1B = λ1G
an ulere al1 = E [ 1B
da quati (p1s
n onfor
1B
E capit
> λ0
= λ1G
= = λ1B
λ1
0
G
, in which no
constraint binds and markets become effectively complete; λ>λ, in which the entrepreneurs hit the
saving limit towards the bad state but not the saving limit or the borrowing
constrainttowardsthegoodsate;andλ1B>λ0,inwhichthesavingslimitisbindingtowards
bothgoodandbadstate. Iwon’tanalyzesincethemhere,sincetheyareempiricallylessrelevant.
Beforesolvingtheplanner’sproblem,itisinstructivetocharacterizethefirstbestallocation.
Whentherearenoborrowingconstraints,aninteriorequilibriumischaracterizedbyλ
+ As)]. Inthatcase,theoutsideconsumersfullyinsure theentrepreneurs,
whocanguaranteeconstantconsumptionsacrossalldatesandstates, whilethe
Eulerequationpreventsarbitragebetweenrisk-freelendingandcapitalinvestment.
andλB
arefixedwhileλ0
15
16MygraphicalrepresentationseemstoimplythatλG
varies;ofcourseallthreemultipliers arejointlydeterminedinequilibrium.
4 Planner’s problem
We know that in the presence of credit constraints, the decentralized equilibrium is trivially not
Paretoefficient. Areasonableyardsticktoevaluatepublicinterventioninsteadisthatofconstrained Pareto
efficiency, this is, can a planner improve over the decentralized outcome by using the same
instruments as the market? Using this approach, I solve two different versions of a constrained
planner’sproblem: inthefirst,theplannerisonlyallowedtomodifytheallocationschosenbythe
agents,whileinthesecond,theplannercanalsodesignex-antetransfersatt= 0 betweenthetwo groups of
agents. I make the distinction between these two cases because giving the planner the opportunity to
redistribute assets ex-ante may be considered by some as giving the planner more
resourcesthanthedecentralizedmarket. Notethatwhensolvingtheconstrainedefficientproblem, the
planner, unlike the agents in the decentralized market, internalizes the effect that changes in
allocationshaveonprices.
Note that this paper always uses the notion of ex-post constrained Pareto efficiency, that is, a
government intervention is only considered to be a Pareto improvement when both entrepreneurs and
outside consumers are (weakly) better off. This is the most demanding welfare criterion. Other
authors, for example, Hart and Zingales (2011) in a related environment, implicitly use the less
restrictive notion of ex-ante constrained Pareto inefficiency. Under that criterion, a Pareto
improvement can be reached by maximizing the sum of the utilities of both types of agents. That
criterion is justified by arguing that, ex-ante, there is an equal chance for an agent to be an
entrepreneuroranoutsideconsumer. Thatviewoftheworldimplicitlyintroducesex-anteinsurance
opportunities between entrepreneurs and outside consumers. Moreover, the ex-ante formulation is
implicitlygrantingParetoweightstoeachsetofagents;Ibelievethatthattaskshouldnotbepart
oftheeconomicanalysisandshouldsimplybelefttothejudgmentofthepolicymaker.
Figure4representsthisargumentgraphically.
16
1 UC
2
UC
+
1 UE
2
=U
U
E
Figure4:
Ex-postversusex-anteconstrainedPa
retoimprovements.
0
Ex-p ost constrained Pareto improvement
◦
4
5
Comp etitive equilibrium
Ex-ante constrained Pareto improvement
Constrained planner’s utility p
ossibility frontie r
Decentralized market utility p ossibility frontie r
denotestheutilityoftheoutsideconsumersandUE
4.1 Transfers at t = 0 not allowe d
U
C
Inthiscase,theplannermaximizesthe
utilityoftheentrepreneurssubjectthea
dditionalconstraint VC P=VC M, where
VC Pand Vdenote the indirect utility
denotestheutilityoftheentrepreneurs.
of the outside consumers under the
planner’s dictate (P) and the
decentralized market outcome (M)
respectively. The utility of the
outsideconsumers,VC jC MCforj=
{M,P},isgivenbythesumoftheirlargee
ndowmentsateachstate eC 0=e+ eC
G+ eC
Bandtheexpectedprofitsmadeintheb
adstateF(k0) -F (k0) k0. Theplanner
acknowledges that profits for
outside consumers are zero in the
good state and that kmust be
equaltokC B0inequilibrium.
Giventheriskneutralityofoutsidecons
umers,anytradingincontingent
assetsdoesnotaffecttheirutility.
Therefore,theindirectutilityforoutside
consumersVC j(k) can
bewrittenasafunctionofonlyk0:0
VC (k0) = eC + pB F(k0) -F
(k0) k0
j
IdenotebyθtheLagrangemultiplierfortheconstraintVC P
17
. Inanequivalentformulation
oftheplanner’sprobleminwhichaweig
htedsumofutilitiesismaximized,θisse
texogenouslyand
controlstheweightassignedtotheouts
ideconsumers.
Imakereferencetothisinterpretationof
the
problemwhendiscussingmyresults.
Theplanner’sproblemcanthenbestat
edas:
=VC
M
U(C0) + pBU(C1B) + pGU(C1G) + θ
max
C
+ pB F(k0) -F
(k0) k0 -VC M
e
The planner faces the same set of constraints that the entrepreneurs, with the caveat that the
effectofallocationonpricesisdirectlytakenintoaccount.
=e
-p a0G G-pBa] (λ0 (pGλ1G
k0+ a 0G
k0,a0G,a
0B,C0,C1G,C1B
1G
=e
0G
=e
00)
+ F
0G
1
B 0B
+A
(pBλ1B
(pBν0B) (23)
k0 + a0B
G
(k
)
k
0
0B
17:
0BG00
C + k ) (17) C1G+ [ 1 + A) (18) C1B (kB) (19) a= LG(pGη0G) (20) a=-[ 1 -ξ(1)] k(p) (21) a= LB(pBη0Bν0G)
0
(22) a=- 1 -ξ F (k0) F0The solution to the planner’s problem without transfers is characterized by
the following optimalityconditions
(C0) = λ0
C1G : U (C1B) = λ1B
C1B : U (C1G) = λ1G
+1 p
0G
= λ1G -η0G + ν0G
Risksharing Externality
(26)
+ν
0B
= λ1B -η0B
F k0
0B
Margin + (1 -ξ)
-ξ F
Collater
al
0
0s
λ
+ As
17
+
ν0
0
B
1
8
C : (24) a:λ0(25) a:λ0
0 U
1=
(p1
)+ν
s
E λ1sλ0
0
[ 1 -ξ(p1s)] p1s
λ0
B
(λ
1B-θ)
(27)
Conditions(24),(25)and(26)areidenticaltotheonesarisinginthedecentralizedproblem. The
terms that differentiate the Euler condition for k(27) from its decentralized counterpart (13) are
thesourcesoftheexternalities.
Expression (27) clearly identifies the three channels through which fire sales can matter. As in the
decentralized problem (see equation (16)), the first two terms in the right hand side of (27)
representtheclassicalassetpricingpresentvalueandthefactthattheagentstakeintoconsideration
theadditionalvalueofkrelaxingtheborrowingconstraint.
Tosimplifythenotation,IoftenwriteF
,F ,ξandξ
forF
(k0),F
(k0),ξ(F
(k0)) andξ
(F (k0)).
The externality term in the planner’s problem has three parts, which I call the risk sharing,
marginandcollateralchannels. TherisksharingchannelcomesfromtheclassicWalrasianfunction of
prices; a reduction (increase) in prices generates redistribution from net sellers (buyers) to net
buyers(sellers)ofanasset. Theplannerknowsthateachmarginalunitofcapitalreducesthewealth
oftheentrepreneursbyF k0inthebadstate,andthattheyvaluethisincreaseinwealthλ. On
theotherhand,theoutsideconsumersmakeacapitalgainwiththischange,andtheirgaininutility isθF k0.
Ifθisanexogenousweightanditsvalueislowerthanλ1B1B, theplanneroptsforalower
amountofcapitalk0,reducingthesizeofthefiresaleinthebadstate. Whenθisendogenous,Ishow
inproposition2thatitadjuststomaketheexternalitytermequaltozero. Thecollateralchannel captures the
externality that an agent causes to other agents when he doesn’t internalize the fact
thatanadditionalunitofcapitaldepressesthedebtcapacityofalltheotheragents. Themargin
channelaccountsforthefactthatagentsdonotinternalizethatchangesinpriceschangethedebt
capacitydirectlythroughatighteningofmargins.
Whichconstraintsbindinequilibriumandhowthethreechannelsinteractdeterminetheoptimal
k0fortheplanner. Therisksharingtermoftheexternalityλ-θhasapriorianambiguoussign, but the second
part (collateral and margin)ν0B(-ξ F 1B18+ (1 -ξ)) is unambiguously nonnegative. When the whole term
that corresponds to the three channels is negative (positive), the marginal social returns of k0are
smaller than those assessed privately and the planner wants to choose a
smaller(larger)amountofcapitalthaninthedecentralizedequilibrium. NotethatF19<0. Figure
(5)showsgraphicallyhowk0 isdeterminedinequilibrium. Theupwardslopinglinerepresentsλ0
(thelefthandsideinequation(27))andthedownwardslopingonesarerespectivelytheprivateand
socialmarginalbenefitofanextraunitofk0(therighthandsideinequation(27)).
1918Theseresultsfollowsdirectlyfromtheassumptionmadeaboutthedeterminationofmargins:
representation holds under modest conditions about F and ξ
19
. See online appendix for more details about uniqueness.
ξ∈[ 0,1] andξThis
<0.
0
k
RHSM ar k et
Mg. U
0
M ar k et 0
LHS(λ0)
k
RHSP l anner
kP l anner 0
Figure5: Overinvestmentcase.
Agraphicalanalysisoftheexternalities.
Thefirstimportantpolicyimplicationofthispaperiscapturedinproposition2:
Proposition2. Ifex-antetransfersarenotallowed,theplannerfindsno Pareto
improvements .
Proof. SeeAppendixA.
Themainintuitionintheproofisthefollowing:
inordertoimprovethewelfareoftheoutside consumers, the amount of capital
chosen by the planner must be larger than the one in the decentralized
market. But this imposes an extra constraint in kfor the entrepreneurs,
reducing their welfare unless k00is the same as in the decentralized market.
