Information Externalities, Funding Liquidity, and Fire Sales

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Information Externalities, Funding Liquidity, and Fire Sales Levent Altinoglu and Jin-Wook Chang 2022-052 Please cite this paper as: Altinoglu,Levent,andJin-WookChang(2022). “InformationExternalities,FundingLiquidity,andFireSales,”FinanceandEconomicsDiscussionSeries2022-052. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2022.052. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Information Externalities, Funding Liquidity, and Fire Sales∗ Levent Altinoglu† Jin-Wook Chang‡ July 22, 2022 Abstract We develop a theory of learning in a model of fire sales and collateralized debt to study howbeliefsaboutfundamentalsareshapedbymarketconditions. Agentsexchangeshort-term debt contracts to invest in a long-term risky asset, and receive shocks to the opportunity cost offunds(costshocks)andnewsaboutthefundamentaloftheasset,bothofwhichareprivate information. Asset prices play a dual role of clearing markets and conveying agents’ private information, but markets are informationally inefficient: Agents can partially, but never fully, infertheircounterparties’privateinformationfromassetprices. Theinformationalinefficiency ofmarketsismoreacutewhenliquidityconditionsareespeciallytightorloose,asthisimpairs ability of prices to reveal private information about fundamentals. As a result, beliefs about fundamentalsareshapedendogenouslybycostshockswhichareorthogonaltofundamentals, leadingtosociallycostlyboomsandbustsinassetprices. Theequilibriumisconstrainedinefficient due to an information externality in which agents do not internalize how their choices affecttheinformationsetofotheragents. Interventionsinfundingmarketscanstabilizeasset pricesbyalteringperceptionsofrisk. JELclassification: D52,D53,E44,G28 Keywords: beliefs,learning,firesales,liquidity,assetprices,informationasymmetry ∗Theviewsexpressedinthispaperaresolelytheresponsibilityoftheauthorsandshouldnotbeinterpretedasthe viewsoftheBoardofGovernorsoftheFederalReserveSystemorofanyoneelseassociatedwiththeFederalReserve System. †BoardofGovernorsoftheFederalReserveSystem. Email: levent.altinoglu@gmail.com ‡BoardofGovernorsoftheFederalReserveSystem. Email: jin-wook.chang@frb.gov. 1

1. Introduction Financial markets are inherently unstable and prone to episodes of panic. An important function of financial markets is aggregating private information about financial assets and facilitating price discovery. Yetintimesoffinancialstress,whenperhapsinformationmattersthemost,thisfunction seems to break down, with sharp and temporary increases in uncertainty and excessive pessimism about fundamentals.1 These episodes of financial panic are characterized by sudden and sharp declines in asset prices and a rise in risk premia, leading to credit and liquidity freezes which may comeatgreatsocialcosts. Moreover,alargeliteraturehasarguedthatassetpricebooms,inwhich spreads are compressed and valuations are stretched, are in part driven by investor exuberance, in which beliefs or expectations about asset returns become divorced from underlying fundamentals. Therefore,anunderstandingofhowbeliefsareshapedoverthefinancialcycleisimportanttobetter understandthenatureoffinancialpanicsandprovideappropriatepolicyrecommendations. The literature has made substantial progress in understanding the role of beliefs in financial markets using models based on information and learning (Grossman and Stiglitz, 1980; Stiglitz, 1981;MorrisandShin,2003;GortonandMetrick,2012;BabusandKondor,2018),sentimentsand sunspots(DiamondandDybvig,1983;GoldsteinandPauzner,2005;Allenetal.,2006;Angeletos andLa’O,2013),deviationsfromrationalexpectations(Brunnermeieretal.,2014;Barberis,2018), among others. A more recent literature has sought to shed light how incomplete risk markets and information production interact (Gorton and Ordonez, 2014; Dang et al., 2020; Asriyan et al., 2021). In a similar spirit, we develop a model to understand how liquidity conditions in funding andassetmarketsaffectbeliefsaboutfundamentals,andtherolethatthisplaysinassetboomsand busts. Our model features collateralized debt, learning, and fire sales. A borrower issues short-term collateralized debt to a lender in order to finance investment in a long-term risky asset. Before the asset matures, agents are subject to an idiosyncratic shock to the opportunity cost of funds (cost shock) and may receive a private signal about the asset’s future return. Both the cost shock and thesignalreceivedbyanagentareprivateinformation. Inresponsetotheseshocks,agentsupdate theirbeliefs,exchangenewdebtcontracts,andcanadjusttheirportfoliosbeforetheassetmatures. Thepriceofdebtandthepriceoftheasseteachplayadualrole: theyclearthemarketfordebtand the asset, and they convey the lender and borrower’s private information, respectively. As a result, agents can learn about the news that the other agent received by observing the price of debt or theasset. However,financialmarketsareinformationallyinefficient: Anagentcanneverperfectly infertheextenttowhichthesepricesaredrivenbytheotheragent’sprivatecostshockversusnews 1Forexample,theseverityoftheliquidityfreezesindebtandmoneymarketsin2008and2019suggestthatthese episodesweredrivenatleastinpartbytemporarychangesintheperceivedriskinessoftheunderlyingassets. 2

abouttheasset.2 This environment leads to the key mechanism of the model: Beliefs about fundamentals are endogenouslyshapedbytheavailabilityofliquidityinfundingandassetmarkets. Namely,shocks to the opportunity cost of funds which are orthogonal to the asset’s fundamental affect the price of debt and the asset, and therefore affect other agents’ beliefs about the fundamental. Hence, the ability of prices to aggregate private information about fundamentals may be impaired by funding illiquidity,orbyanabundanceofliquidity. Moreover,thisinformationalinefficiencymaybemore acute whenliquidity conditionsare especially tightor loose, asthis impairs theability of pricesto revealprivateinformationaboutfundamentals. Thismaycauseagents’beliefsaboutthequalityof assets to become systematically divorced from fundamentals in a manner which generates instability. As a result, liquidity conditions which are orthogonal to fundamentals can lead to socially costlyassetpriceboomsandbuststhroughtheevolutionofbeliefsaboutfundamentals. Our paper makes three contributions. First, we show that liquidity shocks which are orthogonal to asset fundamentals can endogenously shape agents’ beliefs about fundamentals. That is, pessimism and optimism about the fundamental value of assets arises endogenously in our model asaresultoftheavailabilityofliquidityinfundingandassetmarketsinamannerwhichcancause beliefs to become divorced from fundamentals. Second, we show that funding liquidity, market liquidity, and beliefs may interact and lead to asset price booms and busts. In particular, relatively looseliquidityconditionscanleadagentstobecomeoverlyoptimisticaboutthefundamentalvalue of an asset (relative to an appropriately defined counterfactual), leading to overinvestment in the asset and excess losses in bad states of the world. In contrast, tight liquidity conditions can cause agents to be overly pessimistic about asset fundamentals, leading to socially costly fire sales. In thisway,themodelshedslightontheinherentinstabilityoffinancialmarkets. Our normative contribution is to show that the competitive equilibrium is generically constrained inefficient due to the presence of a new externality we term the ‘information externality’, inwhichagentsdonotinternalizehowtheirdecisionsaffecttheinformationsetandbeliefsofother agents, in addition to a more standard pecuniary externality. We characterize optimal policy and show that government interventions in financial markets aimed at tightening funding liquidity ex ante (for example, through regulations which reduce lending in certain asset classes) or loosening funding liquidity ex post (for example, through asset purchases or lender-of-last-resort facilities), may work to stabilize in part through the new channel in our model: the effect of liquidity conditionsonagents’beliefsaboutfundamentals. The mechanism in our model derives from the informational inefficiency of financial markets, 2The underlying assumption is that agents can neither observe one another’s private information, nor trade securities contingent on private information. This assumption is a stand-in for the various frictions, which imply that financialmarketsmaynotbeinformationallyefficientandriskmarketsareincomplete. 3

which itself stems in part from an absence of complete risk markets. Indeed, an old literature has shown that prices may reveal information only imperfectly in the absence of complete risk markets.3 Inourpaper,weshowhowtheinformationalinefficiencyoffinancialmarketsmayvary according to liquidity conditions, and its implications for asset price booms and busts. Moreover, our paper contributes to the debate about the central bank interventions in securities markets in recent decades (Chen et al., 2020). In particular, our model suggests that, by acting as a ‘dealer of last resort’, a central bank can mitigate the informational inefficiency of markets during episodes offinancialturmoilbyreopeningderivativesmarketswhichwouldotherwiseshutdown. Inourmodel,therearethreeperiods—dates0,1,and2—andtworisk-neutralagents—alender and a borrower. There is a consumption good, which can be stored freely as cash. The borrower hasaccesstoariskyassetatdate0,whichhasareturnatdate2subjecttoashock,andhasconstant returns-to-scale. New risky assets cannot be created after date 0. The lender cannot hold the risky asset directly, which creates gains from trade between the agents. The borrower can issue oneperiod debt to the lender in a competitive market, which is secured by its holdings of the risky asset.4 Agentshaveacommonpriorbeliefabouttheasset’sreturn—thatis,thefundamentalvalue oftheasset. At date 1, the borrower must either repay its date-0 debt or default. Since the return on the borrower’s holdings of the risky asset are not realized until date 2, the borrower can finance the paymentofitsdebtatdate1outofitscashholdingsorbyraisingnewdebtinacompetitivemarket. Under the latter option, the borrower effectively rolls over a portion of its date 0 debt at date 1, potentiallyundernewterms,dependingontheequilibriumpriceofdebtatdate1.5 Simultaneouslyatdate1,agentsmayreceiveprivatesignals(news)abouttheriskyasset’sdate 2 return. In addition, agents may receive an idiosyncratic shock to their opportunity cost of funds atdate1,acostshock,whichisuncorrelatedwiththeassetreturn. Foreaseofexposition,webegin with the case in which only the lender receives a cost shock and news about the risky asset, while theborrowerreceivesneither,andwediscussothercaseslater. Boththecostshocksandsignalsareprivateinformationandcannotbeverifiedbyotheragents. The willingness of the lender to lend at date 1 (the price of new debt at date 1) is affected by both the cost shocks and private signal: Cost shocks affect the willingness to lend by altering the lender’s opportunity cost of lending, while the private signal alters the lender’s beliefs about the fundamental value of the asset, which serves as collateral for the loan. Therefore, although the 3Forexample,Stiglitz(1981)showedthatintheabsenceofcompleteriskmarket,pricesmustnotonlyclearmarketsandaggregateinformation,butalsoallocaterisk. Asaresult,assetpricesmayrelayinformationonlyimperfectly. 4Althoughthelendercannotholdtheriskyassetdirectly,itcaresaboutitsreturnsinceitbacksitsholdingsofthe borrower’sdebt. 5The assumption that debt is one-period is meant to capture that, empirically, long-term assets are very often financedusingshort-termdebtorcontracts,whicharesubjecttochangingtermsviamarginrequirementsorcovenants. 4

borrower cannot observe the lender’s private information directly, the market price of date 1 debt is informative of this information through relationship between the lender’s liquidity needs and beliefs and the market price of date 1 debt. Agents are Bayesian, have a common prior belief at date 0 about the asset return, and update their common prior beliefs about the asset’s date 2 return atdate1basedonthevariablestheyobserveandtheirknowledgeofthedistributionofsignalsand costshocks. Inturn,agentsadjusttheirportfoliosatdate1optimallygiventheirposteriorbeliefs. The borrower updates its prior beliefs about the fundamental value of the asset given its observation of the market price of debt at date 1 according to Bayes’ Rule. Namely, the borrower computes the likelihood of all possible realizations of cost shocks and signals to the lender consistent with the observed value of the date 1 market price of debt, based on its knowledge of the distributions the shocks and signals, and forms an expectation over the lender’s private signal accordingly. Importantly,however,theborrowercannotperfectlydisentanglehowthemarketpriceofdate1 debtreflectsthelender’sprivatesignalversusthelender’scostshock. This‘identificationproblem’ arisesbecause,inequilibrium,onlyoneoftheborrower’sobservables—themarketpriceofdate1 debt—is informative about either of the two components of the lender’s private information. That is,theborrowerhasaninsufficientnumberofobservablestodisentanglethelender’sprivatesignal and cost shock. As a result, the equilibrium price of debt serves as a noisy signal to the borrower about the lender’s private signal, where the noise is introduced by the lender’s idiosyncratic cost shock. Thus,theborrowerfacestwolayersofuncertaintyabouttheasset’sreturn: themarketprice of date 1 debt is a noisy signal to the borrower about the lender’s private signal, while this signal itselfisanoisysignalabouttheasset’sreturn.6 The identification problem implies that liquidity conditions in funding markets affect perceptions about fundamentals. Loose liquidity conditions in funding markets, given by a high market price of date 1 debt, cause the borrower to become more optimistic about the quality of the asset. This happens regardless of whether the lender is willing to lend because of low cost shock or because it received positive news about the project. In either case, the borrower cannot identify how muchoflender’sdemandfordebtisdrivenbythelender’sbeliefs,ascribesahigherprobabilityto the lender having received positive news, and becomes more optimistic about the asset’s quality as a result. In contrast, tight funding liquidity, given by a low market price of date 1 debt, causes the borrower to become more pessimistic about the quality of the asset, since it ascribes a higher probability that the lender’s low demand for debt is driven by the lender having received negative news about the asset. Thus, pessimism and optimism about the fundamental value of the risky 6Thepresenceoftwolayersofuncertaintyplaysacrucialroleinourmodel,asitgivesrisetoendogenousbelief disagreement,anditimpliesthatliquidityconditionswillaffecttheborrower’sbeliefsaboutthefundamentalvalueof theriskyasset. 5

asset emerge endogenously as a result of funding liquidity—that is, the lender’s demand for debt atdate1. Themodelfeaturesthepossibilityoffiresalesoftheriskyassetatdate1,similartoLorenzoni (2008). In particular, the date 1 equilibrium features two regimes, which we call ‘normal times’ and‘firesales’. Innormaltimes,whenlender’sdemandfordebtissufficientlyhigh(whichoccurs when the cost shock to the lender are not too large and the signal is not too bad), the borrower has relatively optimistic beliefs about the risky asset’s return. As a result, the price of the risky asset is high andthe borrower holds all ofthe risky asset. Atdate 1, the borrower uses itscash holdings to repay any of its date 0 obligations which are not rolled over, and does not liquidate any of its holdingsoftheriskyasset. In the fire sale regime at date 1, when the lender’s demand for debt is sufficiently low (which occurs when the lender’s cost shock is sufficiently bad or when the lender receives sufficiently bad news about the asset), the borrower has relatively pessimistic beliefs about the risky asset’s return. As a result, the borrower reduces its holdings of the risky asset by liquidating the asset to an outside traditional sector, who has a lower willingness to pay for the asset. Importantly, the nature of fire sales in our setting differs from that in much of the literature on fire sales (e.g. Lorenzoni (2008)): In response to tight funding liquidity conditions, the borrower liquidates the assetnotsimplybecauseofitsneedforliquidfunds,butratherbecausetheborrowerendogenously becomesmorepessimisticaboutthefundamentalvalueoftheasset. The model features asset price booms and busts due to an interaction between funding liquidity, beliefs, and market liquidity. In particular, relatively loose liquidity conditions in the funding market (i.e. a high market price of date 1 debt) leads the borrower to become overly optimistic about thefundamental value of anasset (relative toan appropriately definedcounterfactual), leading to over-investment in the asset. This over-investment reflects a high asset price, and results in the borrower bearing excess losses in bad states of the world. In contrast, tight liquidity conditions in the funding market causes the borrower to be overly pessimistic about about the asset fundamentalatdate1. Asaresult,theborrowermayliquidateitsholdingsoftheriskyassettothe traditional sector, depressing the price of the asset below the value justified by its fundamentals (relative to a benchmark economy where all information is publicly observed). Thus, the model sheds light on the role that liquidity plays in asset price booms and busts, and how agents beliefs evolveendogenouslyduringtheseepisodes. Wealsoperformcounterfactualexercisestoshedlightonhowtheinformationspilloversinour modelaffectthelikelihoodandseverityoffiresaleepisodesinresponsetodifferentshocks. Inparticular,we showthat theinformationspillovers generatedinour modelamplifythe likelihoodand severityoffiresalesdrivenbyadverseliquidityshockstothelender,andreducethelikelihoodand severity of fire sales driven by bad news received by the lender, relative to a benchmark economy 6

in which all information is commonly observed. Thus, the information spillovers may stabilize or destabilizefinancialmarketsdependingontheshock. We then analyze the normative implications of the model and show that the competitive equilibrium is constrained inefficient due to the presence of two externalities. In addition to a standard pecuniary externality along the lines of Lorenzoni (2008) and Dávila and Korinek (2018), the model features a new ‘information externality’ in which agents do not internalize how their decisions affect the information sets, and therefore the beliefs, of other agents. For instance, in choosing how much to lend to the borrower at a given price of debt, the lender does not internalize how its choice affects the borrower’s perceptions of the asset’s fundamental value due to the identification problem that the borrower faces. A social planner, who is subject to the same constraints as private agents—and who, accordingly, does not observe agents’ private information— canengineeraParetoimprovement. Agovernment,whichdoesnotobserveagents’privateinformation,canimplementaParetoimprovementusinginterventionsinfundingandassetmarkets. Inthemodel,thepricesthataggregate privateinformationaboutfundamentalsmaybeimpairedduringepisodesoffundingilliquidityand episodes when funding liquidity is abundant. This causes agents’ beliefs to deviate systematically from fundamentals in a manner which generates instability. Government intervention in funding and asset markets, such as asset purchases or taxes on lending, can stabilize markets by exploiting the identification problem faced by agents and engineering a change in agents’ perceptions of fundamentals. Our model thus shows that, in practice, the government interventions often implemented to stabilize financial markets, such as liquidity facilities and asset purchases, may operate through this additional channel not previously discussed in the literature. Indeed, policymakers have frequently cited the effect of the their interventions on investor confidence as a stabilizing force. We also show that the government can also reduce the inefficiencies associated with the information externality using auctions of derivatives of the risky asset (such as put or call options). In equilibrium, only better-informed agents trade these securities, and therefore the market prices of thesesecuritiespubliclyrevealadditionalinformationaboutagents’privatebeliefsabouttheunderlyingasset. Essentially,byintroducingnewmarketsforderivativesoftheriskyasset,thesepolicies mitigatetheidentificationproblemfacedbyagentsandimprovetheinformationaggregationoffinancial markets. This is similar in spirit to the social value of public information, articulated in Allenetal.(2006),althoughthemechanismandsourceofinefficiencydifferinourpaper. 7