In general, if there is only one
stateinwhichweareconcernedaboutafiresale,orinthecaseinwhichfiresalesinduc
etransfers in the same direction between agents, the Walrasian effect of
prices, which I call the risk sharing
channel,issufficienttopreventtheexistenceofconstrainedParetoimprovements.
Because this model assumes that the amount of equity is fixed (the initial
endowment), any
increaseink0canbeimmediatelytracedtocapitalrequirementsorleverageratios.
Thisproposition impliesthatonlysetting capitalrequirements,
withoutcreatingex-antetransfers, isnotenough to createParetoimprovements.
Proposition2differsfromtherecentresultsinStein(2010),Bianchi(2010),Bianc
hiandMendoza (2010) and Woodford (2011), in which a reduction of the
amount of capital is enough to induce a Pareto improvement. In those
models, a representative agent formulation merges borrowers and lenders to
allow for implicit ex-post transfers. In other words, the authors neglect the
Walrasian effectofpricesandonlyfocusonthecollateralchannel.
Thecrucialassumptioninthosemodelsis
20
that the transfers between borrowers and lenders are welfare neutral, since the marginal
utility of bothgroupsisidentical. Inmyformulation,thetermλ,becomesexactlyzerowhenθ= 1
and λC 1B= λU 1BC 1B-θλU 1B. In that situation, overinvestment in the decentralized market
is the only possible outcome. In general, any model that collapses to a representative
agent formulation will face the sameconcern.
An interesting alternative environment is the one in which we set θ= 0 exogenously,
i.e., the planner just wants to maximize the utility of the entrepreneurs. In that case, the
new solution is alwaystoreducecapitalandimplementakplanner 0lessthankmarket 0.
Thisprovidesaclearrationalefor
aplannerthatcaresmoreaboutentrepreneursthanoutsideconsumerstocurtailex-anteinvestm
ent andsettightcapitalrequirements.
Some may argue that the representative agent framework, which shuts down the risk
sharing channel, is the appropriate one through which to study aggregate models. Note
that the same Walrasian mechanism that usually equates marginal rates of substitution to
marginal rates of transformation is the one that creates the externality here. Arbitrarily
removing that implication would be completely at odds with assuming that competitive
markets determine the rest of the variablesinthemodel.
4.2 Transfers at t = 0 allowed
If the planner can redistribute resources at t= 0 , the solution to the constrained
problem varies considerably.
Inthisnewsetup,theplannerisallowedtodesignatransferT20betweenentrepreneurs
andoutsideconsumersatt= 0;T00>0
impliesapositivetransferfromentrepreneurstoconsumers andT0<0 viceversa.
Withthisnewpolicyinstrument,theproblemtosolvebecomes:
Subjecttothefollowingbudgetconstraintatt= 0:
U(C0) + pBU(C1B) + pGU(C1G) + θ
max
k0,a0G,a0B,C0,C1G,C1B,T0
C + k0 + T0 = e0 -pGa0G -pBa0B
(λ0)
0
Identicalbudgetconstraintsandborrowingconstraintstothosestatedin (18), (19), (20),
(21), (22)and(23)alsoneedtohold.
Therefore, theconditionsforoptimalityareidenticaltothosein(24), (25), (26)and(27), with
theadditionof:
20TheconcavityofU,theexistenceoftwo-sidedcreditconstraintsandthefactthatentrepreneurs’endowmentsare
finiteimplythatT0isbounded. Inasituationwithrisk-neutralityonbothsidesorwithoutborrowingconstraints,it
maybenecessarytointroduceconstraintsintheamountofT0tohaveaboundedsolution.
21
U
e
+ T0 + pB F
T : λ0 = θ (28)
0
Thisconditionpinsdownthevalueofθ. Intuitively,theplannerequalizesthebenefitofshifting
oneunitofconsumptionfromentrepreneurstoconsumers,representedbyθ,tothecostintermsof
utilityforentrepreneurs,λ0. Bycombining (28)withtheEulerequationforcapital,
wecanfindanequivalentexpressionto
market 0Externality
(p1 λ1B + As) + ν0sλ0[ 1 -ξ(p1s)] p1s + pB
(27):
λ0
s
Margin + (1 -ξ)
F k0
>kplanner 0
λE 1BλE 0
1=
E λ1sλ0
λC 0
λC 1B
p
l
a
n
n
e
r
will be positive when k
0
;
22
T
0
Riskshari
ng
Proof. SeeAppendixA.
0
forentrepreneurs,
-
0B
Proposition3. Whenex-antetransfersareallowed,aslongas
λ1Bλ0-ξ F
0
Collateral
λ1B
λ0
(29)
Thesignoftherisksharingtermdependsonthedifferencebetweentheratioofmarginalutilities
,andoutsideconsumers,equalto1 becauseoftheirriskneutrality. AsIdiscuss
withmoredetailinsection6,inthecasewithriskaverseoutsideconsumers,therisksharingterm wouldlooklike.
Witharepresentativeagentformulation,marginalutilitiesaretriviallyequalandtherisksharing
term disappears. In that case, absent any binding price dependent credit constraint, the planner’s
problemhasthesamesolutionasthedecentralizedmarket. Ifmarketsarecomplete,theintertemporal rates of
substitution across states are equalized, ν= 0 and the whole externality term vanishes.
ThiscomesfromtheFirstWelfareTheoremwithcompletemarkets. ThesignofthetransferTis easily
determined after finding kand negative otherwise.
= 1 (i.e.,constraintsbindand
marketsarenotlocallycomplete)andξ>0 ,theplannercanimproveoverthedecentralizedoutcome.
The intuition for this proposition is the following. In case 1, entrepreneurs really value having
resourcesatt= 0 ,sotheyarebetteroffbyenjoyingmorefundsex-ante. Therefore,theplanneris
temptedtocreateapositivetransferfromoutsideconsumerstoentrepreneursex-anteandinducea
largerfiresaleinthebadstatetocompensatetheoutsideconsumers. Inotherwords,entrepreneurs
increase their capital kat t= 0, when they value wealth the most, and then compensate outside
consumers with an increased fire sale in the bad state. Nonetheless, inducing a large fire sale in
thebadstatereducesthedebtcapacityoftheentrepreneurstowardsthatstate,creatingatradeoff
for the planner through the collateral/margin mechanism. In general, the strength of both effects
willdeterminewhetherover-orunderinvestmentwithrespecttotheconstrainedoptimumarisesin equilibrium.
Undertheparticularassumptionsofthispaper,becauseν0B= λ,theexternality termcollapsesto λ1Bλ0-1 [ ξ+
ξ F ]; hence,unless|ξ 1B-λ021|islargeinabsolutevalue,underinvestment would arise. Note also that if
margins are set to zero, the risk sharing and the collateral channel
exactlycompensate,renderingaconstrainedefficientdecentralizedoutcome;thisisaknife-edgecase
thatneednotbetrueinageneralversionwithmoreperiodsordifferenttradedassets.
Finally,aplausibleadditionalconstraintontheplannercouldbeT0=0 ,thisis,ex-antesubsidies to
entrepreneurs are not allowed. In that case, the Pareto improvements will exist only in case of
overinvestment.
In case 2, entrepreneurs are locally unconstrained, the planner cannot improve over the
decentralizedoutcomeandclassicalParetoefficiencyintuitionapplies.
In case 3, ν0B= 0 , so only the risk sharing channel is active. The planner optimally reduces the
amount of capital chosen ex-ante by entrepreneurs, dampening the effect of the fire sale and
makingentrepreneursbetteroffinthebadstate, wheretheyvaluewealththemost. Tomakethis modification
acceptable for outside consumers, a positive transfer from entrepreneurs to outside
consumersisdesignedatt= 0. Notethatinthissituation,theentrepreneurs’borrowingconstraint
isnotbindinginthebadstate: entrepreneursrecognizethattheyvaluewealthalotinthebadstate
andtheyactivelyarrangeinsurancetowardsthatstate.
5 Time inconsistency
22Timeinconsistencyproblemsarecommoninmacroeconomicpolicyenvironments.
Tothebestof my
knowledge, this paper provides a novel treatment of the time inconsistency problem faced by a
planner in an environment with fire sales externalities. I only solve for the constrained efficient
allocation,withoutdiscussingitsexplicitimplementationintheformofcapitalrequirementsortaxes.
However,inasimpleenvironmentlikethis,thoseinterventionsarestraightforwardtocharacterize.
Inordertoproceedwiththisnewanalysis,Imustaddathirdperiodtothemodel. Assumethat t=
0,1,2,withtheneweventtreerepresentingthemarginalutilityofwealthforentrepreneursgiven byfigure6.
21SeeappendixBforanexample.
22TheclassicreferencesfortimeinconsistencyinmonetarypolicyareKydlandandPrescott(1977)andBarroand
Gordon (1983).
Chari and Kehoe (1999) and Golosov, Tsyvinski and Werning (2007) discuss the time inconsistency problem in the Ramsey
and Mirrlees traditions. Chari and Kehoe (2009) have recently studied time inconsistency
problemsinanenvironmentwithbymoralhazard. Theusualmoralhazardstoryplaysnoroleinmymodel.
23
λ1G
Fire Sale
λ2G
pG
λ0
pB
1B
λ2B
Figure6: Threeperiodversion;valuesofmarginalutilityofwealth.
Iassumethatinperiodt=
2,capitalcanbetransformedagainintotheconsumptiongoodona
one-to-onebasis,p2= 1.
Nowoutsideconsumersmustabsorbtheamountk0-k1B23ofthecapitalgood
inthebadstate.
Irestricttheanalysistothecaseinwhichentrepreneursarenetsellersofcapital
inthebadstate,i.e.,k>0;thisassumptionimpliesthatthepriceofcapitalinthebadstate
isp1B= F (k0-k1B0-k1B) =1 ,withF(0) = 1.
Toeasetheexposition,Ieliminatethepricedependence ofmarginsbysettingξ(p1s ) =
ξ,∀p1sandIdirectlyanalyzethesolutionfortheconstrainedplanner. As shown in the
previous section, in order to allow for feasible Pareto improvements, the planner
musthavetheopportunitytouseanex-antetransferT0.
Therestoftheoriginalassumptionsstill holdandthenotationisextendednaturally.
Note that this formulation embeds some notion of within period amplification:
the more the
entrepreneursengageinfiresalesinthebadstatebyincreasingk0-k1B,thedeeperthefir
esalewill
be,reducingtheirnetworthandmakingthemmorewillingtosellevenmorecapitalinorde
rtokeep consumptionhigh.