1.1. Related Literature One of our main contributions is linking the imperfect information aggregation in Grossman and Stiglitz(1980)anditsamplificationthroughthefeedbackbetweenmarketandfundingliquidityin Brunnermeier and Pedersen (2009). Such interaction results in information externality on top of pecuniaryexternalitiesfromfiresalesasinDávilaandKorinek(2018). Gale and Yorulmazer (2013) show that the hoarding of liquidity can shift the equilibrium allocation drastically. Similarly, we show that the liquidity hoarding due to asymmetric information can cause a huge decline in efficiency of the equilibrium allocation, but in the context of a market ofcollateralizeddebtcontracts. Garcia-Macia and Villacorta (2022) show how information frictions between banks can cause freezes in the interbank market and liquidity hoarding. Firms need liquidity for short-term investment opportunities. Banks have heterogeneous lending efficiency, which is private information. Interbank market facilitates the flow of funds from less efficient to more efficient banks, which ultimately lend to firms. However, when bank profitability is too low, interbank trade may not be incentive compatible due to information frictions, causing the interbank market freeze. Under interbankmarketfreeze,firmshaveincentivestohoardliquidassetstobeabletoself-financetheir liquidity needs. This liquidity hoarding reduces the demand for bank loans, which lowers bank profitabilityandmakingtheinterbankmarketfreezemorelikely. Becauseofsuchafeedbackloop, therearemultipleself-fulfillingequilibriaintheirmodel. Unlike the model in Garcia-Macia and Villacorta (2022), our information friction is more direct, as the information asymmetry is between the borrower, who is also the investor, and the lender, who holds private information and collateral. Also, this information is about the fundamentals of the investment opportunity (asset), so there is no information friction that is amplified by coordination failure or multiple equilibria. Therefore, our model focuses on the direct effect of information asymmetry and imperfect learning through indirect signals, which can generate inefficiencyinequilibrium. Astrandofliteratureontheassetmisallocationduetodownward-slopingdemandcurvesforassetsisbasedonShleiferandVishny(1992)andKiyotakiandMoore(1997),followedbyLorenzoni (2008). Under the standard models of this literature, assets have different productivity depending on who holds them. If agents with higher productivity are financially constrained and sell assets toagentswithlowerproductivity,thislowersthemarginalproductandtheequilibriumprice. This price decline would be amplified if agents have large leverage and more financially constrained. Agents don’t internalize the pecuniary externality that falling asset prices impose on financially constrained agents. Therefore, collateral constraints can lead to an amplification of shocks and volatilerealactivity. Theinformationexternalityinourmodelamplifiesthepecuniaryexternality. 8

Kurlat (2016) proposes a model of fire sales with asymmetric information and difference in potential buyer’s expertise in evaluating assets. Buyers with better expertise in detecting bad assets can refrain from buying such assets. Under certain conditions, the equilibrium features a downward-slopingrelationshipbetweenassetsalesandassetprices. Thisisbecausewhenthesale volume is high, the market clearing condition requires less-expert buyers to buy the asset, so the price has to fall. Our model also has a similar mechanism, with which financially constrained borrowers have to sell their assets to less productive agents. However, we separate the effect of asymmetricinformationandtheinefficientfiresales. IfthereisafeedbackmechanismasinKurlat (2016),theequilibriumwillhaveevenlargerfluctuationsinourmodel. BabusandKondor(2018)proposeamodelwithmarketclearingascompetitiveWalrasianauctioneer instead of separate decentralized platforms. Walrasian auctioneer in our model collects both the contract price and the appropriate amount of loans (or collateral) as well as the corresponding asset fire sales. Unlike in the model of Babus and Kondor (2018), we have a mixture distribution, binary state of the asset payoff and continuum of cost shocks, resembling the signal processing models. Also, our model incorporates collateral and simultaneous market clearing to understandtheinteractionbetweendebtmarketsandcollateralassetmarkets. OurpaperisalsorelatedtotheliteratureonleveragecyclesdevelopedbyGeanakoplos(1997), Geanakoplos (2003), Geanakoplos (2010), and Fostel and Geanakoplos (2015). We also use a general equilibrium model with heterogeneous agents and collateralized debt contracts following the literature. Unlike the models in this strand of literature, we highlight the role of asymmetric information in amplifying the leverage cycles. Also, our model has multiple states of uncertainty similartoSimsek(2013),whoproposedamodelwithacontinuumofstates. Ourmodelhasanother sourceofuncertaintyontopoftheassetreturn,whichistheshocktotheopportunitycostoffunds (cost shock). In our model, the mixture distribution of the asset return and the continuum of cost shock realization generate the identification problem in learning as well as more richer interaction betweenpricesandbeliefs. There is a large literature on the role of beliefs, sentiments, and learning in financial markets following Diamond and Dybvig (1983), who propose a model of financial fragility based on multiple equilibria from sunspots. Morris and Shin (2003), Goldstein and Pauzner (2005), Allen et al. (2006),andAngeletosandLa’O(2013)developthisideafurthertohowtheinformationandbeliefs areendogenouslydetermined,andhowfrictionsinlearningcouldpreventefficientinformationaggregation, leading to large swings and crises. Our model differs from many different models in thisliteraturebyprovidingtheinteractionbetweentheformationofbeliefsonfundamentalsofthe assetbyfinancialconditionsandfiresalesoftheasset. Ourmodellingoffiresalesasachangeoftheinformationregimeismotivatedbytheliterature on information production in credit markets as in Gorton and Ordonez (2014), in which lenders 9

could produce information about collateral. Dang et al. (2020) show how collateralized debt can be optimal by alleviating problems of asymmetric information with information-insensitive debt. Dang et al. (2020) argue that short-term debt is designed to provide short-term stores of value by designing the debt such that it is not profitable for any agent to produce private information about the assets (collateral) backing the debt. A financial crisis is a switch of regime from informationinsensitive debt regime to information-sensitive debt regime in which agents produce information aboutthecollateral. Asriyanetal.(2021)alsodevelopamodelwithcostlyinformationproduction and collateral. They show that high collateral values will crowd out information production and information on existing projects gets depleted. As a result, booms driven by collateral end in deep crises and slow recoveries. The crises caused by asymmetric information in this literature are similar to the crises in our model. However, the main driver of the crises in our model is the interaction between cost shocks and endogenous prices of debt and asset instead of information production. Moreover, the information on collateral and projects are interlinked in our model, as borrowerspledgetheirassetreturndirectlyascollateral. Finally, our paper is related to the run on repo markets. The literature on repo runs by Gorton and Metrick (2012), Copeland et al. (2014), Martin et al. (2014), and Infante and Vardoulakis (2021) document and analyze the repo or collateral run dynamics. In most of the repo markets, in particular the repo markets analyzed in the literature, counterparty risks are minimal. Therefore, thefiresalesdynamicsinourmodelwouldbeoneofthemorerelevantmechanismstoanalyzethe fragilityoftherepomarkets. 2. Model Setup Therearethreedatesindexedbyt ∈{0,1,2}. ThereareN =3typesofagents: lenderL,borrower B, and traditional sector T.7 There is a continuum of agents of mass 1 for each type. Agents can observe each other’s type. There is one consumption good (cash) which is storable that implies one unit of consumption good at t can be transformed into τi units of consumption good at t+1. t Foragenti,τi willcontroleachagentsopportunitycostoffundsatt. Forsimplicity,assumeτi =1 t 0 for each i. Each agent gets utility from date 2 consumption,Ci, according toU(Ci)=Ci.8 Each 2 i 2 2 agentisendowedatdate0withei oftheconsumptiongood. 0 There is one risky asset a in positive net supply A with constant returns-to-scale.9 An external un-modeled agent holds all the assets, sells the assets to B, and disappears at the end of date 0. Theassetatransformsconsumptiongoodsinvestedatdate0intogoodsatdate2,withagrossrate 7Thetraditionalsectorisintroducedinordertohaveanotionoffiresales,similartoLorenzoni(2008). 8Therefore,agentsarerisk-neutralandsolvinganinvestmentproblem. 9We can consider this as investment opportunity given by Bertrand competition in the economy with capacity constraintofthepositivenetsupply. 10

of return of R. The return on the risky asset is subject to an aggregate shock at date 2, where is R = R > 1 with ex ante probability (1−π) and R = R < 1 with probability π, where 0 < π < 1 and R<τB <R. We further assume that πR+(1−π)R=1 and τB =τL ≤1 to make investment 1 1 0 in the asset profitable, where τL is the expected value of τL. Uncertainty is fully resolved for all 0 1 agentsatdate2. Importantly,investmentinriskyassetscanbeinitiatedonlyatdate0. Moreover,riskyassetsare illiquidinthattheycannotbeconvertedintotheconsumptiongoodatdate1. Ineachperiod,there isaspotmarketfortheriskyassetinwhichtheborrowerandthetraditionalsectorcanparticipate, where p denotesthepriceoftheriskyassetattimet. t Markets for the risky asset are segmented: Only the borrower can invest in the risky asset at date 0. And only the borrower or the traditional sector can hold the risky asset at any date. The lender, by contrast, can only indirectly invest in risky assets by lending cash (consumption good) at date 0. The traditional sector has a date 1 reduced form inverse demand function of F(cid:48)(aT), 1 where aT is its holdings of the risky asset at date 1. The traditional sector demand function is 1 continuously differentiable with F(cid:48) >0 and F(cid:48)(cid:48) <0. We focus on the case in which the traditional sector always values the risky asset less than other agents do a priori – that is, the expected return of the asset is EB[R]/τB >F(cid:48)(0). Because the risky asset is illiquid at date 1, in equilibrium, the 0 1 onlywayfortheborrowertoconvertitsholdingsoftheriskyassettotheconsumptiongoodatdate 1istosellittothetraditionalsectorinthespotmarketfortheasset. Costshocksandprivatesignals The lender’s opportunity cost of funds τL is a random variable which is drawn at date 1 from 1 a continuous and smooth distribution with probability density function λ (τ) with mean τL and T 0 support[τL,τ¯L]. Wefurtherassumethatλ (τ)isdecreasingin|τ−τL|. Werefertotherealization T 0 ofchangesinτL asthecostshocktoL. ThecostshockisindependenttoR∈ (cid:8) R,R (cid:9) . 1 Agents have a common prior belief about R, where R = R with probability π . Each agent 0 updatestheirbeliefsπL,πBthatR=Rinresponsetonewinformationatdate1accordingtoBayes’ 1 1 rule. At date 1, L gets a private noisy signal about R. Let the signal be denoted as sL = R+ε, 1 where ε is drawn from a continuous and smooth distribution with cumulative density function Λ (ε), which has mean zero and variance σ2, and probability density function λ (ε)with support ε ε ε (−∞,∞). Weassumethatλ (ε)isdecreasingin|ε|—thatis,λ (ε)issingle-peaked. ε ε Both τL and sL are neither observable nor verifiable to the borrower and cannot be contracted 1 1 upondirectly. 11

Contractingenvironment Market segmentation implies that there are gains to trading financial contracts. More precisely, L cannot hold the risky asset at any point, but can gain exposure to the risky asset by lending to B through the use of financial contracts. We assume that, although the risky asset matures after two periods, the borrower finances its holdings of the risky asset using one-period collateralized debt contracts. By financing long-term risky assets with short-term debt, the borrower is exposed to liquidityriskatdate1. Weformalizethisenvironmentandthenatureofliquidityriskbelow. Timeline The timeline is as follows. At date 0, agents simultaneously enter into one-period contracts between date 0 and date 1, and make their investment decisions. At date 1, agents receive the cost shocks and signals about the quality of the risky asset. After the realization of the cost shocks and signals,agentssimultaneouslysettlethedate0contractsandenterintoanotherone-periodcontract at date 1. At date 2, the return on the risky asset is realized, the date 1 contracts are settled, and agentsconsume. Financialcontracts Formally, a date t contract is given by (c , f ), which stipulates a transfer at date t of the risky t t asset of size c from the borrower to the lender as collateral with a promise for the borrower to t repurchase the collateral at date t+1 at a unit price of f , normalized to f = 1. (That is, the t t borrower repays the lender f c =c units of the consumption good at datet+1.) We assume that t t t the collateral on the loan is held by some outside custodian, not modeled explicitly, who either returns the collateral to the borrower if the contract is honored or sells it and pays the proceeds to the lender if the borrower defaults. This ensures that the lender accepts the asset as collateral for theloandespitenotbeingabletoholdtheassetitself. These contracts are traded in a competitive market, where q denotes the market price of the t contractatt =0,1. Therefore,thesizeoftheloanfromthelendertotheborroweratt—thatis,the quantity of the consumption good loaned at date t— is given by q c . At date t+1, the borrower t t decides whether to default on the contract and forgo the collateral or not. If the borrower chooses todefaultonthecontractatdatet+1,thevalueofthiscollateralc p istransferredtothelender. t t Note that both date 0 and date 1 contracts are one-period contracts and involve a repayment of c at date t+1. One may wonder why the borrower would be willing to enter into the date t 0 contract in the first place, given that this will require the borrower to repay the lender c units 0 of the consumption good at date 1 despite the fact that the borrower has no income at date 1. In equilibrium, the borrower will be willing to borrow at date 0 because it anticipates that it can 12

financethisrepaymentc (atleastpartially)byrefinancing(i.e. rollingover)itsdebtusingthedate 0 1contract,albeitunderdifferentterms. Onecaninterpretthiscontractingenvironmentasinvolving long-term(two-period)debtwhichisrenegotiatedintheintermediateperiod. Because the borrower must refinance its date 0 debt under potentially different terms, this renegotiation exposes the borrower to liquidity risk at date 1. If the loan amount q c of the new 1 1 contract is more than the payment amount of initial loan c in equilibrium, then the lender lends 0 moreconsumptiongoodstotheborrowerforthesameamountofcollateral. Incontrast,ifthesize of theloan under thedate 1 contractq c is lessthan the thepayment amount ofinitial loan under 1 1 the date-0 contract c , then the borrower must either pay the difference in cash at date 1 or put up 0 more collateral to borrow more at date 1.10 In either case, the borrower has to pay the difference c −q c in cash. In equilibrium, the borrower will try to pay the debt in full unless the borrower 0 1 1 exhausts all of cash and asset holdings because the lender can require further payments from the borrower’scashholdingsatdate1. Sincetheborrower’sriskyassetsareilliquidatdate1andcannotbeconvertedtotheconsumption good, the borrower only has two means of repaying the debt at date 1: the borrower must either use its own cash holdings to reduce the size of the loan, or it must sell some of its holdings of the risky asset in the date 1 spot market for the asset. The only way for the borrower or the lender to convert the risky asset to cash in aggregate at date 1 is to sell it to the traditional sector. Because the traditional sector’s marginal valuation of the risky asset can be lower than that of the borrower,theborrowercanselltheassettothetraditionalsectorataloss. Inthissense,tightening fundingconstraintsmaybeassociatedwithendogenousfiresalesinwhichtheborrowerliquidates partofitsholdingsoftheriskyassetatacost. Note that the lender has full-recourse on the debt at date 1. Thus, the borrower will try to pay thedebtinfullatdate1unlessitexhaustsallofcashandassetholdings. However,atdate2,there is no further recourse to the borrower’s balance sheet as the borrower can simply walk away from the collateral. Under the date 1 terms of the contract, the borrower will decide whether to default onthecontractandforgothecollateralornot,thus,thepaymentfromtheborrowertothelenderat date2willbedeterminedendogenouslyasmin{p ,1},where1isthepromisedpaymentamount. 2 Wealsomakethefollowingassumptions: Assumption1. eBτB <A (cid:0) EB[R]−1 (cid:1) ,eL >AR/τL−eB,andeT >max aF(cid:48)(a). 0 1 0 0 0 0 a∈[0,A] The first part of the assumption ensures that borrowers cannot fund their asset purchases withoutdebt. Thesecondpartensuresthatlendersendowmentissufficientlylargetosatisfyborrowers’ demand. Finally,thethirdpartoftheassumptionensuresthatthetraditionalsectorhasenoughcash topurchaseanyarbitraryamountsoldtothemfollowingtheirinversedemand. Thisassumptionis 10Onecanconsiderthissituationasamargincall. 13

almost exactly the same as the assumptions in Simsek (2013) and Gottardi et al. (2019), serving thesamepurposes. Assumption2. F(cid:48)(A)≥EB[R]−eB/AandEB[R]/τB >F(cid:48)(0). 0 0 0 1 The first part of this assumption is to guarantee that the borrower can liquidate all their asset holdings A to the traditional sector at price of p = F(cid:48)(A). Then, the total amount of asset fire- 1 sales AF(cid:48)(A) would exceed the maximum borrowing amount AEB[R]−eB. Therefore, the total 0 0 fire-sales of the borrower at date 1 is enough to repay all the initial borrowing amount c at date 0 0.11 The second part of the assumption is formally reiterating the case we are focusing in which theborroweralwaysvaluestheassetmorethanthetraditionalsectordoesatdate0. Informationsets LetIi denotetheinformationsetofagenti∈{L,B}atdatet ∈{0,1,2}. Weassumethat,atdate0, t IB =IL and prior beliefs π are the same for both agents. In addition, the date 0 opportunity costs 0 0 0 of funds for each agent τB,τL are common knowledge. Both agents observe all prices and date 0 0 0 termsofthecontractc ,q . 0 0 Atdate1,lenderobservesτL andsL,buttheborrowerdoesnot. Bothagentsobservepricesand 1 1 terms of contract c ,q ,p . Therefore, the lender’s information set at date 1 is IL = (cid:8) Θ ,τL,sL(cid:9) , 1 1 1 1 1 1 1 while the borrower’s information set is IL = {Θ } where Θ denotes the set of all variables ob- 1 1 1 servable to all agents, which includes prices p ,q and the terms of the contract c . Agents make 1 1 1 decisions at each date conditional on their information set in that date. We denote agent i’s expectationofvariablex,conditionalonitsinformationsetattimedatet,asEi[x]≡E (cid:2) x|Ii(cid:3) . t t t 3. Lender’s Problem The (representative) lender behaves competitively. The lender solves a portfolio choice problem, deciding how much unit of collateral to lend, dL for each date t, and how much cash to hold, κL t t foreachdatet,subjecttoitsbudgetconstraint,takingprices p ,q andavailablecontractsasgiven. t t Lender’sportfoliochoiceproblematdate1 At date 1, the lender’s portfolio choice is to decide how much to lend versus how much cash to hold. Thelender’sopportunitycostoflendingisitsdate1marginalreturnofcash(whichissubject totheliquidityshock)τL,sincethisisthedate2returnthatthelendergetsonitsholdingsofcashat 1 11Thisistoavoidanypeskyissuesarisingwiththepossibilityofdate1defaultoftheborrowersintermsofviolating theirbudgetconstraint. Itcouldbepossiblethattheborrowerjustwantstoselleverythingbecauseitisnotprofitable torollovertheirdebt,butthatliquidationamount p Acanbelessthanq c . Then,thebudgetconstraintisviolated. 1 0 0 14

date 1. If the borrower cannot finance the promised payment to the lender out of its cash holdings or with new borrowing at date 1, then either the borrower has to liquidate some of its holdings of the risky asset to the traditional sector for cash or the lender seizes the collateral and liquidates it to the traditional sector. Both of these actions are payoff-equivalent, so we ignore the confiscation of collateral by the lender and instead assume that the borrower liquidates to the traditional sector inthiscase. Thelender’sproblematdate1istomaximizeitsexpectedutilityEL[CL]fromdate2consump- 1 2 tion by choosing its portfolio (cash versus a loan) subject to budget constraints at date 1 and date 2. Let dL denote the lender’s choice of how many units of loan to invest in (at price q ) at date 1. 1 1 (In equilibrium, the condition for market clearing for the loan will be dL =c if c >0. If c =0, 1 1 1 1 thendL>c ispossiblebecauseofthereasonwediscusslaterinthethelender’soptimaldecision. 1 1 Byassumption1,dL >0isalwayspossibleevenwhenc =0asweshowlater.) 1 1 Lender’s date 0 budget constraint At date 0, the lender allocates its cash endowment betweendate0cashholdingsandthedate0loantotheborrower. κ L + q dL ≤ eL 0 0 0 0 (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) date0cashholdings valueof date0loan date0cashendowment Lender’s date 1 budget constraint At date 1, the flow of payments received by the lender is simply q dL−q dL, the difference in the value of the loan under the contract terms agreed to at 0 0 1 1 date0minusthevalueoftheloanunderthenewcontractterms. κ L + q dL ≤ κ L + dL 1 1 1 0 0 (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) date1cashholdings valueof date1loan date0cashholdings valueof date0loan i.e. κ L−κ L ≤ dL−q dL 1 0 0 1 1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) netincreaseincashholdings netdecreaseinloan Lender’s date 2 budget constraint At date 2, the contract is settled or defaulted upon by the borrower, the lender earns a return from its cash holdings, and consumes. Define the date 2 proceedsfromtheloanasRd ≡min{R,1},whichaccountsforthepossibilityofdefault. 2 CL = τ L κ L + RddL 2 1 1 2 1 (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) date2consumption proceeds fromcashholdings proceeds fromdate1loan Lender’sproblematdate1 15

Taking as given the date 1 contract terms and prices q , and p , the lender decides how much 1 1 ofitsfundstoallocatetotheloanorcash. Thelender’soptimizationproblemis max EL(cid:2) CL(cid:3) 1 2 κL,dL 1 1 s.t. CL ≤τ L κ L+RddL 2 1 1 2 1 κ L−κ L ≤dL−q dL 1 0 0 1 1 withnon-negativityconstraints: κ L ≥0, dL ≥0. 1 1 Both budget constraints will bind at the optimum, so we can replace cash holdings κL using 1 thedate1budgetconstraint. CL =τ L(cid:0) κ L+dL−q dL(cid:1) +RddL 2 1 0 0 1 1 2 1 Let ξL and ξL denote the Lagrange multipliers on the respective non-negativity constraints. The d 1 κ1 Lagrangianforthelender’sdate1problemis (cid:104) (cid:16) (cid:17)(cid:105) LL =EL U τ L(cid:0) κ L+dL−q dL(cid:1) +RddL +ξ LdL+ξ L (cid:0) κ L+dL−q dL(cid:1) 1 1 L 1 0 0 1 1 2 1 d 1 1 κ1 0 0 1 1 Thelender’sfirst-ordercondition(FOC)fordL is 1 EL(cid:2) Rd(cid:3) +ξL 1 2 d q = 1. (1) 1 τL+ξL 1 κ1 This says that the lender chooses how much to lend to equalize the discounted marginal return from lending at date 1 to the discounted marginal return on its opportunity cost from holding cash atdate1. Wealsohavethetwocomplementaryslacknessconditions: ξ LdL =0 d 1 1 ξ L κ L =0 κ1 1 NotethatthereisnodL in(1)asthelender’sproblemislinear. Thelenderisindifferentacross 1 16

differentvaluesofdL aslongasthepriceofthecontractisequaltotheexpectedreturnoftheloans, 1 EL(cid:2) Rd(cid:3) /τL, the ratio of the discounted marginal return from lending at date 1 to the discounted 1 2 1 marginalreturnonitsopportunitycostfromholdingcashatdate1. Notethatassumption1ensures that the lender always holds a strictly positive quantity of cash in equilibrium at date 1, κL >0.12 1 Therefore, the lender’s Lagrange multiplier for the non-negativity constraint satisfies ξL = 0 in κ1 equilibrium. Moreover,assumption1ensuresthat,lendingtakesplaceinequilibriumsothatdL>0 1 andξL =0inequilibrium.13 Therefore,giventheseassumptions,thelender’soptimalitycondition d 1 (1)reducesinequilibriumto EL(cid:2) Rd(cid:3) q = 1 2 . (2) 1 τL 1 Thus, in equilibrium, the competitive spot price of the contract at date 1, q , will ensure that this 1 conditionholds.14 4. Borrower’s Problem The(representative)borroweriscompetitive. Atdate1,theborrowersolvesaportfoliochoiceand chooses how much to borrow, subject to its budget constraint, collateral constraint, and taking as given the price of the risky asset p and the price of the contract q . At date 1, borrowers cannot 1 1 initiatenewriskyassets. Moreover,borrowerscanexchangetheirextantholdingsoftheriskyasset to cash by selling it to the traditional sector, at the market price p and investing the proceeds in 1 cash for total date 2 return of p τB. Hence, the borrower’s opportunity cost of holding the risky 1 1 assetisgivenbyτB/p . 1 1 At date 1, the borrower must repay its date 0 debt by buying back from the lender the c units 0 of the risky asset that were posted at collateral. Recall that the repurchase price defined in the contract was f =1, so that the borrower must pay c units of the consumption good to the lender 0 0 12Thelender’soptimalityconditionimpliesthat,inequilibrium,thelenderisjustindifferentbetweenholdingcash and not, which implies that the lender’s Lagrange multiplier from the non-negativity constraint satisfies ξL =0 in κ1 equilibrium. Otherwise the lender’s expected discounted return from lending would exceed its opportunity cost of funds. 13Notethatassumption1ensuresthatthelender’sdate0endowmenteL islargeenoughtocoveralltheloanseven 0 whenthelenderismaximallyoptimisticaboutthequalityoftheriskyassetandbelievesR=Rwithprobability1. EL(cid:2) Rd(cid:3) 14Toseethisintuitively,supposethatq islessthantheexpecteddiscountedreturnofaloan,q < 1 2 . Then 1 1 τL 1 thelenderwouldwanttomaximizeitsholdingsoftheloansandholdnocash,sothatκL=0.Butassumption1ensured 1 EL(cid:2) Rd(cid:3) that κL >0, so that we have a contradiction. Suppose instead that we have q > 1 2 . Then, the lender would 1 1 τL 1 wanttominimizedL tonegativeinfinitywithoutthenon-negativityconstraintsothatthenon-negativityconstraintof 1 dL wouldbind,dL=0. Butassumption1ensuredthatdL>0inequilibrium,sothatweagainhaveacontradiction. 1 1 1 17

at date 1. However, the borrower obtains no date-1 cash flow from its holdings of the risky asset at date 1. Therefore, the borrower can finance this repayment of c in three ways: out of any cash 0 holdingsatdate1,byraisingnewdebtatdate1,orbyliquidatingtheriskyassetinthedate-1spot marketfortheriskyasset. The borrower can raise new debt at date 1 (i.e. refinance its date-0 debt) using the date-1 contract—that is, by posting c units of the risky asset as collateral in exchange for q c units of 1 1 1 theconsumptiongoodasaloan,whereq isthecompetitivespotpriceofthiscontract. Ifthedebt 1 raised at date 1 q c is insufficient to cover the repayment c the borrower must make at date 1, 1 1 0 this difference must be financed either out of the borrower’s cash holdings stored from date 0, or bysellingsomeportionofitsholdingsoftheriskyassetinthespotmarket. DefineκB astheamountofcashtheborrowercarriesintodate1fromdate0. Nowthisdecision 0 isrelevantonlyinthedate-0optimizationproblem. Forthedate-1problem,theborrowerconsiders itasapredeterminedamountofcashκB.15 0 Collateral constraint at date 1 Let aB denote the borrower’s total risky asset holdings 0 brought to date 1 from date 0, and c denote the borrower’s date 0 risky asset holdings used as 0 collateralatdate0. (Wewilllatershowthat,inequilibrium,c =aB.) LetaB denotetheborrower’s 0 0 1 totalriskyassetholdingschosenatdate1,andc denotetheamountoftheborrower’sdate1asset 1 holdings used as collateral at date 1. (Note that the amount of asset sold by the borrower at date 1 isthenaB−aB.) The‘collateralconstraint’isthenc ≤aB. 0 1 1 1 Borrower date 0 budget constraint At date 0, the borrower has a cash endowment and receives a cash loan from the lender, which it can allocate between date-0 cash holdings and investmentintheriskyasset. q c + eB ≥ p aB + κ B 0 0 0 0 0 0 (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) date0loan date0cashendowment valueof date0assetholdings date0cashholdings Borrower date 1 budget constraint At date 1, if the borrower defaults on its date-0 debt obligations, it loses its collateral and has only its cash holdings to find its portfolio choices at date 1. Inequilibrium,theborrowerneverdefaultsatdate1,becausethesellingallthecollateralwould be sufficient in repaying the debt by assumption 2. For this reason, we omit this case. In the event thattheborrowerdoesnotdefaultatdate1,itsdate1budgetconstraintis 15Wewillseeinthedate-0problemthattheborrowersfinancedtheirpaymenttopurchasec assetswiththeprice 0 p atdate0bytheircashendowmentseB andtheborrowingamountc p (1−m )=c q ,wherem isthemarginof 0 0 0 0 0 0 0 0 thecontract. Therefore,theleftovercashcanbedefinedasκB≡eB+c p (1−m )−c p =eB−c p m . 0 0 0 0 0 0 0 0 0 0 0 18

q c + p aB + κ B ≥ p aB + c + κ B 1 1 1 0 0 1 1 0 1 (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) date1loan valueof date0assetholdings date0cashholdings valueof date1assetholdings date0loan date1cashholdings i.e. q c −c ≥ p (cid:0) aB−aB(cid:1) + κ B−κ B . 1 1 0 1 1 0 1 0 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) netincreaseinloan netincreaseinassetholdings netincreaseincashholdings Borrower date 2 budget constraint The borrower’s date 2 budget constraint limits date 2 consumptionCB bythereturnontheborrower’sportfolioofassetsheldatdate1. 2 CB ≤ τ B κ B + aBR − c Rd , 2 1 1 1 1 2 (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) date2consumption proceeds fromcashholdings proceeds fromassetholdings repaymentof debt whereRd ≡min{R,1}denotestherealized date2repaymentof theloan(includingthepossibility 2 ofdefault). Each borrower takes the price q of the contract at date 1 as given. In equilibrium, the price 1 q will beinformative tothe borrowerabout thelender’s privateinformation atdate 1. Wediscuss 1 how the borrower’s beliefs evolve in section 6.2. We first characterize the borrower’s optimality conditions,conditionalonitsinformationsetatdate1. Borrower’soptimizationproblematdate1 The borrower’s optimization problem at date 1 is to make its portfolio and borrowing decision to maximize expected date 2 utility EB(cid:2) CB(cid:3) , conditional on its date 1 information set, taking prices 1 2 as p andq asgiven. 1 1 max EB(cid:2) CB(cid:3) (3) 1 2 cB,aB,κB 1 1 1 s.t. q c −c ≥ p (cid:0) aB−aB(cid:1) +κ B−κ B, 1 1 0 1 1 0 1 0 CB ≤τ B κ B+aBR−c Rd (4) 2 1 1 1 1 2 c ≤aB, 1 1 c ≥0, aB ≥0, κ B ≥0 1 1 1 Thefirst-orderconditionforc is 1 19

(cid:104) (cid:105) (cid:0) τ B+ξ B(cid:1) q −EB Rd −µ B+ξ B =0, (5) 1 κ1 1 1 2 1 c 1 whilethefirst-orderconditionforaB is 1 − (cid:0) τ B+ξ B(cid:1) p +EB[R]+µ B+ξ B =0. (6) 1 κ1 1 1 1 a 1 Thecomplementaryslacknessconditionsaregivenby µ B(cid:0) aB−c (cid:1) =0 (7) 1 1 1 ξ Bc =0 (8) c 1 1 ξ BaB =0 (9) a 1 1 ξ B κ B =0 (10) κ1 1 We can further characterize the borrower’s behavior by combining the FOCs for the case with risk-neutrality. Supposethatthecollateralconstraintisbindinginequilibriumsothat µB>0. (We 1 showlaterinLemma1thatthismustbethecase.) Then,theFOCsforc andaB holdwithequality. 1 1 Supposealsothatc =aB>0inequilibrium. Combiningthesebindingfirstorderconditionsyields 1 1 (cid:104) (cid:105) EB Rd −q τ B + ξ B (p −q ) = EB[R]−τ Bp (11) 1 2 1 1 κ1 1 1 1 1 1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) shadowvalueof cash expectednetreturnof riskyasset costof financingassetholdings It is convenient to define ν (cid:0) πB,q (cid:1) ≡ EB(cid:2) Rd(cid:3) −q τB and η (cid:0) πB;p (cid:1) ≡ EB[R]−τBp , so that 1 1 1 1 2 1 1 1 1 1 1 1 1 (11)canbeexpressedas ν (cid:0) π B,q (cid:1) + ξ B (p −q ) = η (cid:0) π B;p (cid:1) (12) 1 1 1 κ1 1 1 1 1 1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) costof financingassetholdings shadowvalueof cash expectednetreturnof riskyasset If EB[R]−τBp > EB(cid:2) Rd(cid:3) −q τB, i.e. if the net return from buying asset exceeds the net re- 1 1 1 1 2 1 1 turn from offering collateral (balancing the benefit from a larger loan with the cost of a larger repayment), then we must have that ξB (q −p )<0, which requires that ξB >0 (non-negativity κ1 1 1 κ1 constraint for κB binding) and q < p , which we will show in Lemma 1 holds in equilibrium. In 1 1 1 thiscase,theborrowerisatacornersolutioninitsportfoliochoice. 20