Ianalyzeamplificationanditsrelationtowelfarewithmoredetailinsection6.
Underthesenewassumptions,theproblemsolvedbytheconstrainedplanneratt= 0
is
U(C0) + E [ U(C1s) + U(C)] +θ Ce+ T0+
-k1]) -VC M
pB(F(k0-k1) -F (k02s-k1) [ k0
24
max
a0G,a0B,a1G,a,
C0,C1G,C1B,C2G1B,C,
k0,k1B,k1B,T0,2B
Withbudgetconstraints:
23
1B
Forthistohappeninequilibrium,theendowmentoftheentrepreneursmustbelowinthisstate(perhapsne
gative) aswellastheirproductivityA.
(30
)
Theymustalsohavesmallborrowingcapacityinthatparticularstate.
1G
) C+ k1G+ [ 1 + A
Gλ1G
1B) k1B = e1B + F (k0 ] k1G-k1B) + A1B ]
1B
k1B+ a1B
= e2B
2B
= e2G
1G
2G
) C+ F (k0-k) C+ [ 1 + A2B) C+ [ 1 + A2G+ a
Andcreditconstraints:
(pBλ1B
k0 + a0B -a1B
(p Bλ2B
(pGλ2G)
C
0
) a0B=L0B(pBη0B) a0B=- 1 -ξ F (k00G-k1G=L1G(pGη1G) a1G=- 1 -ξ k1G(p) 0 (p ν
a1B=L1B(pBη1B) a1B=- 1
-ξTheconstrainedplanneroptimalityconditionsare: k1BG(pBν1Gν1B
0 :
(C0) = λ0 C1 : (C1) = λ1s,fors= G,B C2s
: (C2) = λ2s,fors= G,B
s
U
U
U
)a
C
)
B 0B
=L
(pGη0G) a0G
=
1
k
(pGν
a 0G
0
-ξ
1B) k
0
G
k :λ0 = DecentralizedMarketE [ λ1s(p1s+
0
A1s)] + E [ ν0sp1s
] +pB
Externality
k0-k1B]Risksharing + ν
F (k0-k1B)
= λ2G (1 + A2G) + 1 ξ
ν1G
(3
1)
k:
λ
k1B :F (k0 -k1B) λ1B λ= DecentralizedMarket2B(1 + A2B) +
Externality 1 -ξ ν1B
-k1B) (k0 Fk 0B
1G
0B
(λ1B-θ) [
1 -ξ k0Collateral
1G
0
-
(λ1B-θ) [ k0-k1B]Risksharing + ν
a0 : λ0 = λ1G
+ν
a
: λ0 = λ1B
G
0G
0B
0B
-η0G
-η
a1 : λ1G = λ2G -η1G + ν1G a1B : λ1B = λ2B -η1B + ν1B
G
T : λ0 = θ
0
25
1 -ξ
0B
(32
)
Collateral
+ν
( (λ0
p -a
0G1G-pBa0B]
k0+ a
-pGa0G = e0 + T0 +
= e 1G 1G
Thefactthattheexternalitytermenterssymmetricallybutwithoppositesignin(31)and(32) clearly shows
that the constrained planner’s optimal policy entails adjustments both in kand in k1B.
Notealsothattherisksharingtermdependsontheamountoffiresoldcapitalk0-k0butthe
collateralandmargintermsareonlyafunctionoftheamountofthecapitalk01Bheldbetweent= 0 andt= 1.
Insteadofdiscussingallthepossibleequilibriumcombinations,Ifocusonthemostrelevantcases
bymakingthefollowingassumption.
Assumption 2. a) In equilibrium, the marginal utility of wealth in the bad state is strictly larger thanatt=
2: λ1B>λ2B. b)Themarginalutilityatt= 0 islargerthanthemarginalutilityinthe goodstateatperiodst= 1 andt=
2: λ0>λ1G>λ2G.
Assumption2.a)impliesthatentrepreneursvaluewealthmoreatt= 1 inthebadstatethanin
therecoveryatt= 2. Thisassumptioncanonlyariseinequilibriumiftheentrepreneursborrowing constraintatt=
1 inthebadstatebinds. Assumption2.b)simplystatesthattheentrepreneursare
alwayswillingtohittheirborrowingconstraintsinthegoodstateatperiodst= 1 andt= 2. This
lastassumptionplaysnoroleintheresultsandcanbeeasilymodified.
I consider two particular cases. These are analogous to cases 1 and 3 in proposition 1: a) λ1B, in
this case entrepreneurs value wealth the most in the bad state at t = 1 , so they
areactivelyarranginginsurancetowardsthatstateandtheborrowingconstraintdoesnotbind;b) λ0>
λ>λ1B0,inthiscaseentrepreneurshaveveryappealinginvestmentopportunitiesex-ante,sothey
hittheirborrowingconstrainttowardsthebadstate.
Sincetheplannerdoesnotinterveneinthegoodstate,thereisnoreasontochangepolicythere. Let’s set up
instead the new problem for the constrained planner at t= 1 in the bad state. Note that k0is no longer
a control variable and that VC M,1Bdenotes the utility achieved by the outside
consumersunderthedecentralizedmarketconditionalonbeinginthebadstateatt= 1.
0B even when a0Bwillchangethemarketvalueofthecolla
Observethatchangesink1
U(C1B) + U(C2B) + θ
e + eC + F(k0 -k1B) -F
C
1
andguaranteepositiveconsumption.
(k0 -k1B) [ k0 -k1B ] -VC M,1B
2
max
k1B,a1B,C1B,C2B
0B,generating
the possibility that entrepreneurs decide not to repay. To simplify the analysis, I have
opted for assuming that the entrepreneurs honor their inherited debta
under the new allocation. I additionally impose the restriction that entrepreneurs always have
sufficientresourcestorepaya0B
26
2B
1B
1B
1B
= e1B + F (k0 -k1B) + A1B ]
= e2B (η k k1B+ a1B
(ν1B)
=-=L1B
2B
k0 2B + a0B -a1B
1B
1
B
(λ
+
( ) C+ [ 1 + A2B) a) a1 -ξ1B 1B
1
Theoptimalityconditionsfortheconstrainedplanneratt=
k k
1
B
0
1
F
(λ1B
(33)
C
inthebadstatearegivenby:
B
)
k
1
B
:U C
(C1B) = λ1B
k1B :F (k0 -k1B) λ1B
C2B : (C ) = λ2B
U
= λ2B (1 + A2B) + 1 -ξ
a1B : λ1B = λ2B -η
+ν
ν
1B
(λ1BExternality-θ) [ k0-k1B]
F (k0-k1B)
Itis obviousthat thetermcorrespondingtotheborrowingconstraint disappears fromthenew Euler
equation for capital for the planner. The intuition is clear: higher prices during a fire sale
increaseex-anteborrowingcapacity,sincetheyrelaxborrowingconstraintsbyincreasingthevalueof
collateralorrelaxingmargins. Inordertokeephighpricesduringthefiresale,theplannercommits ex-ante to
an ex-post inefficiently high level ofk. Once the fire sale state arrives, a planner has
noincentivetointroduceadistortioninthespotmarket: thisresultsinatimeinconsistencypolicy.
Notethatthiskindoftimeinconsistencyonlyariseswhentheborrowingconstraintsbinds,thatis
whenλ0>λ1B1B. What about the risk-sharing term? We have to consider three different cases
depending on
whetherweallowtheplannertousevarioustransfers:
1. Nolumpsumtransfers: Inthiscase,ananalogousresulttoproposition2forcesθ= λ,
whichmakestheexternalitytermequaltozeroandpreventstheplannerfromfindingaPareto improvement.
Theplannerdecidestosetanewk1B1Bdifferentfromtheonesetintheex-ante problem.
Thisvalueofkcorrespondstothedecentralizedallocationconditionalonagiven
choiceofk0,C0,a0G,a0B1BandT0.
2. A new lump sum transfer T1B: allowing for this transfer would also imply θ= λ1B, so the planner
would implement the same allocation as that without any transfer. Intuitively,
allowingforatransferinthefiresalestateisnothelpful,sincetheplannerwasalreadyableto
redistributeresourcesinthatstatethroughpricechanges.
3. Anex-postlumpsumtransferT2B: thistransferwouldimplyθ= λ2B. Withthispolicy
instrument,itispossiblefortheplannertofindaParetoimprovementfromat= 1 perspective.
Ifλ1B>λ0>λ2Btheplannerwoulddecidetokeepahigherk1thantheoneoriginallychosen
27
andthenintroducealargeex-postcompensationtotheoutsideconsumers. Ifλ,
theplannerwouldchooseak11B>λ2B>λ0smallerthantheonechosenatt= 0 butlargerthantheinduced by
the decentralized market at t= 1. When λ, the planner always chooses
ex-postalevelofk10>λ1Blargerthantheoptimaloneex-ante.>λ2B
Hence,forallthreetypesoftransfers,theex-anteconstrainedefficientpolicyisalsotimeinconsistent
duetorisksharingconsiderations. Theintuitionfortherisksharingmechanismwhentheconstrained planner
initially wants to have a small fire sale is the following: since the price is given by p1B=
F (k0-k1B),aplannercanreducethefiresaleeitherbyreducingtheamountofkheldor
byincreasingtheamountofk1B0. Inordertominimizedistortions,weexpecttheplannertoadjust
onbothmargins. Oncethebadstatearrives,unlesstheplannercanuseanex-posttransferT,the
implicittransferdesignedbytheplannerthroughahighk12Bisoptimallyundonebythedecentralized trading
between agents, so the planner finds it optimal to implement the market solution. When
anex-posttransferT2Bisallowed, theconstrainedplannercanagainredistributeresourcesamong
agentsbyreducingthefiresaleandcompensatingtheoutsideconsumersex-post. Inanycase, the
planner’ssolutionatt= 1 inthebadstatewillnotcoincidewiththeex-anteoptimalone. Whenthe
constrainedplannerwouldliketoinducealargefiresale,theoppositelogicapplies. Inthiscase,the
constrainedplanner’ssolutionasksforalargek0andasmallk1B. Oncethebadstateatt= 1 occurs,
thereisnoreasonforentrepreneurstochooselesscapitalk1Bthaninthedecentralizedallocation,
makingtheoriginalpolicytimeinconsistent.