5. Traditional Sector’s Problem Thetraditionalsectorbecomesrelevantatdate1onlyandenterstheperiodwithanewendowment of cash κT = eT. It can store cash between dates 1 or 2, or buy existing risky assets at date 1 1 0 in the spot market. Existing assets need to be managed subject to some increasing costs. The date-2 return of holding aT of the risky asset at date one, net of these management costs, is given 1 by F(aT) units of the consumption good, where F(cid:48) > 0 and F(cid:48)(cid:48) < 0. Without loss of generality, 1 assume τT = 1. The traditional sector gets utility only from date-2 consumption CT with linear 1 2 utility. The traditional sector’s optimization problem is to choose how to allocate its endowment betweencashortheriskyassettomaximizeitsutility. max CT (13) 2 κT,aT 2 1 s.t. CT =κ T +F(aT) 2 2 1 κ T =κ T +p aT 1 2 1 1 aT ≥0 1 κ T ≥0 2 Forsimplicity,andwithoutlossofgenerality,weassumethetraditionalsector’sendowmentof cashissufficientlylargethatitsnon-negativityconstraintondate-2cashholdingsisneverbinding, κT > 0, by assumption 1. This ensures that, in equilibrium, the traditional sector could buy all 2 existing assets.16 As a result, the traditional sector will always enter date 2 with positive cash holdings κT, so the non-negativity constraint κT ≥ 0 is non-binding. Using binding date-1 and 2 2 -2 budget constraints, the traditional sector’s date-1 problem is to choose aT to maximize κT − 1 1 p aT +F(aT)subjecttoaT ≥0. thetraditionalsector’soptimalityconditionis 1 1 1 1 p =F(cid:48)(aT)+ξ T. (14) 1 1 a 1 16Toseethis,notethatifthetraditionalsectorholdsalltheassets,thenaT =aB. Soitsdate1expenditureis p aB. 1 0 1 0 Notealsothatintheeqwherethetraditionalsectorisholdingassets, wehave p =F(cid:48)(aT), andsothisexpenditure 1 1 is p aB=F(cid:48)(aB)aB. Thentheaboveassumptionguaranteesthatthetraditionalsectorhasenoughcashendowmentat 1 0 0 0 dateonetomakethispurchase. 21

6. Equilibrium at Date 1 6.1. Market Clearing at Date 1 Recallthatthedate-1contractistradedcompetitivelyinamarketforclaimsatdate1. Themarket clearingfortheseclaimsatdate1isgivenby17 dL =c . (15) 1 1 Ananalogousconditionwillholdforthedate-0market. Recall that no new risky assets can be created after date 0. Therefore, the market clearing condition for the date-1 spot market for the risky asset implies that the total quantity of the asset held by the borrower and the traditional sector at date 1 is equal to the total amount of the asset createdatdate0. aB+aT =aB (16) 1 1 0 6.2. Beliefs at Date 1 The previous characterization of agents’ optimal behavior is for given information set or beliefs. In this section, we discuss how agents’ beliefs about the fundamental of the risky asset (the return R)evolveatdate1. Equilibriumwillbepinneddownbythejointdeterminationofagents’actions andtheirbeliefs. Becauseagentshaveprivateinformation,theirbeliefsaboutRevolveindifferent ways: the beliefs of the lender respond directly to the lender’s private signal, while the beliefs of the borrower respond to the equilibrium prices and quantities that the borrower observes, to the extentthattheyareinformativeaboutthelender’sprivatesignal. Lender’s beliefs at date 1 Recall from section 3 that at date 1, the lender’s information set consists of the observed prices and terms of the contract Θ ={c ,q ,p } and its private cost 1 1 1 1 shock and private signal. Note that the bounds of the posterior belief of L are πL ∈ (0,1).18 the 1 lender’sposteriorbeliefsarederivedusingBayes’Rule π λ (cid:0) εL =sL−R (cid:1) π L = 0 ε 1 1 . (17) 1 π λ (cid:0) εL =sL−R (cid:1) +(1−π )λ (cid:0) εL =sL−R¯(cid:1) 0 ε 1 1 0 ε 1 1 This is derived formally in Appendix A.1. Note also that πL is decreasing in sL by the assumption 1 1 on the distribution from which ε is drawn is single-peaked. Denote the reduced form cumulative 17Wehavealreadyderivedthisinthelender’sprobleminsection3. 18ThisfollowsfromthefactthatsL=R+ε withε ∈(−∞,∞). AnyrealizationofsL ispossiblefrombothstateof 1 1 theworldR=R¯andR=R. 22

distributionfunctionofπL asG (·)anditsprobabilitydensityfunctionasg (·). 1 π π Borrower’s beliefs at date 1 The borrower observes all prices and contract terms, but does not observe the lender’s liquidity shock or private signal. Therefore the borrower’s date-1 information set consists only of the observed prices and terms of the contract IB ={Θ }. Recall 1 1 that the borrower has an identical prior belief as the lender π ≡ Pr(R=R|I ). After observing 0 0 equilibrium prices p and q , the borrower updates its beliefs about the risky asset’s return, where 1 1 itsposteriorbeliefisdenotedasπB.19 1 Although the borrower itself does not receive a private signal about the asset’s return, it can partially infer the lender’s private signal based on equilibrium observables, in particular the contractpriceq . Theobservedpriceq reflectsthelender’swillingnesstolend,andthroughequation 1 1 (17), is informative about the lender’s private beliefs about the return on the risky asset. However, the borrower cannot separately identify sL, τL based on the variables it observes, and thus cannot 1 1 perfectly infer the lender’s private signal. This is because q is also affected by cost shocks to the 1 lender,whichareorthogonaltotheprivatesignal. Thus,q servesasanoisysignaltotheborrower 1 aboutthelender’sbeliefsπL,wherethenoiseisgivenbythelender’scostshockτL. Werefertothe 1 1 inability of the borrower to perfectly disentangle the lender’s private signal from its idiosyncratic liquidityneedsasthe‘identificationproblem’facedbytheborrower. Thus,theborrowerfacestwolayersofuncertaintyabouttheasset’sreturnR: q isanoisysignal 1 to the borrower about the lender’s private signal sL, where the noise is introduced by the lender’s 1 idiosyncratic cost shock τL; and sL itself is a noisy signal about R. As we will show, the presence 1 1 of two layers of uncertainty plays a crucial role in our model: it will imply endogenous belief disagreement, and that liquidity conditions affect the borrower’s beliefs about the fundamental valueoftheriskyasset. Identificationproblem Toelaborateontheidentificationproblemfacedbytheborrower,note thattheborrowerobservesq , p ,andc . Thequantityoftheassetinvestedinandcollateralizedc 1 1 1 1 istheborrower’schoice,andreflectsq viamarketclearingc =d ,andthereforedoesnotconvey 1 1 1 any additional information to the borrower beyond q . Similarly, p is determined by the how 1 1 the marginal buyer of the asset values the asset. Since the lender cannot hold the asset until date 2, there are only two potential marginal buyers, the borrower and the traditional sector. Hence, p itself does not convey additional information about the lender’s private signal or cost shock. 1 Therefore, only q contains information on the private information of lender τL,πL through the 1 1 1 equilibriumrelationship(2). 19The assumption that atomistic lenders are competitive, plus the assumption that atomistic lenders are identical andreceivethesameshockandsignal, implythatinequilibriumthereisnostrategicmotiveamongstthelendersto behaveinawayinconsistentwithitstruevaluesbeliefsπLorthetrueshockτL.Rather,thelender’sbehaviorisalways 1 1 consistentwithitstruebeliefsandliquidityshock. Theborroweralsoknowsthisandupdatesitsbeliefsaccordingly. 23

Figure1: IdentificationProblem Note: Thisfiguresillustratestheidentificationproblemfacedbyborrowersatdate1. Thecurveinthefigureplotsthe combinationsofthelender’scostshockandbeliefs, (cid:0) τL,πL(cid:1) ,whichareconsistentwiththeequilibriumq ,basedon 1 1 1 thelender’soptimalityconditionfordL. Theborrower’sobservationofq isinsufficienttoinferthetruerealizations (cid:0) τL∗,πL∗(cid:1) separately. 1 1 1 1 1−πL+πLR q = 1 1 . 1 τL 1 Moreover,theborrowerknowshowthelender’sposteriorbeliefsdependonitsprivatesignal—that is,theborrowerknowsthemapping(17)fromsL toπL.20 Thisidentificationproblemfacedbythe 1 1 borrowerisillustratedinFigure1. Since the borrower has only one observable, which is informative about the lender’s private information, it cannot separately identify the lender’s information set at date 1 (i.e. it’s private signal) and its date 1 cost shock τL. Put differently, the borrower has only one equation (17) 1 to infer two unobservables sL,τL. Indeed, there is a continuum of pairs (τL,πL) that satisfy the 1 1 1 1 relation(17)fortheobservedq ,asillustratedinFigure1. 1 In light of the identification problem faced by the borrower, the borrower’s posterior beliefs evolveaccordingto ˆ ˆ EB(cid:2) π˜L(cid:3) = τ¯ 1 π1 (cid:26) π = 1−τq 1 (cid:27) dG (π)dΛ (τ), (18) 1 1 1−R π T τ 0 20Foragivenvalueofq ,thefunctionalrelationshipbetweenτ˜L andπ˜L iswhereτ˜L∈[τ,τ¯]andπ˜L∈(0,1). 1 1 1 1 1 24

where 1{·} is an indicator function, and Λ (·) is the distribution function of λ (·). By change of T T variables,theaboveequationcanberearrangedasfollows ˆ 1 (cid:18) 1−(1−R)π (cid:19) (cid:26) 1−(1−R)π (cid:27) π B = πλ 1 τ < <τ¯ dG (π). (19) 1 T q q π 0 1 1 Intuitively, the borrower computes the likelihood of all possible realizations of (τL,sL)consis- 1 1 tent with the observed value of q , based on its knowledge of the joint distribution of τL and sL, 1 1 1 and given the mapping (17) between sL and πL. Based on the likelihood of these realizations, the 1 1 borrower forms an expectation over the lender’s private signal sL and updates its own beliefs πB 1 1 accordingly. Proposition1. Borrower’sposteriorbeliefπB hasthefollowingpropertiesinequilibrium: 1 1. Borrower’sbeliefisthesameastheexpectedposterioroflender,i.e. πB =EB[πL]. 1 1 1 2. πB isdecreasinginq . 1 1 Proof. SeeAppendixA.2 Part1ofthepropositionshowsthattheborrower’sposteriorbeliefaboutthefundamentalvalue of the asset is given by the borrower’s belief about the lender’s belief. This is simply a result of Bayesian updating. Part 2 of the proposition shows that if the borrower faces a tighter funding liquidity from the lender at date 1 (that is, a lower contract price q < q ), then the borrower 1 0 becomesmorepessimisticaboutthefundamentalvalueoftheasset. Importantly,thisisregardless ofwhetherthecauseofthedeclinewasanadversecostshockτL orabadsignalsL,becauseofthe 1 1 identificationproblemfacedbyB. Cost shocks and beliefs An important implication of Proposition 1 is that pessimism and optimism and the fundamental value of the asset arises endogenously in our model as a result of funding liquidity. In particular, the borrower’s beliefs are shaped also by the lender’s cost shocks, despite the fact that these are orthogonal to the asset’s return. As we will show, tight funding conditions (a low contract price q ) leads to pessimism (a high πB). By contrast, loose funding 1 1 liquidityconditions(ahighcontractpriceq )leadstooptimism(alowπB). 1 1 6.3. Characterization of Equilibrium at Date 1 Definition of date-1 equilibrium An equilibrium at date 1 is a set of prices p ,q , collat- 1 1 eral amount c , quantities dL,κL,aB,κB,aT,CL,CB,CT, and posterior beliefs πL,πB satisfying the 1 1 1 1 1 1 2 2 2 1 1 agents’ optimality conditions and constraints, market clearing conditions, and the equations characterizing belief formation, taking as given variables determined at date 0 and the date-1 shocks 25

τL,sL. Notethatauniqueequilibriumalwaysexistsbecauseofthelinearityoftheagents’objective 1 1 functions and the constraints of the borrower’s optimization problem, and the smoothness of the inversedemandfunctionofthetraditionalsector,whichdeterminesthemarketclearingcondition. The spot market for the risky asset is competitive, so the price p will be pinned down in 1 equilibrium at the marginal valuation of the marginal buyer. Recall that there can be two potential marginalbuyers: (continuumof)borrowers,orthetraditionalsector. Let pB denotetheborrower’s 1 marginalvaluationoftheasset,whichisgivenbytheborrower’soptimalitycondition(11): 1 (cid:16) (cid:104) (cid:105) (cid:17) pB = EB[R]−EB Rd +q τ B . (20) 1 τB 1 1 2 1 1 1 Recall that the traditional sector has the inverse demand function F(cid:48)(aT), so that it’s marginal 1 valuation is given by F(cid:48)(aT). (Recall also that the traditional sector will buy any amount of as- 1 sets up to its marginal valuation F(cid:48)(aT) because the traditional sector has enough amount of cash 1 endowmentsbyAssumption1.) Moreover,thepriceischaracterizedinequilibriumby (cid:26) (cid:27) 1 (cid:16) (cid:104) (cid:105) (cid:17) p =max EB[R]−EB Rd +q τ B ,F(cid:48)(aT) , (21) 1 τB+ξB 1 1 2 1 1 1 1 κ1 where the maximum willingness to pay determines the identity of the marginal buyer in equilibrium. Notethat p < pB canoccurinequilibrium,iftheborrowerisliquidatingassetsforliquidity 1 1 needs,whichimpliesthattheborrowerisconstrainedincash,i.e. ξB >0. Itisconvenienttoderive κ1 someadditionalpropertiesthataresatisfiedinequilibrium,summarizedinthelemmabelow. Lemma1. Inequilibrium,thefollowingholds: (A) Theborrower’scollateralconstraintisgenericallybinding,sothatc =aB. 1 1 (B) Thepriceoftheassetexceedsthepriceofthecontract, p >q . 1 1 Proof. SeeAppendixA.4 The date 1 equilibrium features two regimes, which we call ‘normal times’ and ‘fire sales’. Thepartitionofthestatespaceintothesetworegimescanbecharacterizedbytheprice p andthe 1 identity of the marginal buyer in equilibrium. Part (A) of the proposition below describes the two regimes, while Part (B) shows that fire sales are more severe for worse private signals and also for worseliquidityshocks. Proposition2. Inequilibrium,thefollowingholds: (A) Tworegimesindate-1equilibrium: When p ≥F(cid:48)(0),theeconomyisinthe‘normalregime’ 1 in which p = pB, aB =c =aB, and aT =0. When p <F(cid:48)(0), the economy is in the ‘fire 1 1 1 1 0 1 1 saleregime’inwhich p =F(cid:48)(aT),aB =cB <aB,andaT >0. 1 1 1 1 0 1 26

(B) Fire sales are more severe for tighter funding liquidity: In the fire sale regime, a lower realization of sL or a higher realization τL, at the margin, is associated with larger aT and 1 1 1 lower p inequilibrium. 1 (C) Fire sales may be driven by liquidity needs or pessimism: In a ‘liquidity driven fire sale’ in which the borrower is optimistic but is forced to liquidate assets to repay debt, we have pB > F(cid:48)(0) > p . In a ‘belief driven fire sale’ in which the borrower liquidates due to its 1 1 pessimism,wehave pB <F(cid:48)(0).21 1 Normal Regime In normal times, when cost shocks are not too large and the signal is not too bad, all of the risky asset is held by the borrower. In this situation, the borrower has relatively optimistic beliefs. As a result, the borrower is the marginal buyer of the asset and the asset price p is relatively high. the borrower is at a corner solution in its portfolio choice and 1 wants to hold only the risky asset, and so its holdings of the risky asset remain unchanged from date 0, aB =aB. (Moreover, since it is always optimal for the borrower to collateralize all its asset 1 0 holdings, we have c =aB =c .) If the size of the loan under the date-1 contract q c is less than 1 1 0 1 0 the promised payment of the loan under the date-0 contract c — that is, if the borrower faces a 0 tightening funding liquidity from the lender—then, the borrower finances the difference c −q c 0 1 0 outofitscashholdingsstoredfromdate0. Asq falls,theborrowerbecomesmorepessimistic,sothatπB fallsinaccordancewithPropo- 1 1 sition 1. However, in normal times, this pessimism is not sufficient to trigger a change in its portfolio. As a result, in equilibrium, the risky asset is held entirely by borrowers, aT = 0. The 1 marginal buyer of the asset at date 1 is still the borrower, and so p is given by the borrower’s 1 marginalvaluationoftheasset pB. 1 Fire Sale Regime In the fire sale regime, the equilibrium at date 1 features some strictly positivefractionoftheriskyassetheldbythetraditionalsector,whiletheremainderisheldbythe borrower. In particular, when the cost shock is large (τL is high) or the signal is bad (sL is low), 1 1 the tightening of funding liquidity causes the borrower to become sufficiently pessimistic about thereturnontheassetthat,andtheborrower’sportfoliodecisionswitches: theborrowerprefersto alter its portfolio from holding the risky asset to cash. As a result, the borrower meets the tighter funding liquidity by liquidating some of its holdings of the risky asset to the traditional sector at a fire-saleprice. Toseethis,recalltheborrower’sdate1optimalitycondition(12) ν (cid:0) π B,q (cid:1) = η (cid:0) π B;p (cid:1) , 1 1 1 1 1 1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) costof financingriskyassetholdings expectednetreturnof riskyasset 21The price p can be less than pB in this case, if the borrower is extremely pessimistic and pB <F(cid:48)(aB). See 1 1 1 0 AppendixA.6forthefullcharacterizationofthiscase. 27

wherewehaveimposedtheresultthatξB =0,sothattheborrowerholdsapositiveamountofcash. κ1 Atagiven p ,atighterfundingliquidity(lowerq relativetoq )worsensthetradeofftoinvesting 1 1 0 in the risky asset by making the borrower’s beliefs more pessimistic. As a result ν rises and η 1 1 fallsbyProposition1. Iftheshockorsignalissufficientlybadthattheborrower’svaluationisless thanthetraditionalsector’s(i.e. pB <F(cid:48)(0),theninordertomeetthetighterfundingliquidity,the 1 borrowerpreferstoliquidatetothetraditionalsector). Giventhisbehaviorofborrowers,howdoesthespotprice p oftheassetchangeinequilibrium? 1 Since all borrowers are equally pessimistic and trying to switch to holding cash, the only way for borrowerstoconvertassettocashatdate1inaggregateistoselltheassettothetraditionalsector, who have higher marginal valuation. Therefore, the traditional sector, T, becomes the marginal buyer of the asset, and price is given by T’s marginal valuation p =F(cid:48)(aT). Recall that F(cid:48)(cid:48)(·) is 1 1 a decreasing function, and so the price is then decreasing in quantity aT of the asset held by the 1 traditional sector. Hence, equilibrium is restored when p falls sufficiently to restore equality in 1 the borrower’s optimality condition at an interior solution in which the borrower holds both cash andtheasset. Part (B) of Proposition 2 indicates that a lower q is associated with more severe fire sales— 1 thatis,higheraT andalower p . Hence,moreseverecostshocksandworsenewsaboutthequality 1 1 oftheassetleadtomoreoftheriskyassetbeingliquidatedtothetraditionalsectorbygeneratinga largerdecreaseinfundingliquidityandthereforeagreaterdate-1liquidityneedfortheborrower. Part (C) of Proposition 2 describes two different cases of fire sales. In a liquidity driven fire sale, the borrower is still optimistic about the asset’s fundamental value but the borrower is forced toliquidateassetstorepaydebtc . Thisfiresaleisstillinefficientbecausetheborrowervaluesthe 0 asset more than the traditional sector does, i.e. pB >F(cid:48)(0)≥F(cid:48)(aT). On the contrary, in a belief 1 1 driven fire sale, the borrower is pessimistic about the asset’s fundamental value, which is even belowthemarginalvaluationinthetraditionalsector,i.e. pB <F(cid:48)(0). Inthiscase,theborroweris 1 happy to sell the asset to the traditional sector, up to the point either the marginal valuation of the traditionalsectordecreasestoF(cid:48)(aT)= pB ortheborrowersellsalltheasset,i.e. aT =aB. 1 1 1 0 We can partition the state space into the Normal and Fire Sale Regimes by defining the set of states (cid:0) τL,πL(cid:1) consistent with pˆ ≡F(cid:48)(0), the threshold such that the economy enters the Fire 1 1 1 Sale Regime when the asset price is below the threshold pˆ . In the Baseline Economy, which we 1 constructed so far, there is a one-to-one mapping from q to p , and hence, the threshold asset 1 1 price pˆ corresponds to a threshold value qˆ . The lender’s optimality condition defines the set of 1 1 (cid:0) τL,πL(cid:1) consistentwithqˆ ,givenby 1 1 1 1−τLqˆ π L = 1 1 . (22) 1 1−R 28