Wecansummarizetheresultsofthissectioninproposition4:
Proposition4.Theoptimalpolicyforaconstrainedefficientplanneristimeinconsistent. Allthree
channels,risksharing,collateralandmargin,inducetimeinconsistency.
Proof. Itfollowsfromthediscussioninthetext.
A natural step after identifying the time inconsistency problem is to characterize the time consistent
constrained efficient policy. In this simply environment, the natural way to tackle this problem is by
backwards induction: first, the planner solves for its optimal policy at period1 and then takes into
account its solution in the ex-antet= 0 stage. In practical terms, the problem to
solveisidenticaltotheonediscussedinsection4.2,withtheexceptionthatweneedtoaddequations
(33)and(34)asconstraintsintheplannerproblem. Unfortunately,thecharacterizationofthetime
consistentpolicyneednotbeparticularlyintuitive. Ithusrelegatethesolutionofthetimeconsistent
problemtotheonlineappendix. Themainupshotofthatexerciseisthatthetimeconsistentex-ante
policyinthecasethatoriginallyfeaturesoverinvestmentcanimplyaklargerorsmallerthantheone
inthedecentralizedmarket. Theintuitionforthisresultisthatk10nowdependsendogenouslyonk,
andakeyquantityfortheplanneristhedifferencek0-k1B. Ifk1B0reactsmorethanproportionally
withk0,itmaybeoptimaltoallocatemorecapitalex-antetotheentrepreneurs,sincek0-k1Bwill
28
be smaller and the fire sale reduced. In general, the final effects will depend on productivityA1B
and A2B, borrowing ability ξand the entrepreneurs’ attitude towards risk. The reverse reasoning
canapplywhenthetimet= 0 policypresentsunderinvestment.
Finally, how should we interpret these time inconsistency results in actual policy terms? The actual
implications for time inconsistency of both the risk sharing and collateral externalities are
extremelydifferent. Thereasoningbehindthetimeinconsistencyforthecollateralchannel,although natural
inside the model, seems counterintuitive in reality: we don’t see policymakers reducing prices in crisis
states because this high prices had already fulfilled their duty of allowing ex-ante borrowing. A
compelling reason not to observe this policy behavior is that, by reducing prices in
stresssituations,thepossibilityofdefault,whichIhaveassumedaway,mayrisesubstantiallywith
negativeconsequencesforotherreasonsunmodeledhere.
Fromapolicyperspective,therisksharingmechanismhasamuchmoreappealinginterpretation. Assume
that we are in the most realistic situation, which corresponds to case 3 in proposition 1;
entrepreneursparticularlyvalueresourcesinthebadstate. Inthiscase,sincetheconstrainedplanner wants
to keep high prices in the bad state, he will decide to implement two different measures: a
capitalrequirementtoreduceinitialinvestmentkandsomethingequivalenttoalenderoflast
resortpolicytoincreasethelevelofk10(i.e., reducingtheamountoffiresoldcapital)duringthe crisis. Once the
crisis state actually arrives, if the planner can shift even more resources towards
entrepreneursbyraisingpricesandcompensatingoutsideconsumerswithanex-posttransfer,hewill
gladlyadoptthispolicy. Thisisthebehaviorthatweusuallyobserveinactualpolicymaking. Ifthis
transferisnotavailable,theplannercannotfindParetoimprovementssoheisforcedtoreplicatethe
decentralized market solution, which implies a lower k1than the originally expected. In this case,
anactualpolicymakerwouldprobablydisregardtheParetocriterion,downweightingthewelfareof
outsideconsumersandeffectivelyavoidingthefallinprices.
6 Discussion of the results
Table1summarizesthemainresultsofthepaper:
Cases λ0
andλ1B
k
0
Case1: Maximumex-anteborrowing λ0
>λ1B >λ1G
= λ1B >λ1G
>λ0 >λ1G
kmarket 0 Q kplanner 0
kmarket 0
= kplanner 0
kmarket 0 >kplanner 0
Overinvestmen
t
29
Over-orUnderinvestm
ent Case2: Constraintsdonotbindlocally λ0NoExternalities Case3: Insufficientinsurance λ1B
Table1: Summaryoftheresults
Ingeneral,thecollateralandmarginchannelsalwaysworkinthedirectionofoverinvestment,but
therisksharingchannelcanworkineitherdirection. Aslongastherisksharingchanneliseliminated,
asitiswhenweassumearepresentativeagentformulation,theplanner’ssolutionimpliesareduction
intheamountofk0chosen. Observethatintherepresentativeagentcase,thisreductionwilltrivially
generateaParetoimprovement,evenintheabsenceoftransfers.
An important conclusion of this paper is that Pareto improvements are hard to find in fire sale
environments when the planner properly accounts for heterogeneity and lacks the option of
arrangingex-antetransfers. PreviousworkinthisliteraturehasemphasizedtheexistenceofPareto
improvements,butthispapershowsthatthoseonlyariseunderveryspecificassumptions. However, the lack
of Pareto improving policies does not mean that the externalities described here are not important or
that capital regulations should be disregarded. My results highlight nonetheless that
distributionalconsiderationsshouldbetakenintoaccountbythepolicymakerandthatacost-benefit
analysisapproachshouldbeused;thediscussioninDrezeandStern(1987)abouttheParetocriterion
directlyappliestothissituation.
Togaindeeperintuition,IpresenttheconstrainedplannerEulerequationforcapitalinthethree
periodcasewhereanex-antetransferisallowedandwherebothentrepreneursandoutsideconsumers
areriskaverseagents. Thisrequiresanendogenouschoiceofk1Bdifferentfromk241B= 0 ,asinthe
timeinconsistencysection. Theequivalentexpressiontoequation(29)isnow(35). Themultipliers
oftheentrepreneursarerepresentedwithasuperscriptEandthoseoftheoutsideconsumerswitha
superscriptC.
C 1B
1 = E λE
(p1s + As E 0s) +
[ 1 -ξ(p1s)] p1s [
] E 0B+ +
λE
1sλE 0
νλE 0
1BλE 0
k
0
νλE 0
0
Externality
λC
Margin + (1 -ξ)
-ξ F
0
1
1B
∆p 1B
+ pB
λ -k1BRisksharing
CollateralThe two key differences with respect to (29) are the following:
a) the amount of asset fire sold
k-kisthekeyvariablethatdeterminestherisksharingexternalitybut,sincekplaysnorole
indeterminingex-anteborrowingcapacity,onlytheamountofk0E 1Bheldaffectsboththemarginand
collateralchannels;b)thedifferencebetweentheratioofintertemporalmarginalratesofsubstitution of
entrepreneurs with respect to outside consumers is the key term that determines the size of the risk
sharing externality. (35) clearly shows that when markets are complete, i.e.,E 0and ν0B= 0,
firesaleexternalitiesdonotmatter. ξ F =∆ξ/p 1Bλλ,whichdenotesthesemi-elasticityof
marginswithrespecttopricesseemstobeaninterestobjectforfutureempiricalwork. AlsoF ,the
sensitivityofprices,and1 -ξ,thepledgeabilitycoefficient,areempiricallyrelevantterms.
24Seetheonlineappendixformoredetailsonthederivation.
30
λ
Another significant conclusion of this paper is that amplification mechanisms and welfare
implicationsareconceptuallytwoindependentphenomena. Thisstatementisnotclearintheprevious
literature. Thesimpletwoperiodversionofmymodelpresentsthethreetypesofexternalitiesbut
lacksanykindofamplificationmechanism,sincetheentrepreneursarealwaysforcedtosellalltheir capital in
any event. When I introduce the three period formulation, both demand and supply of
capitalk0-k1B25aredownwardsloping,creatingamplification: alowerendowmentinthebadstate forces
entrepreneurs to sell more capital, which reduces its price and makes entrepreneurs poorer, which
makes them fire sell a larger amount, reducing prices even more, and so on. However, this
amplification mechanism will also exist when credit constraints are not binding and markets are
effectively complete in equilibrium (the equivalent situation to case 2 in proposition 1). We know
that,inthatcase,theeconomyisefficient: thesetwocasesshowhowassociatingamplificationand
welfaremaybeseriouslymisleading.
Thisdecouplingresultemphasizesthattheexistenceofa“crisis”withverylowpricesdoesnot justify a
government intervention per se: a particular market failure has to be addressed by the policymaker.
That said, the quantitative importance of the externalities will be larger if there are amplification
mechanisms that generate deeper fire sales. Since p0= 1 is fixed in my model, no dynamic forward
looking mechanisms such as those presented in Kiyotaki and Moore (1997) play a role. In that model,
a binding price dependent borrowing constraint can simultaneously create
amplificationandexternalities. Thisdynamicmechanismwillgenerateamplification,butthesources
oftheexternalitieswillbeexactlytheonesdescribedinthispaper.
Throughout the paper, I have adopted the assumption that entrepreneurs follow price taking
behavior. If some entrepreneurs were aware of their non-infinitesimal size, they would internalize part
of their effect on prices, modifying the effect of the externality. The larger the size of those agents, the
greater the welfare transfer from the outside consumers. In my model, theθ= 0 case is a monopolist
that fully internalizes its impact on prices and doesn’t care about the welfare of others. Therefore, he
would always set a smaller k0than in the price-taking case, hurting outside consumers.
Ingeneral,othercombinationsofnoncompetitivebehaviorcouldeasilybeexploredinside myframework.
In appendix B, I discuss the case with a single risk-free asset, instead of two state contingent
securities. That case is closer to reality than the main formulation of the paper in a particular aspect: in
that situation, it is possible to have an entrepreneur who hits the borrowing constraint and who also
believes that having resources in the crisis state is more valuable than at the initial period (λ1B>λ0).
This is possible because now the entrepreneurs evaluate the benefits of riskless
borrowingbyconsideringtheexpectedrisk-adjustedreturns,asinλ0= pGλ1G+ pBλ1B-η0+ ν,
insteadofmakingthatcomparisonstatebystate,asinλ0= λ1B-η0B+ ν0B0. Underthisparticular
25Section6intheonlineappendixshowsaclarifyinggraph.
31
equilibriumconfiguration,aslongastheentrepreneurshittheirborrowingconstraints,thereisalways
goingtobeoverborrowing. Thisresultisveryintuitive: entrepreneursdonotinternalizethatthefire
saleinthebadstatedepressesitsborrowingcapacityatthesametimethatitreducestheirwealth
inthestatewheretheyvalueitthemost.
Twoimportantlessonscanbedrawnfromthecomparisonbetweentheleadingcaseinthepaper
andtheonediscussedinappendixB.First,theoverallwelfareeffectsduetopecuniaryexternalities
cruciallydependonthesetoftradableassets. Hence,quantitativeworkmustbeparticularlycareful
whenspecifyingthesetoftradingopportunities. Second,thefactthatagentslackstatecontingent
insuranceinfiresalestatesmakesthedesiredoverborrowingresultmorelikely, sinceboththerisk sharing and
collateral/margin channels work in the same direction. This last result is of extreme
importance,sincetheremaybeastrongerrationaleforsettingtightcapitalrequirementswhenthere
arenoopportunitiestoarrangestatecontingentborrowing.
26Finally, thispaperhasfocusedonlyontherealeffectsoffiresales. Anunexploredandfruitful avenue for
future research would be to append a nominal side to the model and to study how a
monetaryauthority’sinfluenceonpricesandwagesinteractswithallthechannelsdescribedhere.
7 Conclusion
This paper has shown how pecuniary externalities matter for fire sales. Two main mechanisms,
pricedrivenredistributionandborrowingcapacity,throughcollateralvalueandmargintightening,
determinetheextenttowhichaconstrainedplannercanachieveParetoimprovementswithrespect
tothedecentralizedcompetitivemarket. Eventhoughtheultimaterootofallthreechannelsisthe presence of
some form of market incompleteness, the way this incompleteness affects decentralized
outcomesispatentlydifferent.
Thispaperisthefirsttoidentifythetimeinconsistencyproblemfacedbyaconstrainedplannerin
anenvironmentwithpecuniaryexternalities. Allthreechannelsaresubjecttothetimeinconsistency concern.
Iadditionallyemphasizehowfinancialamplificationmechanismsandpecuniaryexternalities
areingeneralindependentphenomena.
My results show that a planner cannot improve the decentralized outcome unless he decides to
disregard distributional considerations or is allowed to use ex-ante transfers. For instance, the
exclusive use of capital requirements is not enough to create Pareto improvements as long as the
welfareofthebuyersoffiresoldassetsisproperlytakenintoaccount.
A necessary first step to guide the much needed financial regulatory reform is to identify the
failuresofthemarketmechanism: Ihopethatthetractabilityandscalabilityofmyresultshelpto
focusfuturediscussionsaboutfiresalesandpecuniaryexternalitiesinthiscontext.
26Inthispaper,Ihavenotconsideredproductionandlaborsupplydecisionsorwagedetermination.
model,thoseeffectscanpotentiallyinteractwiththeonespresentedhere.
32
Inalargerscale
APPENDIX
8 App endix A: Pro ofs
8.1 Pro of of prop osition 2:
Proof. Withouttransfers, theindirectutilityoftheoutsideconsumersdependsdirectlyonk. We can therefore
write V C j(k0) = -pBF kC 0 kC 00>0 by using the envelope condition. This implies
thatinordertofindaParetoimprovement, anecessaryconditioniskPlanner 0. However,
weknowthat,atthedecentralizedequilibrium,amarginalchangeink0Market 0=kmodifiestheindirectutility
oftheentrepreneursbypBλ1BF (k0) k0<0 inanyofthethreeequilibriadiscussedinproposition1.
Thisresultcanbeeasilyderivedusingagaintheenvelopecondition. Therefore,ifk,
priceswillbelowerinthefiresale,necessarilyhurtingentrepreneurs. Thus,theonlyfeasiblepolicy
fortheconstrainedplannerwithouttransfersistosetkPlanner 0= kMarket 0.Planner 0>kMarket 0
8.2 Pro of of prop osition 3:
λ1BProof.
Iproceedbyanalyzingtheanalogouscasestoeachofthethreecasesdescribedinproposition 1.
Incase1,λ0<1 ,ν0B>0 andη0B= 0. Itisusefultorewrite,bysubstitutingν0B= λ, the externality term in (27) as
pB λ1Bλ0-1 [ ξ+ ξ F ] F k0. If margins are fixed and ξ 0-λ1B= 0, or more generally when -, then we would
expect the externality term to be negative. If this happens, then kmarket 0ξ F <ξ<k planner 0, T0<0 and the
entrepreneurs underinvest in the decentralized market. If we constrain our solution to T0=0, then the
solution would imply kmarket 0= kξ F >ξ ,thenkmarket 0<kplanner 0andT0planner 0. When->0. Thecasewithξ= 0
istheknifeedgesituation inwhichtherisksharingandcollateralchannelsexactlycancel.
λ1BIncase2,noconstraintsbindtowardsthebadstate,therefore,marketsarelocallycomplete,this is, λ0= 1
andν0B= η0B= 0. Consequently,kmarket 0= kplanner 0andTλ1B= 0. Incase3,λ0>1 ,ν0B= 0,η0B>0.
Thisimpliesthatkmarket 0>k0andapositivetransfer T0planner 0>0 isimplemented.
ThiscaseisanalogoustotheleadingcaseinLorenzoni(2008).
33
9 App endix B: The case of a single noncontingent asset
In this formulation, instead of having two state contingent securities, I assume that entrepreneurs
can only use a risk-free bond. The rest of the notation and the assumptions are identical to main
case discussed in the paper. Since this bond is traded by the outside consumers and they are risk
neutral with discount factor of 1 , both its price and return must be equal to 1 . Using again an
argumentàlaHartandMoore(1994),Iassumethattheamountentrepreneurscanborrowex-ante is limited
by the maximum amount that can be recovered by the outside consumers in the worst case
scenariop1Bk0. I append once again the margin correction1 -ξ(p) to the borrowing limit and appeal to
section 2 of the paper to justify its relevance. Sincep1B1Bis always smaller than p1G
by assumption, the borrowing constraint can be written as a0 =-min {[ 1 -ξ(p1s)] p1sk0
1s)] p1sk0}= [ 1 -ξ(p1B)] p1Bk0
U(C0) + pGU(C1G) + pBU(C1B)
max
k0,a0,C0,C1G,C1B
= e0 -q0a0+ [ (λ0
+a
p
1G
=e
+ AG] k
1G
=e
+[
p1B
(p λ1G
(p
+a
+ AB] k
1B
0
)
C
0)]
C
p1Bk 0
C
0
=
λ
k λ0 = E [ λ1s (p1s + As)] + ν0 1
0
-ξ
( )
F k
F (k0)
0
1B
1B
34
1B
0
0
. Thedecentralizedversionofthisproblemisthen:}, where min {[ 1 -ξ(p
1B ) C1G
0
0
C0 + p0k0
0
0
0
(ν
: U (C
) = λ1G
G
B
1B
)λ
) C) a= L (η) a=-[ 1 -ξ(p1B0
Withthefollowingfirstorderconditionsforoptimalityinequilibrium:
0
1G : U (C1B) = λ1B
1B
1G
:U (C0
1
0
+ ν0 ] -η = E [
a0 :λ0
s
:p0
λ
As described in the paper, depending on the value of λ0, λ1G and λ
>λ
or with λ0 <λ1B. This last situation, when λ0
<λ
there can be different type of equilibria. The most interesting case to consider here is the one with a
binding borrowing constraintν>0 .
Notethatwithasinglerisk-freeassetwecanhaveabindingborrowingconstraint at t= 0 with λ, appears to
be the mostinterestingonefromanempiricalviewpoint. Inthatcase,theentrepreneursvaluewealththe
mostinthebadstate. Hence,iftheyhadaccesstostatecontingentsecurities,theywouldarrange
insuranceforthebadstate,however,theystillhittheirborrowingconstraintex-antesincetheycan
onlyuseanassetthatisnotstatecontingent.
Proceedinginthesamewayasinthesection4inthepaper,wecanwritetheEulerequationfor
capitalfortheconstrainedplanneras:
0
0= DecentralizedMarketE [ λ1s(p1s+ B λ0
λ0
p B
As)] + ν0(1 -ξ) F
p+ ExternalityB(λ1B-ξ) F k0
0,equaltoλ0-λ1B.
0
Let’sthenrewritethe
0
1B
Gλ
Theotheroptimalityconditionsremainidenticaltotheonesinthedecentralizedmarket. Ifwe
allowforex-antetransfers,λ= θ. Thevalueofνinthecasewithstatecontingent assets,nowbecomesν=
λ0-pGλ1G-pBλ1B
-1 + pG λ1G + p λ1B -1
-ξ F + (1 -ξ)
F k
λ0
λ1Bλ0
Theimportanceofhavingasinglecontingentassetisthatnowwecanhavethecasewithν>0 and λ0. In that
situation, there is always going to be overinvestment. The intuition is thefollowing:
theentrepreneursvaluehavingresourcesinthebadstatemorethanex-anteandthan in the good state:
λ1B. Nonetheless, they borrow as much as they can and have to repay a= -[ 1 -ξ(p>λ1B)]
p0>λ1Bk01Gin the bad state. Why? For this to happen we must have that λ1G+ pBλ1B. This means that
if carrying wealth in the risk-free asset towards the good state is not very valuable (λ1Glow), and the
bad state is somewhat not very likely (psmall), it may be ex-ante optimal to borrow as much as
possible. There is no policy tradeoff in this case: a constrained planner will always set a kplanner
0Bsmaller than the one in the decentralized market,
implementingforinstancetheappropriatelevelofcapitalrequirements.
>0
>0
>0
>λ
0
0
>p
>0)
λ0
Figure7showsthiscasegraphically:
Borrowing constraint binds (ν0
pGλ1G + pB λ1B
λ1G
λ1B λ0 >λ1B
<λ1B
λ
0
Figure7: Equilibriawithasinglenoncontingentasset.
35
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39
FOR ONLINE PUBLICATION ONLY
Dissecting Fire Sales Externalities
Online App endix
EduardoDávila
August 19, 2011
1 Understanding the mechanism: Risk Sharing Externality and
CollateralExternality
Thesenotespresentasimplerversionofthemodelthantheonedevelopedinthepaper. Inthesenotesthere
isasinglestateandthereisnoprice-dependentmarginmechanism. Mostoftheinsightsremainthesame.