Figure2: TwoRegimes Note: Thisfigureillustratesthebisectionofthestatespaceintotworegimesatdate1. TheequilibriumisintheFire SaleRegimeifandonlyiftheequilibriumvalueof p isbelowathresholdvalue pˆ (or,equivalently,theequilibrium 1 1 value of q is below a threshold value qˆ ). The curve in the figure plots the combinations of the states (cid:0) τL,πL(cid:1) 1 1 1 1 consistentthethreshold pˆ andmarksthefrontierbetweenthetworegimes. 1 This equation (22) defines the frontier partitioning the state space into the Normal and Fire Sale Regimes,andisillustratedinFigure2. 6.4. Downstream versus Upstream Information Spillovers Thus far, we have focused on the case in which the lender gets the cost shock and private signal, but the borrower does not. This implies that information spillovers flow downstream from the lender to the borrower. Here, we briefly touch on the alternative case in which the borrower gets a cost shock and private signal, but the lender does not, leading to upstream information spillovers. Overall,whilethemechanismdiffersslightly,thefundamentalinsightsarethesame. SupposetheborrowergetsacostshockτB andprivatesignalsB,butthelenderdoesnot(i.e. τL 1 1 1 is a fixed parameter known to both agents). How would the borrower’s choices reflect its private information τB,sB? In this case q still only reflects the lender’s first-order condition according to 1 1 1 (2), and is therefore not informative about the borrower’s private information. However, the asset price p will reflect the borrower’s private information through the equilibrium price (12). Using 1 thisequation,thespreadbetweentheborrower’svaluationandthepriceofdebtatdate1is EB[R]−EB(cid:2) Rd(cid:3) p −q = 1 1 2 (23) (cid:124) 1 (cid:123)(cid:122) (cid:125) 1 ξ κ B 1 +τ 1 B spread 29

(Notealsothatthequantityoftheborrower’sinvestmentintheriskyassetaB=c isjustdetermined 1 1 residuallyfromthisoptimalityconditionandthemarketclearingcondition,andhencecontainsno additionalinformationaboutborrower’sprivateinformationoverthespread.) Therefore,thelender uses the observation of p −q to form its posterior belief πL, but cannot separately infer πB and 1 1 1 1 τB duetoasimilaridentificationproblemaswhatwehadinthecasewithdownstreaminformation 1 spillovers. Supposethattheborrowerreceivesacostshock(highτB)orbadprivatesignalsothattheasset 1 price is low. The lender then observes that the spread p −q is low and will update its belief, 1 1 which becomes more pessimistic. As a result, as can be seen from the equilibrium expression for q , the equilibrium price of date-1 debt will fall—that is, there will be a tightening of funding 1 liquidity. Thisworsenstheborrower’sliquidityposition,andforcesittoliquidatemoreoftherisky asset, further lowering the asset price. Hence, the information spillover from the borrower to the lender still amplifies the fire sale through a feedback loop between market illiquidity and beliefs, inamannerwhichgivesrisetoassetpriceboomsandbusts. 7. Equilibrium at Date 0 Giventheequilibriumatdate1,wecansolverecursivelyforthedate-0equilibrium. Fortractability, weassumeherethatF(cid:48)(a)=α foralla≥0,althoughwedidnotassumethisincharacterizingthe equilibrium at dates 1 and 2 above.22 The date-0 equilibrium is derived in Appendix A.7, and the keypropertiesaresummarizedinthefollowingproposition. Proposition 3. Suppose that assumptions 1 and 2 hold. In date 0 equilibrium, the borrower holds zero amount of cash and full amount of asset, using all of them as collateral to borrow from the lender—that is, (c ,a ,κB)=(A,A,0)— and the prices of the asset and contracts are (p ,q )= 0 0 0 0 0 (1+eB/A,1). 0 Proof. SeeAppendixA.7. The date-0 loan from the lender to the borrower is strictly positive, the borrower invests only inassets,andusesallofitsholdingsoftheriskyassetascollateralfortheloan. 8. Information Contagion and Fire Sales IntheFireSaleRegime,theevolutionofagents’beliefsplaysanimportantroleinthedynamicsat date 1. Moreover, the nature of fire sales here differs from that in much of the literature: in a fire 22Effectively,thisassumptionimpliesthat,inafiresaleatdate1,theborrowerliquidatesallofitsholdingsofthe riskyasset. 30

sale, the borrower liquidates not simply because it is liquidity constrained (indeed, the borrower could finance some of the decrease in funding liquidity out of its cash holdings, as it does in the normal regime), but also because the borrower endogenously becomes more pessimistic about the fundamental value of the asset. Importantly, pessimism can occur even if the lender receives relativelygoodsignal. In this section, we show how misinformation affects the likelihood and severity of crises, and how it affects the distribution of states at date 1 across the two regimes. In our model, the interaction between beliefs and liquidity arises from two related but distinct features of the learning process: 1. (learning) In response to new information, agents update their prior beliefs about the fundamentalvalueoftheriskyasset. 2. (heterogeneous beliefs) Because information is private, agents’ beliefs evolve differently to newinformation. To understand how each feature of beliefs interacts with market and funding liquidity, we conduct twoseparateexercises. In the first exercise, we compare the date-1 equilibrium to that in an alternative benchmark economy in which agents do not update their beliefs, and we show that the endogenous response of beliefs to changes in funding liquidity exacerbates fire sales by worsening market liquidity. In the second set of exercises, we compare the date-1 equilibrium to another benchmark economy in which all information is publicly observed, so that beliefs are homogeneous. This counterfactual exerciseallowsustounderstandhowtheheterogeneityinagents’beliefsaffectsoutcomesthrough liquidityconditions. 8.1. Interaction between pessimism, funding liquidity, and market liquidity To understand how learning interacts with funding and market liquidity, we first compare the response of our baseline economy to shocks to the response of an alternative benchmark economy in which beliefs are stale—that is, we assume agents’ date-0 beliefs are never updated in response tonewinformation. Wemeasuretheseverityofafiresalebytheextenttowhichtheborrowerfire sellsitsholdingsoftheriskyassettothetraditionalsector: thegreater(and,equivalently,thelower theassetprice p ),themoreseverethefiresaleatdate1.23 ThisresultissummarizedinLemma2 1 below. 23Aswediscussinthenormativesectionofthepaper,Section9,theliquidationoftheriskyassetthatoccursina firesaleisnotnecessarilyassociatedwithconstrainedParetoinefficiency. 31

Lemma 2 (Learning amplifies the severity of fire sales). For any set of shocks (πL,τL) such that 1 1 the equilibrium is in the Fire Sale Regime at date 1, the price of the risky asset p is lower and 1 the extent of misallocation of the asset aT is higher in equilibrium compared to an alternative 1 benchmarkeconomyinwhichagents’posteriorbeliefsareexogenouslysettotheirdate-0priors. Proof. SeeAppendixA.8. We showed in Section 6 that pessimism and optimism and the fundamental value of the asset arises endogenously in our model as a result of funding liquidity. Lemma 2 above shows that this increased pessimism contributes to market illiquidity, by depressing asset prices and making it more costly to raise funds through liquidation. Therefore, the model yields an adverse feedback between pessimistic beliefs about fundamentals, funding liquidity, and market illiquidity—a dynamic,whichexacerbatescrises. 8.2. The interaction between belief disagreement and liquidity conditions In order to isolate the role of belief disagreement, and misinformation in particular, in the equilibriumallocationatdate1,wecompareourequilibriuminourBaselineEconomywithabenchmark casewithoutprivateinformation. Wefirstbrieflydescribethisbenchmark. CommonInformationBenchmark Inthebenchmarkeconomywithcommoninformation, we assume that the borrowers can directly observe both the lender’s signal sL and the lender’s 1 shock to the opportunity cost of funds τL.24 In this sense, neither agent has private information. 1 A corollaryof Proposition 1is that, underthis common information benchmark, the borrowerand lender have identical posterior beliefs, πB = πL in every state. The form of the equilibrium is 1 1 otherwisethesameasthatinthebaselineeconomywithprivateinformation. By comparing the date-1 equilibrium allocation under the baseline economy with that under thisbenchmark(inbothcases,takingdate-0variablesasgiven),wecanisolatetheeffectofmisinformationontheequilibriumallocationatdate1.25 First, we characterize the frontier, which is partitioning the state space between Normal and Fire Sale Regimes under Baseline Economy and the Common Information Benchmark. qˆ is 1 the threshold q separating the Normal and Fire Sale Regimes in the Baseline Economy, defined 1 implicitly by (cid:0) 1−πB(qˆ ) (cid:1)(cid:0) R−1 (cid:1) +τBqˆ = τBF(cid:48)(0), and pˆ is the corresponding asset price of 1 1 1 1 1 1 the frontier. As in the Baseline Economy, the frontier under the Common Information Benchmark isdefinedbythesetofstates (cid:0) τL,πL(cid:1) consistentwith p = p˜ ≡F(cid:48)(0),whichisgivenby 1 1 1 1 24Wecouldequivalentlyassumethatthedate-1signalaboutRispubliclyobservedwhilethelender’sopportunity cost, τL, isnotobservedbytheborrower. Thisisequivalentbecause, inequilibrium, theborrowerwouldbeableto 1 perfectlyinferτL givenitsobservationofq anditsknowledgeofπL. 1 1 1 25Ingeneral,differencesintheequilibriumallocationsatdate0acrossthebaselineandbenchmarkmayalsomanifestasdifferencesinthedate1allocation. Wecanshowthatthedate-0equilibriumallocationsofthetwocaseswould bethesameunderassumptions1and2. 32

(cid:0) R−1 (cid:1) − (cid:20) (cid:0) R−1 (cid:1) + τ 1 B (1−R) (cid:21) π L+ τ 1 B =τ BF(cid:48)(0). (24) τL 1 τL 1 1 1 This condition defines the set of states, (cid:0) τL,πL(cid:1) |p˜ , and characterizes the partition between the 1 1 1 Fire Sale and Normal Regimes in the Common Information Benchmark. The following lemma showshowthisfrontiercomparestothatundertheBaselineEconomy. Lemma 3 (Frontier in the Common Information Benchmark). The frontier (cid:0) τL,πL(cid:1) |p˜ in the 1 1 1 Common Information Benchmark is strictly convex and has a negative slope in the domain for τL. Moreover, the y-intercept 1 of the frontier (cid:0) τL,πL(cid:1) |p˜ is the same as the y-intercept for 1 1−R 1 1 1 the frontier (cid:0) τL,πL(cid:1) |pˆ in the Baseline Economy, and the x-intercept τ 1 B in the Common 1 1 1 τBF(cid:48)(0)−R+1 1 InformationBenchmarkexceedsthatintheBaselineEconomy 1 . qˆ 1 Proof. SeeAppendixA.10. For the given qˆ of the frontier in the Baseline Case, there exists a point τˆL such that the 1 1 correspondingπL =1as 1 1−τˆLqˆ π L =1= 1 1 . 1 1−R Similarly,thereexistsapointτ˜L suchthatthecorrespondingπL =1as 1 1 τBf +τ˜L(cid:0) R− f −τBF(cid:48)(0) (cid:1) π L =1= 1 1 1 1 1 . 1 τ˜L (cid:0) R− f (cid:1) +τB(f −R) 1 1 1 1 Some of the results, which follow, depend on whether the frontier partitioning the state space into the Normal and Fire Sale Regimes in the Baseline Economy intersects the frontier in the Common Information Benchmark. If the following condition holds, then the frontier in the Baseline Economy is above the frontier under the Common Information Benchmark up to some point (τL∗ ,πL∗ )suchthatπL∗ <1,wherethetwofrontiersmeet. 1 1 1 Condition1. τˆL >τ˜L (25) 1 1 Lemma4. ThefrontiersintheBaselineCaseandtheCommonInformationBenchmarknevermeet inthedomainsuchthatπL ≤1ifandonlyifτˆL ≤τ˜L. 1 1 1 Proof. SeeAppendixA.11. 33

Condition 1 implies that the frontier in the Baseline case starts declining linearly at a point (τˆL,1),whereasthefrontierintheCommonInformationBenchmarkgoesthroughthepoint(τ˜L,1). 1 1 Because both frontiers started at the same y-intercept, the average slope of the frontier in the Common Information Benchmark is above that in the Benchmark case. Because the slope of the frontierintheCommonInformationBenchmarkisincreasing(bystrictconvexityshowninLemma 3), the frontier in the Baseline Economy is always below the frontier in the Common Information Benchmark. Figure 3 illustrates how the partitions of the state space between the Normal and Fire Sale Regimes compare across the Baseline Economy and the Common Information Benchmark, when Condition 1 holds.26 Figure 6 in Appendix A.13 illustrates this for the case in which Condition 1 doesnothold. Proposition 4 summarizes how misinformation affects the severity of fire sales and the allocation in the Normal Regime in the baseline economy, relative to those of the Common Information benchmark,whiletakingdate-0variablesasgiven. Proposition 4 (Effect of misinformation on the allocation in the Normal and Fire Sale Regimes). Giventhedate-0allocation,thefollowingholds: (A) The allocation of the risky asset in the Normal Regime is identical in the Baseline Economy andthetheCommonInformationBenchmark: aB =aB inbotheconomies. 1 0 (B) If Condition 1 holds in equilibrium, then misinformation amplifies the severity of liquidity driven fire sales and dampens severity of belief driven fire sales at date 1. Formally, for any state (cid:0) τL,πL(cid:1) in the Fire Sale Regime, aB is lower in the Baseline Economy compared 1 1 1 to the Common Information Benchmark if τL is sufficiently high (exceeding some threshold 1 τL(πL)),andislowerifπL issufficientlyhigh(exceedingsomethresholdπL(τL)). 1 1 1 1 1 (C) If Condition 1 does not hold in equilibrium, then misinformation unambiguously amplifies the severity of fire sales. Formally, for every state (cid:0) τL,πL(cid:1) in the Fire Sale Regime, aB is 1 1 1 always lower in the Baseline Economy compared to the Common Information Benchmark. (For the knife-edge cases in the Baseline Economy in which πB =πL, aB is the same across 1 1 1 thebaselineandbenchmarkeconomies.) 26Note that, while the slope of (cid:0) τL,πL(cid:1) |pˆ in the Baseline Economy is constant, the slope of (cid:0) τL,πL(cid:1) |p˜ in the 1 1 1 1 1 1 CommonInformationBenchmarkisnotconstant. Toseewhy,notethat,intheBaselinecase,thefrontierbetweenthe tworegimesisdeterminedentirelybyathresholdq ,andthethelender’sFOCimpliesthattherelationshipbetweenπL 1 1 andτLislinear.Bycontrast,intheBenchmark,q isnolongersufficienttodeterminewhichregimetheeconomyisin; 1 1 theexactrealizationofshocks(becausetheexactrealizationdeterminesπB)alsoaffectsthedeterminationofregime. 1 Thus, the fact that there’s more than one variable determining the equilibrium regime implies that the relationship betweentheπL andτL)pairsconsistentwiththefactthatthethreshold p isnonlinear. Relatedly, weshoulddefine 1 1 1 thefrontiernotasqˆ butratheras pˆ ,sincethereisnosuchwell-definedqˆ intheCommonInformationBenchmark. 1 1 1 34