Therearetwodates, t= 0,1. Therearetwoagents: entrepreneurs(mainactors)andoutsideconsumers (they provide
the downward sloping demand for assets in the fire sale). There are two goods: consumption goods (fruit) and
capital goods (trees). Entrepreneurs are risk averse, maximize U(C0) + U(C) and have
endowmentsofconsumptiongood(fruit)ofe0ande11. Theyhavetochooseconsumption(offruit)C0andC,
theamountofcapitalk01toholdbetween0 and1 (thenumberoftreestoplant),andwhethertosaveorborrow a0in a
risk-free asset denominated in consumption good. Each unit of capitalk0held by the entrepreneurs
producesAunitsofconsumptiongoodsatt= 1.
The consumption good is the numeraire, so p0 and p1
00 1 = F
= k0), generates p1 = F (k0
kC 1 andthat,inequilibrium(sinceallcapital
1
2
is sold, yielding kC 1
k
C1
- p kC 1, with FOC p
1
=F
kC 1 .
0
1
2
denote the prices of capital at t= 0 and t= 1
respectively. Iassumethatatt= 0,consumptiongoodsandcapitalgoodscanbetransformedonaone-to-one
basis,whatpinsdownthepriceofcapital(trees)att= 0 top= 1. Outside consumers are risk neutral, do not discount
the future and have large endowments, so they pin
downinequilibriumreturnontheriskfreeassettoq= 1 (netinterestrateis0 ). Inperiodt= 1,theyruna
concavetechnologywithcapitalthatimpliesapricep
). Any other assumptions that imply a downward sloping
demandcurvewouldalsowork(e.g.,thebuyersoftheassethavelimitedrisk-bearingcapacity). Notethat,in periodt=
0,theoutsideconsumershavenoproductiontechnology,andtheyareonlyrelevanttodetermine
theamountaofborrowing/lending. Thetimesubscript0 or1
denotestheperiodinwhichanendogenousvariableischosen.
We usually assume this in, for instance, the basic version of the neo classical growth mo del.
1
1
They solve the problem max C1 F k
U(C0) + U(C1) (1)
max
1.1 Decentralizedproblem
,a0,C0,C1
Entrepreneurssolvethefollowingproblem:
k0
Subjecttothefollowingconstraints:
1
0
= e0 -q0a0+ [ (λ0 + a0
= e1 p1
(λ1
atpricep0
0
0
0
p
0
0
k0
(
ν
0,investink0
0
0
1
0
0
1
0
0
0
0
p1k0
1
1k
0
p
2
Notethat,ifinsteadofusinga
Lagrangian:
+ p0k0 ) (2) C+ A] C
k) (3) a=0 (η) (4) a= 1 ξ1) (5)(2)istheinitialbudgetconstraint.
0
TheentrepreneurscanconsumeC(givenmy assumptions,p= 1)withtheendowmente,saveifa>0 orborrowifa<0.
(3)isthebudgetconstraintinperiod1 . Sincetheworldends,allcapitalissold(k= 0). Theentrepreneurs
consumetheirendowment,plusthefruitAkandsellthetreesatthepricep. Theyrepaytheirdebtsifthey borrowed(a<0)
orconsumetheamountsavedifa>0. (4)impliesthatagentscannotsave.
ThisequationcanbemotivatedàlaHolmstrom-Tirolebythelackof
stores of value. The outside consumers that provide credit cannot commit and can steal everything. In the
paper,Ihavetheconstrainta0= Linsteadofa=0. Lcanbethoughtastheamountofstoresofvaluein theeconomy.
(4)isaclassicborrowingconstraintalaHart-Moore94. Thefruitcanbestolen,soisnotpledgeable. Only
i default,sotheactualborrowingcapacityis 1 ξ.
borretreep0eis
owin
salloweDependingonwhichconstraint(4)or(5)bindsinequilibrium,wewillhavedifferentwelfareimplications.
gupt
d.
The collateral channel is due to the fact that pappears on the constraint (5), so only when that constraint
othe
Iassume
marthatafra
ketvction (5)binds that channel will app ear. However,as long as any of the two credit constraints (4) or (5)
ξislostw
alue
binds, the risk sharing channel will play a role in equilibrium
ofthhenther
= 1ξ
1k asborrowingconstraint,weusea
1.1.1 Optimality conditions
0
0
=0 (thisis,thetrees cannot
be pledged at all by the entrepreneurs), the collateral channel would immediately disappear. That would imply
that both borrowing and lending are forbidden (i.e., in that case there are no asset markets). That is exactly the
assumption that we would find in the literature on exogenous incomplete markets. In
otherwords,exogenouslyspecifiedincompletemarketscanbeunderstoodastheparticularcaseofborrowing
constraintsthatarecompletelytight,a= 0 inmyformulation. The main problem of using the one-state version of the
model is that in order for (4) to bind, we need
tohaveentrepreneursthatsave. Withtwostates, theentrepreneurscouldborrowatt= 0 butstillarrange
insuranceatthesametimeforthebadstateatt= 1.
L= U(C0) + U(C1) -λ0
FirstsetofFOC’s:
:U (C0) = λ0
(C1) = λ1
C0
[ + p0k0 + q0a0] -λ
C0
1
[ -[ p1 + A] k0 -a0] -η0a0
C1
+ ν0
a+
1 -ξ p1k0
0
:U
C1
a : λ0
= 0 q
Imp ortant FOC (Eulerequationforcapital):
-η0 + ν0
0
λ
1
(A+ p1) + ν0
1 ξ p1
1.1.2 Optimality conditions in equilibrium
p
0
λ
0
Imposingtheequilibriumconditions,thedecentralizedsolutionisfullycharacterized. Thisis:
a :λ 0 = λ1 -η0 + ν0
0
C0 :U (C0) = λ0
C1 :U (C1) = λ1
= λ1 (A+ F
1
C (k0))
+ ν0
1ξ F
0(k0
0
CM
CM
Ce
NotethatVC M
k0,a0,C0,C1,T0
And
) Thislastequationisthecrucialone,anditwillbecomparedtotheplanner’ssolution.
1.2 Constrainedplanner’sproblem
Theplannersolvesthesameproblemasthedecentralizedagent,butheaddstheconstraint:
λ
0
+ (F(k0) -F
(k0) k0
0
Utilityforoutsideconsumersindecentralizedproblem
0
+ T0
max
)
Utilityforoutsideconsumersunderplanner’ssolution =
Visagivennumber,sothisconstraintcanbethoughtofasaparticipationconstraintforthe outsideconsumers.
Alsonoticehow,giventheriskneutralityofoutsideconsumers,anyborrowingandlending plays no role in their
welfare. Let’s define θas the Lagrange multiplier of this constraint, that will bind in equilibrium.
Theplannermaximizes:
) + U(C ) + θ e + T0 + (F(k ) -F (k ) ) -V
3
U(C
k
C
e
isthelargeendowmentofconsumptiongoodofoutsideconsumers(wecanthinkthateC
isapossibleex-antetransfer. T0
>0 impliestransfersforentrepreneurstoconsumers,andT0
<
0
vice
vers
a.
Asdi
scus
sedi
nthe
main
text,i
fexante
tran
sfers
aren
otall
owe
d,th
erea
reno
feasi
bleP
aret
oimp
rove
men
ts.
S
ub
je
ctt
ot
he
fol
lo
wi
ng
co
ns
tra
int
s:
0
0
+[
F
0
F
0
+a
(k0) + (ν0
A] k
1
1
0
= eC + eC 1).
0
T0
0
0)
k0
λ
(
1
0
N
ot
et
ha
tth
ep
la
nn
eri
m
po
se
st
he
eq
uil
ibr
iu
m
pri
ce
sb
ef
or
e
m
ax
im
izi
ng
.
N
ot
ea
ls
ot
ha
tth
isf
or
m
ul
ati
on
im
pli
cit
all
o
w
st
he
pl
an
+k
) C= e C
(λ -a = e + T0 0
=0 (η) a= 1 -ξ
0
0
)
(k
)
a
ne
rto
tra
ns
fer
re
so
ur
ce
so
fth
ec
on
su
m
pti
on
go
od
.
If
w
et
ak
eli
ter
all
yt
he
int
er
pr
et
ati
on
th
at
co
ns
u
m
pti
on
go
od
s
ca
nn
ot
be
pl
ed
ge
d,
w
e
ha
ve
to
ad
mi
t
th
at
w
e
ar
e
im
pli
cit
ly
gi
vi
ng
th
e
pl
an
ne
r
an
ad
dit
io
na
la
bil
ity
wi
thr
es
pe
ctt
ot
he
m
ar
ke
tp
art
ici
pa
nt
s.
1.2.
1
Opti
mali
ty/E
quili
briu
m
con
ditio
ns
Fu
llL
ag
ra
ng
ia
n:
L= U(C0) + U(C
-λ0 [
C0
)+
θ
e+ q0a0C ] -λ1
0
+ p (F(k
B
-V
0
)
-F
CM
0
) (k
k
StandardFOC’s:
C0 :U (C0) = λ0
C1 :U (C1) = λ1
FOCforT0:
In
tu
iti
o
n:
in
or
de
rto
ha
ve
an
int
eri
or
so
lut
io
nf
ort
he
tra
ns
fer
s,t
he
m
ar
gi
na
lc
os
tof
m
ov
in
g
w
ea
lth
fro
m
en
tre
pr
en
eu
rs
0
a
a :λ 0 = λ1 -η0 + ν0
0
θ
=
λ
0
0
+T
0
10
(k )
k
0a] -η -a ) + A] k (k0 -[ F [
+ p0k0
C1
λ0
m
us
tb
ee
qu
alt
ot
he
m
ar
gi
na
lb
en
efi
tof
th
ee
ntr
ep
re
ne
ur
θ(
si
nc
et
he
ya
re
ris
kn
eu
tra
l).
Im
p
or
ta
nt
F
O
C
(E
ul
er
eq
ua
tio
nf
or
ca
pit
al)
:
= DecentralizedTerm 1(F (k0
(λRisk-Sharing +)
λ0 k0F (k0
4
C
oll
at
er
al
λ) + A) + 1 ξ ν0F (k0) + Externality1-θ)
1 -ξ ν0k0F (k)
0:0
S
ub
sti
tut
in
gθ
=
λ
Collateral
Twointerestingcasesariseinequilibriumdependingonwhichconstraintbinds:
•Case 1: Only savings constraint binds (ν0= 0 ) - OVERINVESTMENT - ONLY RISK SHARING
CHANNEL
Entrepreneurswanttosavefrom0 to1 . Theyhittheirsavingsconstraint,therearenostoresofvalueinthe economy.
Since they want to save, this means that they value wealth more at t= 1 than at t= 0, this is λ>λ0.
Thefirstorderconditionsinthatcaselookslike
) k0F (k0 + Externality1-λ0)
=
λ 0
DecentralizedTerm 1(F (k0
λ)
k
0F (k
(λ)
1-λ
00
+ A) + 1 ξ ν0p1
0
0) + A) +
1 -ξ ν0p1
k0F
ν
1
λ
0
= DecentralizedTerm 1(F (k0
λ) + A) + 1 -ξ ν0F (k0) + Externality1-λ0)
k0F (k0
(λ)
Risk-Sharing + 1 -ξ ν0k0F (k0)
λ0
1
Risk-sharing
Sincetheexternalitytermisnegative,rememberthatF<0 andλ>0,theplannerwantstoreduce
investmentex-ante,thereisoverinvestment.
Intuition: entrepreneursreallyvaluehavingresourcesinthefuture,buttheylackstoresofvalue. Since they do not
internalize their effect in prices, they fire sale too much in t= 1 . A planner would reduce k,
reducingthefiresaleandtransferringwealthviathispricechangetotheentrepreneursint= 1. Tocompensate
theoutsideconsumers, wholosewiththisincreaseinprice, aplannerwouldgivensomepositivetransferT
ex-ante. •Case 2: Collateral constraint binds (ν>0) - OVER OR UNDERINVESTMENT - BO TH RISK
SHARING AND COLLATERAL CHANNELS
<λEntrepreneurs want to borrow at t= 0 and repay at t= 1. They hit the borrowing constraint due to their limited
commitment. Since they want to borrow, this means that they value wealth more at t= 0 than at t= 1,thisisλ0.
Thefirstorderconditionsinthatcaselookslike:
= DecentralizedTerm 1(F (k
λ+ Externality1-λ0
(λ)
Intuition: theexternalitytermhastwoparts. 1.
Now the risk sharing term is positive, since λ1Risk-sharing +-λ0< 0 and F 1 -ξ0(kCollateral (6))
(k00) < 0.
This term leads to underinvestment. This is natural, since agents are borrowing constrained and really
would like to investmoreex-ante. Theplannerwouldgiveatransferex-antetoentrepreneursandthencompensate
theoutsideconsumersex-postwithaverylargefiresale.
2. Thecollateraltermisalwaysnegative,sinceν0>0 andF (k0) <0. Thistermleadstooverinvestment.
Theintuitionhereisthattheentrepreneursdonotinternalizethefactthatbyreducingpricestheyworsen
theborrowingcapacityoftheotherentrepreneurs,whowillbeabletoborrowless.
5
Inthisparticularcase,wehavethatν0
= λ0
DecentralizedTerm 1(F (k0
λ0 C0 + V=
+ A) + 1 -ξ ν0p1
C1
-λ1. Bysubstitutingin(6):
λ)
+ Externality1-λ0
(λ)
-ξ k0F (k0
1- 1
)
Inthiscase,giventheassumptionsabouttradedassets,theexternalitytermbeco
mesalwayspositive,that
is,therewouldbeunderinvestment.
AswecanseeinappendixBinthetext,thisconclusioncruciallydepend
onthesetofavailableassetstraded.
Note:
IfIhadallowedforarepresentativeagent,orsomekindofex-postrisksharingmecha
nismbetween theparties,therisksharingchannelcanbeshutdown.
Mostmacropapersthatstarttousetheseideasassume that,butthisisworrisome.
DoingthisisshuttingdownthestandardWalrasianmechanism. Aren’ttheoutside
consumers buying cheap assets making a great deal? A different question is
whether we (the planner) care aboutthem.
2 Riskaverseoutsideconsumers
The model is identical to the one described above, with the exception that
outside consumers are now risk aversewithutilityV ˜ ˜outsideconsumers.
Theirbudgetconstraintsare: ,insteadofbeingriskneutral.
Iuseatildetodenotechoicevariablesof
= eC 0+ T0 -q0˜a0
˜λ0 0
˜1
0
0
C0
˜ C0
˜C
˜ = eC + F(k
C 1
) -p1 Bk
+
˜a
λ
1
Notethatwedon’thavetospecifyborrowingconstraintsforoutsideconsumers.
Theequilibriumconditionsforoutsideconsumersaregivenby:
0 :
C ˜1 ˜
V =
λ0
˜ 0 0:
= ˜ →q0 = ˜˜
a
q
λ1
λ1
1 :
˜ λ1
˜
V
λ
˜
= 0
C
˜
λ
0
The conditions for the entrepreneurs are identical. The problem to solve for
the constrained planner becomesnow:
V e + T0 -q0˜a0 + V e + F(k0) -F
10,00C,C,akmax, ˜a0, ˜ C0, ˜ U(C0) + U(C1) + θ
C1,T0
C0
(k0λ) + Externality1-θ ˜
(k0
λ1 k0F (k0Risk-Sharing +)
1 ξ ν0k0F
λ0
6
Therefore,theconstrainedplanneroptimalityconditionforcapitalbecomes:
= DecentralizedTerm
λ
Collateral)
1
(F (k0) + A) +
1 -ξ ν0F
(k0) k0 + ˜a0 -VC M
C1
SubstitutingthefirstorderconditionforT0: λ0
=θ˜
λ0:
ν0F
(k0
1
=
D
e
c
e
n
t
r
a
l
i
z
e
d
T
e
r
m
1
(
F
(
k
0
λ
)
+
A
)
+
1
ξ
3.1 Decentralized
3
I
nSubjecttothefollowing
t
r
o
d
u
c
i
n
g
a
t
h
i
r
d
p
e
r
i
o
d
λExternality
λ10˜λ˜
λ10 k0F (kRisk-Sharing0)
ξ ν0k0F (kCollateral0)
0
+
1
-k
-k1
+ p0k0 = e0 -q0a0+ [ p (λ0 + a0 -q1a1( (λ1
λ2
+ p1k1 = e1 p1
1] k0 + a1
1
(ν0
+ [ p2
2] k1
2
0
0
= e2
0
1
k0
1
Astraightforwardchangeofnotationyieldsequation(36)inth
epaper.
Themodelisexactlythesameasbefore,butnowweintroduceathird
periodt= 2. Iassumethatinperiod
2capitalcanbeeatenagain,sop= 1. In this case outside
consumers run the concave technology on the amountk01, this
is the amount of capitalsoldbytheentrepreneurs.
Icouldassumethatwhenk<0,thenthepriceisone(transforming
consumptiongoodsincapital).
Iwillassumethatweareinthecasewithk0-k1>0,sotheentrepreneur
s arefiresellingassets.
U(C0) + U(C1) + U(C2) (7)
C ) (8) C+ A) (9) C+ A) (10) a=0 (η) (11) a= 1 ξ1) (12) a=0 (η) (13) a1
0
1
(k0 (ν1) (14)
In equilibrium, q0 = 1, q1 = 1 , p0 = 1 ξ= 1, p1 p2k=
min {F
-k1) ,1}, p2
= 1 so the optimality conditions become:
C : (C0) = λ0
C :
(C1) = λ1
C :
(C2) = λ2
0
1
2 U
U
U
k0) k1:p1λ1= λ2(1 + A211
-k (k0 -k1) + A1) + 1 ξ ν0F
a :λ0 = λ1 -η0 + ν0 a) + 1 -ξ1:
-η1 + ν1
0
λ ν1= λ2
7
(k0 (F= λ1 0:λ
3.
2
C
o
n
st
ra
in
e
d
pl
a
n
n
er
’s
pr
o
bl
e
m
max
U(C0) + U(C
1)
+ U(C2) + θ
eC + T0 + (F(k0 -k1) -F
(k0 -k1) [ k0 -k1]) -VC M
k0,k1,a0,a1,C0,C1,C2,T0
1
C0
C0 + k0 + T0 = e0 2-a0+ [ (k0 (λ0
F
1) k1 = e1
+ a1
(λ2
1
(ν0
0
1
0
= e2
0
0
1
1
1
11
0
0
1
2
=0 (η) a= 1
ξ F=0 (η) a1
) C+ F (k-k
0) a
) C+ [ 1 + A2] k) a
(λ -a + a ) + A1] k -k
) -k (k0
k
1
= 1 -ξ k
(ν1)
Withoptimalityconditions:
:U (C0) = λ0
C :
(C ) = λ1
C :
(C2) = λ2
1 U
U
k0 :λ0 = DecentralizedTerm 1(F (k0-k1) + A1
λ) + 1 -ξ ν0F (k0-k1) + Externality(λ1-θ) [ k0-k1
]
+ 1 -ξ ν0k0 F (k0-k1)
)
a0
:λ0
=
λ1
0+
ν1:
λ=
λ1
+
ν0
-k
1
T :λ0 = θ
0
k1 :F (k0 -k1) λ1
-η
0
a
= DecentralizedTerm 2(1 (λ1Externality-θ) [ k0-k1
+ A2
λ) + 1 ξ ν1 ξ ν0k0 F (k
1
2 -η
1
]+
1
3.
3
Ti
m
eI
n
c
o
n
si
st
e
n
c
y
Pr
ob
le
mt
ha
tth
ep
la
nn
er
so
lv
es
att
=
1.
ma
x
k1,a1C1,C2
= e1 +
1
F
) + U(C
C
1U(C
2)
k -k1 + A1
k1+ a1(λ2
+
+
θ e
F
k0] + a0 -a1
k -k1 -F
0
k -k1 k0
-k1 -VC M
0
(λ1
02
= e2
2
1
C +F
k -k1 k1 + T1 ) C+ [ 1 + A) a=0 (η) a1
= 1ξ k
Withoptimalityconditions:
8
1
0
1
(ν1)
C1 :U (C1) = λ1
C2 :U (C2) = λ2
-θ) k0-k1 F k0-k1
: 1=
+ (λ1
=
η
Decentraliz
a λ
1
Notethatthecollateralexternalitytermclearlydisappears.
Notealsothatgivenk
0,nowfixed,aplanner
λ
ν
λ
2
cannotinduceanewconstrainedParetoimprovement,evenifhehadthechancetousetransfersinperiodt=
1.
Ifhehadthechancetoarrangeex-postredistribution,att= 2 theremaybeanewpossibleimprovement,but
thereisstilltimeinconsistency. Theagentthatwassupposedtobecompensatedbythepriceredistribution willlose.