Figure3: TwoRegimesundertheBaselineCaseandtheCommonInformationBenchmark Note: This figure illustrates the bisection of the state space into two regimes at date 1 in the baseline case in which thelender’sliquidityshockτLandbeliefsπLareprivateinformation,andundertheCommonInformationBenchmark 1 1 caseinwhichthisinformationisdirectlyobservablebytheborrower. TheequilibriumisintheFireSaleRegimeif and only if the equilibrium value of p is below the threshold value pˆ . (For the baseline case, this corresponds to 1 1 the threshold value of qˆ .) The solid curve in the figure plots the combinations of the states (cid:0) τL,πL(cid:1) consistent the 1 1 1 threshold pˆ ,basedonthelender’soptimalityconditionfordL,anddenotesthefrontierbetweenthetworegimes. The 1 1 dashedcurveplotsthesamefrontierintheCommonInformationBenchmarkinwhichtheborrowerdirectlyobserves thelender’sprivateinformation,andhenceπB=πL allalongthiscurve. Forbothcases,theregiontothesouthwest 1 1 ofthesecurvesistheNormalRegime,whilethenortheastistheFireSaleRegime. Thedottedlineisconstructedbytracingouttheborrower’sposteriorbeliefs (cid:0) EB(cid:2) τL(cid:3) ,πB(cid:1) |q atdifferentrealizations 1 1 1 1 of the equilibrium value q . For a given q , the borrower’s posterior beliefs (cid:0) EB(cid:2) τL(cid:3) ,πB(cid:1) |q about the state are a 1 1 1 1 1 1 point onthe set (cid:0) τL,πL(cid:1) |q of allpossible states consistentwith itsobservation of q (illustrated inFigure 1). At a 1 1 1 1 givenq , allpointsonthecurve (cid:0) τL,πL(cid:1) |q tothenorthwestofthepoint (cid:0) EB(cid:2) τL(cid:3) ,πB(cid:1) |q featureπB>πL , while 1 1 1 1 1 1 1 1 1 1 all points on the curve to the southeast feature πB >πL. Thus, the dotted curve in the figure demarcates the region 1 1 ofthestatespaceinwhichπB<πL inequilibriumtothenorthwestofthecurve,andtheregioninwhichπB>πL to 1 1 1 1 thesoutheast. Inthenorthwestregion,theborrowerisoptimisticabouttheriskyassetrelativetothebetter-informed lender,whileinthesoutheast,theborrowerisrelativelypessimisticabouttheriskyasset. Proof. SeeAppendixA.12. Part (A) shows that misinformation does not affect the allocation of the risky asset in the Normal Regime at date 1. While the beliefs of the lender and borrower may diverge in a way which does not reflect fundamental shocks, this has no bite on the allocation of the risky asset in the Normal Regime as long as the risky asset is not sold to the traditional sector. (Nevertheless, the allocationofcashbetweenborrowersandlendersmaydiffer,ingeneral.) 35

Part (B) indicates that, if Condition 1 holds, misinformation amplifies the severity of liquidity driven fire sales and dampens severity of belief driven fire sales at date 1. Therefore, the mechanism by which beliefs respond endogenously to conditions in funding markets implies that the economy features more severe fire sales when fire sales are driven by severe shocks to the opportunity cost of funds (high realizations of τL), while fire sales are less severe when driven by bad 1 news. Intuitively, for fire sales driven by bad news about the asset’s fundamental value, the borrower is more optimistic about the fundamental of the asset relative to the better-informed lender, and this relative optimism reduces the extent to which the borrower liquidates the asset to the traditional sector. In contrast, when the fire sale is driven by a negative cost shock to the lender, the borrowerismorepessimisticabouttheassetsinceastheborrowercannotinferthelender’sprivate information. As a result, this pessimism causes the borrower to liquidate more of the asset than it otherwise would. Note that the overall effect on severity of a fire sale relative to the Common InformationBenchmarkisambiguous,asitdependsontherelativeimportanceofcostshocksversus newsindrivingfiresales. Part(C)indicatesthat,ifCondition1doesnothold,thenmisinformation featuresunambiguouslymoreseverefiresales. Effectofmisinformationonthelikelihoodoffiresales Next, we consider how misinfromation affects the distribution of the state space across the NormalandFireSaleRegimes. ComparingthefrontiersintheBaselineEconomyandtheCommon Information Benchmark reveals that misinformation increases the likelihood of liquidity-driven fire sales and decreases the likelihood of signal-driven fire sales, if Condition 1 holds. However, if Condition 1 does not hold, the likelihood of fire sales is unambiguously higher in the Baseline Economy than that in the Common Information Benchmark. These results are summarized in Lemma5. Lemma5(Effectofmisinformationonthelikelihoodoffiresales). (A) If Condition 1 holds, then in the Baseline Economy, the likelihood of entering a fire sale given a large cost shock is higher than in the Common Information Benchmark, while the likelihood of entering a fire sale given a bad news shock is lower in the Baseline Economy. Moreover, the overall effect on the unconditional probability of a fire sale in the Baseline EconomyrelativetotheCommonInformationBenchmarkisambiguous. (B) If Condition 1 does not hold, then the unconditional probability of a fire sale is unambiguouslyhigherintheBaselineforanystate (cid:0) τL,πL(cid:1) intheFireSaleRegime. 1 1 Proof. TheresultsfollowfromProposition4. SeeAppendixA.13formoredetailsonPart(B). 36

Part(A)saysthat,ifCondition1holds,misinformationincreasesthelikelihoodoffiresalesin response to cost shocks, since cost shocks lead borrowers to become more pessimistic about the fundamental value of the asset, despite being orthogonal to this fundamental value. By contrast, misinformation reduces the likelihood of a fire sale in response to bad news about fundamentals. In this sense, the misinformation mechanism reduces the probability of fire sales in response to badnewsaboutfundamentals,butincreasestheprobabilityoffiresalesinresponsetocostshocks. Thus, when Condition 1 holds, the effect of misinformation on the unconditional likelihood of crises is ambiguous and depends on the relative importance of cost shocks versus news in driving firesales. On the other hand, if Condition 1 does not hold, for any shock, the economy is more likely to entertheFireSaleRegimeintheBaselineEconomythanintheCommonInformationBenchmark. See Figure 7 in Appendix A.13 for an illustration. As a result, the misinformation mechanism unambiguouslyincreasesthelikelihoodoffiresaleswhenCondition1doesnothold. This characterizes how our mechanism by which funding liquidity, or conditions in funding markets, affects perceptions of risk and beliefs about fundamentals, affects the likelihood and severityoffiresales. 9. Normative Implications Inthissection,weexaminethenormativeimplicationsofthemodel. Fornow,wefocusonexpost efficiency—that is, we compare competitive equilibrium at date 1 to allocation at date 1 chosen by social planner, taking date-0 variables as given. We define the date-1 constrained efficient allocationasthesolutiontotheproblemofaconstrainedsocialplannerwhotakesdate-0variables as given. The planner faces uncertainty about R at date 1 and forms beliefs about R through Bayesian updating in response to news at date 1. Although the planner observes the lender’s private information (and hence has the same posterior beliefs as the lender), it respects that the borrower does not observe this private information but must infer this information imperfectly through observables.27 In other words, we assume the planner cannot disclose the lender’s private information to the borrower. And finally, the planner faces the borrowing constraint between the lender and borrower. However, the planner internalizes how choices about the terms under which to lend and borrow at date 1, and the choice to sell the risky asset, affect the allocation across the distributionofstates. Weshowthatthecompetitiveequilibriumatdate1isgenericallyconstrainedinefficientdueto the presence of two externalities. In choosing how much to lend at date 1, lenders do not internalize how their choice affects the information set of borrowers through the price q . Therefore, the 1 27Section10analyzesthesetupinwhichthegovernmentcannotobservethelender’sprivateinformation. 37

Figure4: Optimistic/PessimisticRegionandNormal/FireSaleRegime Note: Theleftpanelillustratesthebisectionofthestatespaceintoregionsbasedontheborrower’sbeliefsaboutthe asset relative to those of the lender. The dotted curve demarcates the region of the state space in which πB <πL 1 1 in equilibrium to the northwest of the curve, and the region in which πB >πL to the southeast. In the northwest 1 1 region, the borrower is optimistic about the risky asset relative to the better-informed lender, while in the southeast, theborrowerisrelativelypessimisticabouttheriskyasset. TherightpaneloverlaystheleftpanelwithFigure2andillustratesthedemarcationoffourregionsofthestatespace bywhethertheequilibriumisintheNormalRegimeorFireSaleRegime,andbywhethertheborrowerisoptimistic orpessimisticabouttheriskyassetrelativetothebetter-informedlender. economy features an information externality, which is new to the literature. In addition, there is a pecuniary externality which is essentially the same as in Lorenzoni (2008): atomistic borrowers do not internalize how their asset sales at date 1 affects the budget constraints of other borrowers through the asset price p . The planner can engineer a Pareto improvement over the competitive 1 equilibrium by addressing both externalities.28 Since policies designed to address pecuniary externalities have been analyzed extensively in the literature, we first focus on policies designed to address the information externality. Nevertheless, we show that a single policy tool alone cannot addressbothexternalitiessimultaneously. We can divide the state space (πL,τL) at date 1 into two regions, illustrated in the left panel 1 1 of Figure 4, in which we define the optimism of the borrower relative to the information set of the lender. In the ‘optimistic region’ of the state space, the borrower is overly optimistic relative to the better-informedlender,i.e. πB <πL. Theeconomyentersthisoptimisticregionwhenthelender’s 1 1 28Similarly, there are two broad sets of ex ante policies (policy interventions at date 0) which could implement a Paretoimprovement: policieswhichreducethelikelihoodandseverityoffiresale(e.g. leveragerestrictions,capital requirements), and policies which reduce opacity and improve information diffusion. In this section, we focus on date-1interventionsonly. 38

costshockisrelativelygood(lowτL). Inthe‘pessimisticregion’,theborrowerisoverlypessimistic 1 relative to the better-informed lender, i.e. πB ≥ πL. The economy enters this pessimistic region 1 1 when the lender’s cost shock is relatively bad (high τL). The partition between these two regions, 1 marked by the dotted line in both panels of Figure 4, is defined by tracing out the borrower’s posterior beliefs (cid:0) EB(cid:2) τL(cid:3) ,πB(cid:1) |q at different realizations of the equilibrium value q . Since the 1 1 1 1 1 social planner at date 1 observes the true state, the planner knows which region the economy is in atdate1. Notethattherelativefrequencywithwhichtheeconomywillendupinoneregionorthe otherdependsonthedistributionofshocksτL andnews. 1 Howdotheplanner’sdate-1choicescomparetothoseofagentsinthecompetitiveequilibrium? The social planner would make choices on behalf of agents in a way which increases the size of the total surplus and use transfers between agents to ensure that all agents remain at least as well off. Fire Sale Regime First we focus on the Fire Sale Regime. Suppose that the economy happens to be in the pessimistic region of Figure 4. Then, the borrower would be better off in expectation if it held more of the asset (i.e. liquidated less of it), so that aB is higher. This is for 1 tworeasons. First,inexpectation,theborrowerwouldbebetteroffifitbehavedasifπB=πL since 1 1 its portfolio choice would be making use of all information available to agents at date 1. Given that the borrower is risk-neutral, this implies that it would have a higher expected utility at date 1. Second,thereisapecuniaryexternalityatdate1inwhichborrowersdonotinternalizehowselling theassetdepressesthepriceoftheasset,makingitmorecostlytosellforotheragents. Therefore, in response to shocks, which leave the borrower relatively pessimistic, the planner would have the lender roll over more of the debt (i.e. the planner would choose a higher q than 1 whatthelenderwouldchoose). Thisisfortworeasons: theborrowerwouldbelesspessimisticand wouldselllessoftheriskyasset(thusmitigatingtheadverseeffectoftheinformationexternality); and by increasing the price of the asset p , which would reduce the need for the borrower to 1 sell (thus mitigating the adverse effect of the pecuniary externality). As a result, the borrower’s expected date-2 return will be higher, since the borrower’s holdings of the risky asset would now reflecttheadditionalinformationavailabletothelenderandplanner. For both of these reasons, the expected net return at date 2 on the borrower’s portfolio will be higher, and the planner can transfer portion of this higher return from the borrower to the lender. Since,inexpectation,date2outputwillbehigher,thistransferfromtheborrowertothelenderwill bepositiveinexpectation. Hence,thelenderwillalsobebetteroffatdate1. (Totheextentthatthe planner would reduce fire sales at date 1 ex post rather than ex ante, this would be an additional waythatplannerwouldincreasethetotalsurplus.) Suppose instead that the economy is in the optimistic region of Figure 4. Then the borrower would be better off in expectation if it held less of the asset (lower aB). Therefore, in response 1 39

to shocks, which leave the borrower relatively optimistic, the planner would choose a lower q . 1 The planner would then transfer some of the increased surplus which accrues to the lender to the borrowertoensuretheborrowerisnoworseoff. NormalRegime NowconsidertheNormalRegime. IntheNormalRegime,theallocation of aB is irrelevant since it is held fully by the borrower. However, q affects the allocation of cash 1 1 between the borrower and lender. By changing q in a way which internalizes this choice on the 1 borrower’sbeliefs,theplannercanallocatecashmoreefficientlybasedonτL andτB (thereturnto 1 1 cash of each agent). This will increase the total surplus. The planner can also design a lump-sum transfer Φ between the borrower and lender to share this surplus in a way which leaves neither 2 agentworseoffinexpectation. Thus,thedate-1competitiveequilibriumisgenericallyconstrained inefficient. Interpretation Inthecompetitiveequilibrium,thepriceofthecontractq (orequivalently 1 theinterestrate1/q )playstworoles: itfacilitatesareallocationofliquiditybetweentheborrower 1 and lender in response to news and shocks at date 1, and it (imperfectly) diffuses new information about the fundamental value of the asset from the better-informed lender to borrower.29 However, becauseq diffusesinformationonlyimperfectly,itleadsbeliefsoftheborrowertosystematically 1 diverge from fundamentals. Therefore, from a social perspective, there is a tradeoff associated with the price of lending q between reallocating liquidity in response to shocks and facilitating 1 informationsharing. Inthecompetitiveequilibrium,theequilibriumpriceq facilitatestheprivatelyoptimalsolution 1 tothistradeoff. However,thepresenceofinformationandpecuniaryexternalitiesmeansthatthese choicesarenotconstrainedefficient. Incontrast,theplannerinternalizestheconsequencesofboth roles of q on the allocation. Effectively, the planner’s solution decouples the two roles played by 1 q fromoneanother: theplanner’schoiceofq internalizestheeffectontheborrower’sinformation 1 1 set, while the date 2 transfer from the borrower to the lender is used to compensate the lender for thetimeandriskassociatedwithlendingtotheborrower. Pecuniary externality In addition to the information externality, the model features a pecuniary externality, similar to that outlined in Lorenzoni (2008), in which agents do not internalize how choices at date 1 affect the budget constraints of other agents through the asset price p . The planner can address this pecuniary externality using the tax γL on the lenders’ return on 1 cash holdings. By reducing the opportunity cost of lending, this tax forces lenders to internalize how their choices to lend affects liquidity needs of borrowers and therefore the effect on the price oftheriskyasset. Muchoftheliteratureonfinancialcrisesandsystemicriskhasfocusedontheroleofpecuniary 29Moregenerally,q servesaspriceoflendingandcompensatesthelenderfortheopportunitycostofitsfundsand 1 theriskassociatedwithlending. 40

externalities and the policies needed to address them. In addition to a pecuniary externality, our setting features an information externality. Since the constrained inefficiency of the competitive equilibrium is driven by two externalities, we need two policy tools to implement the constrained efficientallocation. Moreover,usingonlyonetoolisnotonlyinsufficient,butalsomayevenmake theallocationworseintermsofParetoefficiency. Therefore, policies, which address only pecuniary externalities (such as leverage or margin requirements) without addressing the information externality cannot implement constrained efficiency,andtheymayevenmakeagentsworseoffinsomestatesduetoinformationspillovers. For example, in states in which the borrower is optimistic relative to the better-informed lender, the borrower would be better off liquidating more of the asset and holding more cash. But a higher leverage or margin requirement would not incentivize this. Similarly, implementing a policy that addresses the information externality without also addressing the pecuniary externality may leave agentsworseoffinsomestates. Forexample,duringoptimisticfiresales,bothagentsmaybemade better off by the borrower’s relative optimism, as this reduces the extent to which the borrower liquidates the asset, and thus partially offsets the negative effects associated with the pecuniary externality. Thus, implementing the constrained efficient allocation requires a combination of policies designed to eliminate both externalities, such as the option facilities (discussed in Section 10) and Pigouviantaxesonliquidationmentionedabove. Liquidityfacilities Inpractice,policymakershaveimplementedregulatorypoliciesdesigned to reduce the inefficiencies associated with pecuniary externalities through ex post interventions such as liquidity facilities and asset purchases. Importantly, our model shows that these interventions may work to stabilize financial markets through an additional channel not discussed in the literature—theinformationchannel. Viewed through the perspective of our model, these episodes of financial turmoil are states of the world in which information becomes blurred by funding and market illiquidity. These dynamics increase pessimism about fundamentals. Government interventions, which directly increase market and funding liquidity, may boost investor confidence—that is, reduce pessimism about the fundamental value of financial assets—through the mechanisms we have outlined. Greater investor confidence, in turn, further increases funding and market liquidity. Indeed, several of the government interventions during periods of financial market turmoil appear to stabilize markets by restoring confidence in financial assets.30 Our model suggests the interactions between beliefs 30For example, a notable feature of some of the US government interventions made during the financial turmoil in 2020, such as the Primary Market Corporate Credit Facility and the Secondary Market Corporate Credit Facility, is that they seemed to stabilize financial markets even with very little transaction volume. To explain this, some policymakers have suggested that these policies restored confidence in these financial assets, which then stabilized liquidityconditionsandassetvaluations. 41