4 Timeconsistentpolicy
This section of the online appendix analyzes the same case as in the paper, with 3 periods and 2 states. It
characterizesthetimeconsistentproblemforaconstrainedplanner. Forsimplicity,Ionlydiscussthecaseof a planner
who can just use an ex-ante transferT. This assumption implies that at t= 1 in the bad state,
theplannerwillnothonorthepredeterminedk10; hewillreplicatethesamesolutionasthemarket. Wecan
combinetheoptimalityconditionsfromat= 1 perspectivetoyieldequation(15)andthenfindtheex-ante time
consistent solution. If we allow the constrained planner to use an ex-post transfer T2, there would be
additionaltermtakingthatintoaccount,butthereasoningwouldbeidentical.
F (k0-k1 BFromequations(33)and(34)inthetext,theadditionalconstraintthattheplannerfacesatt= 0 is: ) 1
ξ λ1 BOrsubstitutingthemarginalvalueofwealth:= λ2 B A2 B+ 1 1 ξ (pBψ) (15)
0(e1 B + A1 Bk0 + F (k0 -k1 B) [ k0 B-k1 B] + a0 Bψ)-a1 B
+ a1 B
(C0
(k0 -k1 B)
(C2
1 -ξ U
F
0
1s
2s
C :U (e2 B+ [ A2 B+ 1] k1 B) -U) A2 B+ 1 1 -ξ = 0 (
:U pTherefore,theex-anteproblemfortheconstrainedplanneryieldsthefollowingoptimalityconditions:
C1 s
(C1 ) = λ ,fors= G,B C2 s:U ) = λ,fors= G,B
9
)
=λ
a0 G :λ0 = λ1 G -η0 G + ν0 G
-η
+ν
:λ = λ
:λ0 = θ
a0 ) (16) a1 B2 B1 B1 B+ ψ
a1 G : λ1 G
η
λ
1
2
F
1B
0=
0
+ ν1 G
+1
1 -ξ
1B
U
G
(C1 B)
U
(C2
B)
A2 B
G
-(k-k1 B)
(C 1
1 -ξ(17) T
1
G
(As
)
B
B
+ p1 s)] + E [ ν0 sp1 s] +
pB
1ξ U
Externality
(λ1 B-θ) [ k0-k1 B] + ν0 B 1
ξ k0 F (k0-k1 B) 1 B01 B
= E [ λ1 s
2G
F (·) λ1 +
(·) 1 ξ U
B
F
:λ = λ
k0 :λ0 (18) + p
1
k1
ψ
(A2 G + 1) + 1 ξ ν
B
(C
)[
F
(·) [
k
-k
] + F (·) + A1
B]
G
G
(·) λ:F
k1
(·) λ1
1
B
B
+ 1) +
001 B1 B0 B001 B
B
+ ψ -F
(C (λ1 B -θ) [ k -k
]+ν
1ξ k
(k -k
)
)
F
F
(·)
1ξ
U
Bysubstitutingv1 B
Ex
ter
na
lit
y
(λ
1
B-
θ)
[
k0
-k
1
B]
+
ν0
B
1
-ξ
k0
F
(k
0k1
B)
] + F (·)] A2 B + 1 1 -ξ
(·) [ -k
F
k
from(17)into(19)andimposing(15),wecanfindthat:
1 B)
[
U
2
B
+
1]
(19
1ξ ν
1
B
Ext
=
ψ -F
(·) λ1
F
(·)
1ξ U
B
+ψ 1 ξ
Thus
:
Ex
ter
na
lit
y
F
(·)
(C1 B) [
F
1 -ξ U
(·) [ -k1 B] + F (·)] A2 B + 1 1 -ξ
k0
(C1 B) + A2 + 1 1 -ξ U
(C2
B
B)
U
(C2 B) [ A2 +
B
1]
(λ
1
B-
θ)
[
k0
-k
1
B]
+
ν0
B
1
ξ
k0
F
(k
0k1
B)
=
ψ -F
(·) λ1
F
(·)
1ξ U
B
A
nd
th
en
su
bs
tit
uti
ng
ba
ck
int
o(
18
):
(As
F
λ0 = E [ λ1 sSubstitutingnow(15):
(C1
B)
F
(·) [
k0
-k1 B] + F
(·)
1 -ξ
1
(C
B
ψ + p1 s)]+E [ ν0 sp1 s]+pB
A2 B
A2 B
(·)
10
+1
1 -ξ
U
(C
1ξ U
(C2 B) A2 B
) A2
B
1 B)
1 -ξ + A
+1
1ξ U
λ0 = E [ λ1 (As + p1 s)] + E [
s
(λ0
ψ A2 B
F
λ0 + 1
-λ1 s + η1 s) p1 s] + (20)
U
(C2 B )
+pB
Bψ
1 B)
1B
U (C
1 -ξ
U
(C1 B
) (As
U
0)
2 B(C(C2 B)
A2 B
1 -ξ + A
1 s)
)U
-
(C
) A2 B
p1 s
A
= E [ λ1
Ifν0 B
s
)A
-λ ispositive,weneedtosubstituteν0 Bfrom(16)into(21)tohavetheappropriatecomp
+ p1 s)] + E [
(λ
2
B
U ] + +p(C1 B1 B-F
Substitutingback(15):
)
λ0 = E [ λ1 (As + p1 s)] + E [
s
(λ0
0
1
(·)
-
2
B
(·) 1 -ξ
1 -ξ
-λ1 s) p1 s] + (21)
1
0 1 B(C(C1 B)
0 B
1 -ξ
+ A-F
1
(·)
0
A+
p
B
ψ
W
hen
wear
einth
eove
rinve
stme
ntca
se,ψ
isne
gativ
ean
dpos
itive
othe
rwis
e.
Hen
ce,fr
2
B
+1 1
ξ U
+A
(
C
2
B
U) U
isafunctionofk
)U
- U (C2 B
(C2 B
2
B
+1
1 -ξ
U
omei
ther(
20)o
r
(21)i
tisno
tobvi
ous
whet
hert
hepl
ann
erisg
oingt
ocho
osea
klarg
eror
smal
lerth
anint
hed
ecen
traliz
ed
mar
ket.
Thei
ntuiti
onfo
rthis
resul
tsist
hatk,
andt
heco
rrect
ionte
rmfo
rthe
cons
train
ed
plan
nerd
epe
ndso
nk-k
1(k0)
.
Eve
nifw
eare
inthe
situa
tion
with
overi
nves
tme
nt,it
may
beth
ecas
etha
t an
incre
ase
in
kincr
ease
s
end
oge
nous
ly
k(·)
mor
e
than
prop
ortio
nally
,
redu
cing
the
fire
sale
and
crea
ting
thed
esire
dins
uran
cee
ffect
thro
ugh
price
s.
5
U
n
i
q
u
e
n
e
s
s
I
sh
o
w
ho
w
to
fin
d
su
ffi
ci
en
t
co
nd
iti
on
s
for
un
iq
ue
ne
ss
in
th
e
de
ce
ntr
ali
ze
d
pr
ob
le
m.
Fo
r
si
m
pli
cit
y,
I
fo
cu
s
on
th
e
si
m
pl
es
t
ca
se
de
sc
rib
ed
in
thi
s
ap
pe
nd
ix,
wi
th
on
ly
tw
o
pe
rio
ds
,
on
e
st
at
e
an
d
no
t
pri
ce
de
pe
nd
en
t
m
ar
gi
ns
.
Si
mi
lar
re
su
lts
ca
nb
efi
nd
int
he
m
or
eg
en
er
al
ca
se
s.
1
= e0 -q0a+ [ (λ0 + a0
p1
+ A] k0
= e1
(λ1
0
0
0+
C
0
p0k ) (22) C) (23) a=0 (η) (24) a0
0
)
bi
nd
s,
th
e
w
or
st
ca
se
in
pri
nc
ipl
e
to
sh
o
w
un
iq
ue
ne
ss
,
th
e
eq
uil
ibr
iu
m
ca
pit
al
ch
oi
ce
k0i
sd
et
er
mi
ne
db
y:
e0+
1 ξ0 F0 (k0) k0 -k0 1 -0
11
U
Assuming that a
0
=
1ξ
1 0
1 ξ F0 (k0)
= 1 ξ0 p1k
= (26)0
0(νk
(ν0) (25)
p
=U
1
e+[
F
(k ) + A] k
1 -ξ p1k
A+ F
(k )
1 -ξ F
(k )
Thelefthandsidederivativewithrespecttok0
(·) 1 <01 -ξ F (k0) 0
0
U
0
0
0
isgivenby:
<0
-U
(·)
<0
1 -ξ F (k
>0
0
)
Thisshowsthatthelefthandsidein(26)isincreasingink
)
]
in(26)isgivenby:
1
A+ ξ[ F (k ) + F (k0)] + U (·) ξ (k )
k0
F
,asdepictedinfigure5inthetext.
Therighthandsidederivativewithrespecttok
U<0
(·) A
( )
+ ξF
k
0
Thistermmustbepositive0
0
Thisshowsthatasufficientconditionfortherighthandsidein(26)toben
egativeisthat:
( )] >0
(k ) k0 +
k 0
F
A+ ξ[ FIn practical terms this implies that F
0
1
2
cannot be too large,
this is, small changes in the amount of fire sold assets cannot
depress prices too quickly. Once kis determined, the rest of the
variables are uniquely determinedtoo.
Ananalogousapproachcanbeusedtoderiveconditionsfortheconst
rainedplanner’sproblem. Usingthe
samereasoning,wecanalsoprovidesufficientconditionsforuniquenes
sforthemoregeneralcasesdiscussed inthepaper.
6 Illustratingthelackofamplification
Figure1clearlyshowshowthetwoperiodformulationlacksanykindof
amplificationinthebadstate. When
theamountofcapitalsoldisfixedinthebadstate,astheinelasticsupply
ofcapitalkrepresents,thereisno roomforamplification.
However,whenthesupplyofcapitalbytheentrepreneursisdecreasin
ginprice,there
isroomforspiralsandcyclesinwhichlowpricesinducemoresalesofas
sets,whichdepresspricesandsoon.
ObservethatIhaveassumedawaymultiplicityproblems,whichcanpo
0
( +
k
F
) (
k 1
k
-ξ
[
tentiallyexistintheseenvironments.
Myanalysiscanbeeasilyadaptedtothosesituationsbyusinglocalarg
uments.
p1B
p1B
= F (·)
Supply of capital
Capital sold 0k0
Figure1: Showingthelackofamplification.
13