aboutfundamentalsandfundingandmarketliquiditymayplayaroleduringsuchepisodes. 10. Improving Informational Efficiency during Fire Sales The inefficiencies in this model arise in part from the informational inefficiency of financial markets: prices cannot fully reveal agents’ private information.31 In our setting, this informational inefficiency stems in part from the incompleteness of the risk markets. Moreover, our model shows that this informational inefficiency may be more acute when liquidity conditions are especially tightorloose,asthisimpairsabilityofpricestorevealprivateinformation. In this section, we consider the effect of policies, which may improve the informational efficiency of financial markets during liquidity dry-ups by creating new markets. To that end, we introduce a government, who can offer insurance contracts, which are contingent on the asset’s payoff. Unlike private agents, the government can commit to honoring its obligations and can enforce payment from its counterparties, even when the asset returns are low, giving it an advantage overtheprivatesectorinofferingthesecontracts. This exercise is motivated in part by policy interventions during recent episodes of financial turmoil in which central banks acted as a ‘dealer of last resort’ by acting as counterparty to securities trade when these markets froze, and effectively re-opening the markets for state-contingent securities, which shut down during the turmoil. In our setting, these policies may improve social welfare by improving the ability of prices to convey private information, rather than simply by reallocating risk from the private to the public balance sheets.32 To highlight this belief channel, we focus on policies in which the quantity of these insurance contracts supplied by the government to the private sector is infinitesimally small, so that the quantity of risk transferred from the private sector to the government’s balance sheet is negligible. Even these policies are sufficient to eliminatetheinefficienciesassociatedwiththeinformationexternality. To proceed, we first modify the setup by introducing a government. The government does not observe agent’s private information. At date 1, the government has access to lump-sum taxes and two distortionary taxes: the government can tax each unit revenue that the borrower obtains from liquidating the risky asset at date 1 at the rate γB; and the government can tax the date-2 return thatthelenderreceivesonitscashholdingsfromdate1attherateγL. Inaddition,thegovernment can issue two securities with state-contingent payoffs, each in a competitive market. One contract 31Itiswell known, thatintheabsence ofcompleteriskmarket, pricesmust notonlyclearmarketsand aggregate information,butalsoallocaterisk. Asaresult,assetpricesmayrevealinformationimperfectly(e.g. Stiglitz(1981)). 32Notethat,ingeneral,theallocationimplementedusingthispolicyinterventionwillnotcoincidewiththesolution totheconstrainedplanner’sproblemoutlinedinSection9. Thisisbecause,whiletheplanner’ssolutionstillfeatures privateinformation,thepolicyinterventionoutlinedinthissectionpubliclyrevealsallprivateinformation. 42

resembles a put option on the risky asset while the other resembles a call option.33 This is a parsimonious way to capture the provision of insurance thatmany of the government interventions in credit and asset markets observed in the past brought about.34 Modeling government intervention in this parsimonious way is a simple way to capture the insurance aspect of government intervention, which can play a role in stabilizing markets through agents’ information and beliefs, a channelnotheretoforeidentifiedintheliterature. p TheputoptioncontractinvolvesatransferofT unitsoftheconsumptiongoodatdate2tothe 2 buyerofthecontractifR=Rand0otherwise. Thecontractissoldinacompetitivemarketatdate 1 at a spot price p p . The call option contract involves a transfer of Tc units of the consumption 1 2 good at date 2 to the buyer of the contract if R=R and 0 otherwise, and is sold in a competitive market at date 1 at a spot price pc. We assume that the prices p p ,pc are publicly observable. 1 1 1 Therefore,ifthesesecuritiesaretradedinpositivesupply,theequilibriumpricesoftheoptionwill beinformativetotheborroweraboutthelender’sprivateinformation. Thus, the government has three types of tools to intervene in financial markets at date 1: the distortionary taxes γL,γB allow it to reallocate liquidity at date 1, the put and call options allow it to provide insurance at date 1, and the lump-sum transfers allow it to redistribute wealth. The government’sbudgetconstraintatdate1is (cid:0) aB−aB(cid:1) γ B+p p ω p +pc ω c+Γ =κ G (26) 0 1 1 1 1 1 1 1 where γB is the proportional tax on the borrower’s liquidation of the asset at date 1, ω p and ωc 1 1 is the total quantity of the put and call options sold to private agents (each of which is defined as ω p ≡ω L,p +ω B,p andωc ≡ω L,c +ω B,c ),respectively,Γ arelump-sumtaxesatdate1,andκG is 1 1 1 1 1 1 1 1 thegovernment’sdate-1cashholdings,Thus,thegovernment’sdate-1cashholdingsconsistofthe proceedsofsalesoftheputandcalloptionsandtaxescollectedfromtheborrower’sliquidationof theriskyassetandlump-sumtaxes. Thegovernment’sdate-2budgetconstraintis κ G+Γ +γ L κ L =ω p T p1{R=R}+ω cTc1(cid:8) R=R (cid:9) . (27) 1 2 1 1 2 1 2 33We assume that, due to limited enforcement, private agents cannot replicate these securities option facilities or trade other financial contracts (more than collateralized lending) in equilibrium. The government, by contrast, can enforcepaymentandsocanofferthesecontracts,reducingthemissingmarketsproblem. 34For example, consider the credit and liquidity facilities implemented by the Federal Reserve and backed by the Treasury in 2020, such as the Secondary Market Corporate Credit Facility or the Commercial Paper Funding Facility, enacted during a period of severe market turmoil in which the prices of many financial assets fell dramatically. BackstoppedbytheTreasury,theFedpurchasedqualifyingsecuritiesfrominvestorsontheopenmarket. Ifthevalue ofthesesecuritiesroseinthefuture,investorscouldbuythembackatthemarketprice,whileifnot,investorswould besavedfromincurringfurtherlosses. Therefore,atsomelevel,theFedandTreasuryprovidedimplicitinsuranceto investors. 43

Thus, the government’s date 2 expenditure must be met out of its cash holdings stored from date 1, any lump-sum taxes at date 2, or tax revenue on the lender’s cash holdings from date 1. The marketclearingconditionsforeachoptionatdate1are p G,p ω =ω 1 1 ω c =ω G,c , 1 1 G,p G,c whereω ,ω arethetotalquantitiesofeachoptionsuppliedtothemarketbythegovernment. 1 1 The lender’s date 1 and date 2 budget constraints, respectively, adjusted for the lender’s holdingsoftheoptionfacilityare κ L−κ L+p p ω L,p +pc ω L,c ≤dL−q dL 1 0 1 1 1 1 0 1 1 CL ≤τ L κ L+RddL+ω L,p T p1{R=R}+ω L,c Tc1(cid:8) R=R (cid:9) . 2 1 1 2 1 1 2 1 2 While the lender’s first order condition for lending to the borrower is similar to (2), it reflects a higheropportunitycostofholdingcashduetotheunittaxγL onthereturnoncashatdate2: EL(cid:2) Rd(cid:3) q = 1 2 . (28) 1 τL−γL 1 Thelender’sfirstorderconditionforinvestingintheputandcalloptions,respectively,are T p πL p p = 2 1 (29) 1 τL−γL+ξL 1 κ1 and Tc(cid:0) 1−πL(cid:1) pc = 2 1 . (30) 1 τL−γL+ξL 1 κ1 p Tp πB p Similarly, the borrower’s first order conditions for each option are given by p = 2 1 and p = 1 τB+ξB 1 1 κ1 Tc(1−πB) 2 1 . Inequilibrium, thelender willbe themarginal buyerfor exactlyone ofthe twooptions, τB+ξB 1 κ1 and the borrower will be the marginal buyer for the other, depending on the relative optimism and liquidity shocks of each agent.35 Therefore, in equilibrium, either p p or pc will be priced by the 1 1 35In particular, if π 1 L < π 1 B , in which case the borrower’s marginal valuation for the put option is higher than the τL τB 1 1 (1−πL) (1−πB) lender’s,thenitmustbethat 1 > 1 ,sothatthelender’smarginalvaluationforthecalloptionishigherthan τL τB 1 1 theborrower’s(andviceversa). Andasaresultwewilleitherhavethatωp=ωL,p andωc=ωB,c,orthatωp=ωB,p 44

better-informed lender. Let po ∈ (cid:8) p p ,pc(cid:9) denote the price of the option, which is priced by the 1 1 1 better-informedlenderinequilibrium. NotethatthecompetitiveequilibriumoutlinedinSection6isidenticaltothecompetitiveequilibrium in this modified economy when the government is completely passive and issues 0 securitiesortaxes: ω G,p ,ω G,c ,Γ ,Γ ,γL,γB,κG =0. 1 1 1 2 1 The government can engineer a Pareto improvement by supplying a strictly positive (but arbi- G,p G,c trarily small) quantity of the the put and call options to the market, ω ,ω >0. As we show 1 1 below,settingthesequantitiesarbitrarilyclosetozerosufficestoreapthefullsocialbenefitsassociatedwiththeseoptionsbyeliminatingtheinefficienciesassociatedwiththeinformationexternality by incentivizing agents to credibly and publicly reveal their private information at date 1. As a result, the market prices of these options reveal agents’ private information, thereby eliminating the identificationproblem,whichgivesrisetotheinformationexternality. To see this, note from lender’s optimality conditions (29) and (30) that the observed price po 1 will be informative to the borrower about the lender’s private information πL,τL, over and above 1 1 the information already provided by q in (28). That is, because the payoff profile of the option 1 differs than that the loan from the lender to the borrower, the lender’s optimality conditions for the option and the loan are linearly independent. (In contrast, the price of the other option—for whichtheless-informedborroweristhemarginalbuyer—willbeuninformativetotheborrower,as it will simply reflect the borrower’s beliefs.) Therefore, the borrower’s knowledge of the lender’s optimality conditions, together with the observations of q ,po, can be used to separately identify 1 1 τL and πL. This is illustrated in Figure 5, where po is the price of the option priced by the better- 1 1 1 informedlender. Hence, the publicly observable price of the option facility po allows the borrower to perfectly 1 identify the lender’s private information. According to the characterization of the borrower’s posterior belief (18), the borrower’s posterior is then identical to that of the lender πB =πL. In this 1 1 manner, by publicly revealing private information, the option facility eliminates the relative pessimismandoptimismoftheborrowerassociatedwiththeinformationexternality. Discussion By selling a put option on the risky asset, the government is effectively bearing risk associated with the aggregate shock at date 2.36 The extent to which the government bears this risk is proportional to the quantity of the option it supplies to the market ωG. Importantly, however, the price po fully reveals the lender’s private information even as the quantity ωG of 1 options sold is infinitesimally small (as long as ωG > 0). Therefore, the full social benefits of the option facility can be realized even if the government sells an arbitrarily small quantity of the andωc=ωL,cinequilibrium,sothateachoptionisheldexclusivelybyoneagent. 36Sincethegovernmentbearsriskthroughtheoptionfacility,thisfacilitywouldbeakintothefacilitiesoperatedby theFederalReserveundersection13(3)oftheFederalReserveActinwhichtheFederalReserveprovidesfinancing totheprivatesectorbuttheTreasuryprovidesabackstopforlossesincurredontheseoption. 45

Figure5: Effectoftheoptionfacility Note:Thisfigureillustratestheeffectoftheoptionfacilityontheborrower’sidentificationproblematdate1.Thesolid curveinthefigureplotsthecombinationsofthelender’sliquidityshockandbeliefs, (cid:0) τL,πL(cid:1) ,whichareconsistentwith 1 1 theequilibriumq ,basedonthelender’soptimalityconditionfordL. Thedash-dottedcurveplotsthecombinationsof (cid:0) τL,πL(cid:1) whichar 1 econsistentwiththeequilibriumpriceoftheopt 1 ion po,basedonthelender’soptimalitycondition 1 1 1 forinvestmentinthefacility. Theborrower’sobservationsofbothq and po,togetherwithknowledgeofthelender’s 1 1 optimalityconditions,aresufficienttoinferthetruerealizationsofτL∗andπL∗separately. 1 1 46

option, and bears negligible risk. This is because the social benefit of the facility arises not from transferring risk to the government’s balance sheet, but rather in the information provided by the priceofthesignal. WhenωG>0isarbitrarilyclosetozero,theeconomyapproachesthecommon informationbenchmarkeconomyanalyzedinSection8. The identification problem in our setting ultimately derives from a missing markets problem: Because of the absence of the insurance markets against cost shocks and the shock to the risky asset, there is only one price q , which conveys information about two orthogonal shocks.37 By 1 introducing a market for insurance against the shock to risky asset, the government adds a price signalwhich,togetherwithq ,fullyrevealsagents’privateinformation.38 1 11. Conclusion Wedevelopedamodelofheterogeneousagents,collateralizeddebt,learning,andfiresalestoshed light on how the availability of funding liquidity affects agents’ perceptions about fundamentals, and the role that this may play in asset booms and busts. The model shows that beliefs about fundamentalsareendogenouslyshapedbytheavailabilityofliquidityinfundingandassetmarkets. As a result, cost (liquidity) shocks, which are orthogonal to asset fundamentals can cause beliefs to become systematically divorced from fundamentals. Moreover, this informational inefficiency may be more acute when liquidity conditions are especially tight or loose, as this impairs the ability of prices to reveal private information about fundamentals. As a result of this mechanism, loosefundingliquiditycangenerateover-optimismaboutfundamentalsleadingtoover-investment, whiletightfundingliquiditycanleadtoexcessivepessimismaboutfundamentalsandfiresales. We showed that the competitive equilibrium is generically constrained inefficient due to the presence of a new information externality, and we characterized interventions in funding markets, which can generate a Pareto improvement. Finally, we showed that, by acting as a dealer-of-last-resort, a central bank can mitigate the informational inefficiency of markets by reopening derivatives markets,whichwouldotherwiseshutdownduringepisodesoffinancialturmoil. 37Forsimplicity,weassumethatmarketsareincompleteinoursettingandweleaveasidethemicro-foundationfor thisassumption. Inpractice,financialmarketsareindeedincompleteandprivateinformationcannotbefullyinferred frompricesignals. Wetakethisasgivenanddrawouttheimplicationsofthisforhowbeliefsareformed. 38Inpractice,theremaybeseveralreasonswhythegovernmentisinabetterpositiontoofferthistypeofinsurance. For example, the government, due to its ability to tax, may be able to absorb the risk associated with the aggregate shockbetterthanprivateagents. 47

A. Appendix: Omitted Proofs A.1. Derivation of the lender’s posterior beliefs First, note that the bounds of the posterior belief of the lender is πL ∈ (0,1). The bounds are 1 derived from the fact that sL =R+ε with ε ∈(−∞,∞), so any realization of sL is possible from 1 1 bothstateoftheworldR=R¯ andR=R. Thelenderupdatesposteriorbeliefsasin(17) π λ (cid:0) εL =sL−R (cid:1) π L = 0 ε 1 1 . 1 π λ (cid:0) εL =sL−R (cid:1) +(1−π )λ (cid:0) εL =sL−R¯(cid:1) 0 ε 1 1 0 ε 1 1 This expression of πL is possible by using Bayes’ rule for events with a positive mea- 1 sure, and then taking limits. For any δ > 0, the events (cid:8) sL−R≤εL ≤sL−R+δ (cid:9) and 1 1 1 (cid:8) sL−R≤εL ≤sL−R+δ (cid:9) arewelldefinedandhavepositiveprobability. Therefore,Bayes’rule 1 1 1 implies P(R=R|sL−R≤ε L ≤sL−R+δ) 1 1 1 π P (cid:0) sL−R≤εL ≤sL−R+δ (cid:1) = 0 1 1 1 (cid:0) (cid:1) (cid:0) (cid:1) π P sL−R≤εL ≤sL−R+δ +(1−π )P sL−R≤εL ≤sL−R+δ 0 1 1 1 0 1 1 1 ´ π sL 1 −R+δ ε(cid:48)dΛ (ε(cid:48)) 0 sL−R ε = ´ 1 ´ , (31) sL−R+δ sL−R+δ π 1 ε(cid:48)dΛ (ε(cid:48))+(1−π ) 1 ε(cid:48)dΛ (ε(cid:48)) 0 sL−R ε 0 sL−R ε 1 1 where P is the probability function and both the denominator and numerator are positive. As δ →0,wehave P(R=R|sL−R≤ε L ≤sL−R+δ)→P(R=R|ε L =sL−R)=π L 1 1 1 1 1 1 ˆ sL−R+δ 1 ε (cid:48)dΛ (ε (cid:48))→λ (ε L =sL−R) ε ε 1 1 sL−R ˆ 1 sL−R+δ 1 ε (cid:48)dΛ (ε (cid:48))→λ (ε L =sL−R). ε ε 1 1 sL−R 1 Thus,takingthelimitofδ →0onbothsidesof(31)resultsin(17). A.2. Proof of Proposition 1 Proof. 1. B’ s information at date 1, IB, is given by the set of all variables observable to all agents, 1 48

which are prices p and q . In contrast, L has additional information of τL and sL, denoted by IL, 1 1 1 1 1 ontopofthepubliclyavailableinformation. Bythelawofiteratedexpectation EB(cid:2) EL[x] (cid:3) =E (cid:2) E (cid:2) x|IL(cid:3) |IB(cid:3) =E (cid:2) x|IB(cid:3) =EB[x]. 1 1 1 1 1 1 Applyingthedefinitionofposteriorbeliefsπi =Ei[1{R=R}]fori=L,Bleadstostatement1. 1 1 2. BythefirststatementofProposition1and(19),wehave ˆ 1 (cid:18) 1−(1−R)π (cid:19) (cid:26) 1−(1−R)π (cid:27) π B(q )= πλ 1 τ < <τ¯ dG (π), 1 1 T q q π 0 1 1 foragivenq . 1 First,notethatthereexistsπˆ(q)suchthat 1−(1−R)πˆ(q) =τ 0 q for any q<q . Also, note that πˆ(q) is decreasing in q. From (2), we can rearrange the difference 0 betweenthepriorforτ,τ ,andrealizedτ foragivenπ andequilibriumpriceqas 0 (cid:12) (cid:12) (cid:12)1−(1−R)π (cid:12) |τ−τ 0 |=(cid:12) −τ 0(cid:12), (cid:12) q (cid:12) whichcanbefurthersimplifiedas (cid:12) (cid:12) (cid:12)1−(1−R)π (cid:12) 1 1 (cid:12) −τ 0(cid:12)= |1−(1−R)π−1+(1−R)πˆ(q)|= |(1−R)(π−πˆ(q))|. (32) (cid:12) q (cid:12) q q Weclaimthatforanyπ andq(cid:48) >q, (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)1−(1−R)π (cid:12) (cid:12)1−(1−R)π (cid:12) (cid:12) (cid:12) q −τ 0(cid:12) (cid:12) <(cid:12) (cid:12) q(cid:48) −τ 0(cid:12) (cid:12) (33) andtherefore, (cid:18) (cid:19) (cid:18) (cid:19) 1−(1−R)π 1−(1−R)π λ −τ >λ −τ (34) T q 0 T q(cid:48) 0 bythefunctionalformassumptiononλ . T Byequation(32),(33)istrueif 1 1 |(1−R)(π−πˆ(q))|< |(1−R)(π−πˆ(q(cid:48)))| q q(cid:48) 49

holds. Thepreviousequationholdsif q(cid:48) |π−πˆ(q(cid:48))| < q |π−πˆ(q)| holds,whichisequivalentto 1−(1−R)πˆ(q(cid:48)) 1 −πˆ(q(cid:48)) τ 1−R |π−πˆ(q(cid:48))| 0 = < (35) 1−(1−R)πˆ(q) 1 |π−πˆ(q)| −πˆ(q) τ 1−R 0 bythedefinitionofπˆ(q). First,considerthecaseinwhichπ iseitherπ >πˆ(q)orπ <πˆ(q(cid:48)). Then,(35)becomes 1 −πˆ(q(cid:48)) 1−R π−πˆ(q(cid:48)) < . (36) 1 π−πˆ(q) −πˆ(q) 1−R x−b Notethatforanyx and0<b<c, isdecreasinginx,because x−c (cid:18) (cid:19) x−b ∂ x−c b−c = <0. ∂x (x−c)2 1 Thus,(36)holdsbecauseπ ≤1< . 1−R Finally,considerthecaseinwhichπ isπˆ(q(cid:48))<π <πˆ(q). Then, π−πˆ(q(cid:48)) |π−πˆ(q(cid:48))| < π−πˆ(q) |π−πˆ(q)| holdsandby(36),(35)alsoholds. Therefore,(33)and(34)hold. By the claim, the support of the conditional expectation is decreasing in q for each given 1 realizationofπ andforthegivenpdfλ as T (cid:18) (cid:19) (cid:26) (cid:27) (cid:18) (cid:19) (cid:26) (cid:27) 1−(1−R)π 1−(1−R)π 1−(1−R)π 1−(1−R)π λ 1 τ < <τ¯ >λ 1 τ < <τ¯ T q q T q(cid:48) q(cid:48) foranyq<q(cid:48). Thus,πB isdecreasinginq . 1 1 50

A.3. Solving the Borrower’s Optimization Problem max EB(cid:2) CB(cid:3) (37) 1 2 c 1 ,f 1 ,aB 1 ,κ 1 B s.t. q c −c ≥ p (cid:0) aB−aB(cid:1) +κ B−κ B, 1 1 0 1 1 0 1 0 CB ≤τ B κ B+aBR−c Rd(f ) 2 1 1 1 1 2 1 c ≤aB, 1 1 c ≥0, aB ≥0, κ B ≥0 1 1 1 Notethatthedate2budgetconstraintimplies CB =τ B κ B+aBR−c Rd(f ), 2 1 1 1 1 2 1 whilethedate1budgetconstraintimplies κ B =q c −c −p (cid:0) aB−aB(cid:1) +κ B. 1 1 1 0 1 1 0 0 Combiningthesetwoyields CB =τ B(cid:0) q c −c −p (cid:0) aB−aB(cid:1) +κ B(cid:1) +aBR−c Rd(f ) 2 1 1 1 0 1 1 0 0 1 1 2 1 ThereforewecanwritetheLagrangianas LB =EB(cid:2) CB(cid:3) +µ B(cid:0) aB−cB(cid:1) +ξ BcB+ξ BaB+ξ B (cid:0) q cB−cB+p (cid:2) aB−aB(cid:3) +κ B(cid:1) (38) 1 1 2 1 1 1 c 1 1 a 1 1 κ1 1 1 0 1 0 1 0 where we have Rd(f )≡min{R, f }, and E (cid:2) Rd(f ) (cid:3) =(1−π )f +π min{R, f }. Replacing q 2 1 1 2 1 1 1 1 1 1 yields CB =τ B(cid:0) q cB−cB−p (cid:0) aB−aB(cid:1) +κ B(cid:1) +aBR−cBRd(f ). 2 1 1 1 0 1 1 0 0 1 1 2 1 Then,theexpectationisgivenby (cid:104) (cid:105) EB(cid:2) CB(cid:3) =EB τ B(cid:0) q cB−cB−p (cid:0) aB−aB(cid:1) +κ B(cid:1) +aBR−cBRd(f ) 1 2 1 1 1 1 0 1 1 0 0 1 1 2 1 (cid:104) (cid:105) EB(cid:2) CB(cid:3) =τ B(cid:0) q cB−cB−p (cid:0) aB−aB(cid:1) +κ B(cid:1) +aBEB[R]−cBEB Rd(f ) 1 2 1 1 1 0 1 1 0 0 1 1 1 1 2 1 Hence,theLagrangianis: 51

LB =EB(cid:2) CB(cid:3) +µ B(cid:0) aB−cB(cid:1) +ξ BcB+ξ BaB+ξ B (cid:0) cBq −cB+p (cid:2) aB−aB(cid:3) +κ B(cid:1) (39) 1 1 2 1 1 1 c 1 1 a 1 1 κ1 1 1 0 1 0 1 0 whereEB(cid:2) CB(cid:3) isgivenbytheabove. 1 2 FOCforcB: 1 dEB(cid:2) CB(cid:3) 1 2 −µ B+ξ B +ξ Bq =0 dcB 1 c 1 κ1 1 1 (cid:104) (cid:105) τ Bq −EB Rd(f ) −µ B+ξ B +ξ Bq =0 1 1 1 2 1 1 c 1 κ1 1 (cid:104) (cid:105) (cid:0) τ B+ξ B(cid:1) q −EB Rd(f ) −µ B+ξ B =0 1 κ1 1 1 2 1 1 c 1 FOCforaB: 1 dEB(cid:2) CB(cid:3) 1 2 +µ B+ξ B −ξ B p =0 daB 1 a 1 κ1 1 1 −τ Bp +EB[R]+µ B+ξ B −ξ B p =0 1 1 1 1 a 1 κ1 1 − (cid:0) τ B+ξ B(cid:1) p +EB[R]+µ B+ξ B =0 1 κ1 1 1 1 a 1 A.4. Proof of Lemma 1 Proof. ProofofPart(A): ByreplacingµB in(5)with(6)andusingthecontractpriceq thatmakesLtolendinapositive 1 1 amount: q (cid:0) τ B+ξ B(cid:1) +EB[R]−EB[Rd]−τ Bp +ξ B −ξ B p +ξ B =0, 1 1 κ1 1 1 1 1 a 1 κ1 1 c 1 whichcanberearrangedas (cid:104) (cid:105) EB R−Rd +ξ B +ξ B = (cid:0) τ B+ξ B(cid:1) (p −q ). 1 c 1 a 1 1 κ1 1 1 ConsiderthecaseinwhichBborrowsapositiveamountsoa ,c >0. Then,theaboveequality 1 1 52

canberearrangedas EB(cid:2) R−Rd(cid:3) (1−πB)(R−1) τ B+ξ B = 1 = 1 , (40) 1 κ1 p −q p −q 1 1 1 1 which implies that the expected return of holding the asset with the collateralized debt net of B’s shadow value of asset equals to the sum of cash return of B and B’s shadow value of cash. The returnoftheleveragedassetholdingontheright-handsideof(40)canbegreaterthanreturnofB’s storagetechnology,butthenBhastoexhaustallthecash,ξB >0,andBcannotpayforadditional κ1 down payment (or cash collateral or variation margin), p −q (f ). In addition, B might want to 1 1 1 generate more asset because the return of the leveraged asset holding is exceedingly profitable. If thisisthecase,thenthepricewilladjusttothepointthattheequalityholds. Now we show that the collateral constraint is generically binding. If B is borrowing zero amount—that is, c =0, B does not have enough cash to repay the loans to L unless B liquidates 1 all the assets, implying aB =c =0. B borrows a positive amount c >0 only if the return from 1 1 1 rollingoverthedebtisgreaterthanorequaltothecashreturn. Thus, EB[R−Rd] τ B+ξ B = 1 (41) 1 κ1 p −q 1 1 holdsinanyequilibriumwithc >0. Ifthecollateralconstraintisnotbinding,thenby(6), 1 EB[R] τ B+ξ B = 1 (42) 1 κ1 p 1 holds. (41)and(42)implythat EB[Rd] (1−πB)+πBR τ B+ξ B = 1 = 1 1 , 1 κ1 q q 1 1 whichholdsonlyatnon-genericrealizationofq asπB isdecreasinginq . Thereturnfromlever- 1 1 1 aging the asset purchase, (41), should exceed the return from the asset purchase without leverage, (42), because otherwise B will not purchase with leverage—that is, c = 0—which is a contra- 1 diction. Even if the previous equation holds, which is a not a generic case of parameters, B is indifferent between purchasing the asset with leverage and without leverage. Therefore, we can impose a tie-breaking rule for B, choosing to maximize c when indifferent.39 Hence, B leverages 1 allofitsassetpurchaseasc =aB,andthecollateralconstraintisbinding. 1 1 ProofofPart(B): 39Thistie-breakingrulecanbejustifiedbyassumingthatBobtainsinfinitesimallysmallutilityofholdinganasset ascollateral. 53

Theborrower’soptimalitycondition(11),whichcanberearrangedas (cid:104) (cid:105) (cid:0) ξ B +τ B(cid:1) (p −q )=EB[R]−EB Rd(f ) κ1 1 1 1 1 1 2 1 1 (cid:104) (cid:104) (cid:105)(cid:105) p −q = EB[R]−EB Rd(f ) . 1 1 ξB +τB 1 1 2 1 κ1 1 SinceξB ≥0andτB >0,wehave p >q ifandonlyif κ1 1 1 1 (cid:104) (cid:105) EB[R]>EB Rd(f ) 1 1 2 1 (1−π B)R+π BR>(1−π B)f +π BR 1 1 1 1 1 (1−π B) (cid:0) R− f (cid:1) >0 1 1 ThisholdssinceR>1byassumption. A.5. Proof of Proposition 2 Proof. The proof is based on the full characterization of date-1 equilibrium in Proposition 5 in AppendixA.6. ProofofPart(A): As in Case 1 in Proposition 5, p > F(cid:48)(0) implies that the borrower is not selling any assets 1 to the traditional sector, and the asset price equals the fundamental value of the asset based on the borrower’s beliefs. Whenever p <F(cid:48)(0) in equilibrium, the asset is sold to the traditional sector 1 inapositiveamount,i.e. therearefiresales,asmoresalestothetraditionalsectordepressesprices asF(cid:48)(aT)<F(cid:48)(0). 1 ProofofPart(B): By(17),πL increasesassL decreases. Also,q isdecreasinginπL andτL by(2),implyingthe 1 1 1 1 1 firsthalfofthestatement. InCase2inProposition5,lowerq resultsinlargeraT andlower p ,as 1 1 1 c −q aB−κB q + 0 1 0 0 1 aT 1 isdecreasinginq becauseaB ≥aT,and 1 0 1 c −q aB−κB p =F(cid:48)(aT)=q + 0 1 0 0 . 1 1 1 aT 1 Therefore, a decrease in q will increase the liquidity shortage of the borrower, leading to a larger 1 54

amountoffiresalesandlowerassetprice. Moreover, the borrower’s posterior belief πB is decreasing in q by Proposition 1. In Case 1 1 3 in Proposition 5, higher πB results in lower price through this belief channel. This is because 1 the borrower’s lower valuation of the asset should be met by the lower marginal valuation of the traditionalsectorthroughthedecreaseinF(cid:48)(aT),i.e. largeraT. 1 1 ProofofPart(C): In Case 2 in Proposition 5, the borrower sells the asset because the borrower has to repay the date-0 debt contract and their cash holdings are not sufficient as q aB < cB−κB. However, 1 0 0 0 the borrower still believes the fundamental value of the asset is above the marginal valuation of the traditional sector evaluated at 0, F(cid:48)(0), which is always above the market price as F(cid:48)(0) > F(cid:48)(aT)= p . Therefore, we have pB >F(cid:48)(0)> p in a liquidity driven fire sale. In Cases 3 and 1 1 1 1 4 in Proposition 5, the borrower’s valuation of the asset is below the marginal valuation of the traditional sector evaluated at 0, as pB < F(cid:48)(0), which is why the borrower sells the asset, i.e. 1 belief driven fire sale. In Case 3, the borrower has an interior solution, so that pB = F(cid:48)(aT) = 1 1 p . However, in Case 4, the borrower values the asset even less than the traditional sector does, 1 implyingthat pB <F(cid:48)(aB)= p . 1 0 1 A.6. Full Characterization of Date-1 Equilibrium IntheNormalRegime,wehave aT,ξ L,ξ B,ξ B,ξ L,ξ B =0 1 d 1 c 1 a 1 κ1 κ1 EL(cid:2) Rd(f ) (cid:3) q = 1 2 1 1 τL 1 EB[R]−EB(cid:2) Rd(f ) (cid:3) p = 1 1 2 1 +q 1 τB 1 1 aB =cB =dL =aB 1 1 1 0 κ B =q aB−cB+κ B 1 1 0 0 0 κ L =κ L+dL−q aB 1 0 0 1 0 κ T =κ T −p aT 1 0 1 1 55

CB =τ B κ B+aBR−aBRd(f ) 2 1 1 1 1 2 1 CL =τ L κ L+RddL 2 1 1 2 1 CT =κ T +F(aT) 2 1 1 µ B = (cid:0) τ B+ξ B(cid:1) p −EB[R]−ξ B 1 1 κ1 1 1 a 1 µ T =F(cid:48)(0)−p 1 1 π λ (cid:0) εL =sL−R (cid:1) π L =Pr (cid:0) R=R|sL,I (cid:1) = 0 ε 1 1 1 1 0 (1−π )λ (cid:0) εL =sL−R (cid:1) +π λ (cid:0) εL =sL−R (cid:1) 0 ε 1 1 0 ε 1 1 ˆ 1 (cid:18) 1−(1−R)π (cid:19) (cid:26) 1−(1−R)π (cid:27) π B = πλ 1 τ < <τ¯ dG (π) 1 T q q π 0 1 1 IntheFireSaleRegime,wehave µ T,ξ B,ξ B,ξ L,ξ L,ξ B =0 1 c 1 a 1 d 1 κ1 κ1 EL(cid:2) Rd(f ) (cid:3) q = 1 2 1 1 τL 1 p =F(cid:48)(aT) 1 1 (cid:32) EB[R]−EB(cid:2) Rd(f ) (cid:3) EL(cid:2) Rd(f ) (cid:3)(cid:33) aB =aB−F(cid:48)−1 1 1 2 1 + 1 2 1 1 0 τB τL 1 1 aT =aB−aB 1 0 1 cB =dL =aB 1 1 1 κ B =q aB−cB+κ B−p (cid:0) aB−aB(cid:1) 1 1 1 0 0 1 1 0 56

κ L =dL−q dL+κ L 1 0 1 1 0 κ T =κ T −p aT 1 0 1 1 CB =τ B κ B+aBR−aBRd(f ) 2 1 1 1 1 2 1 CL =τ L κ L+RddL 2 1 1 2 1 CT =κ T +F(aT) 2 1 1 µ B =τ Bp −EB[R] 1 1 1 1 π λ (cid:0) εL =sL−R (cid:1) π L =Pr (cid:0) R=R|sL,I (cid:1) = 0 ε 1 1 1 1 0 (1−π )λ (cid:0) εL =sL−R (cid:1) +π λ (cid:0) εL =sL−R (cid:1) 0 ε 1 1 0 ε 1 1 ˆ 1 (cid:18) 1−(1−R)π (cid:19) (cid:26) 1−(1−R)π (cid:27) π B = πλ 1 τ < <τ¯ dG (π) 1 T q q π 0 1 1 Belowisthefullsolutionofthedate-1equilibriumandderivation. Proposition5. Theequilibriumatdate1canbecharacterizedasthefollowing: (1−πB)(R¯−1) (1−πL)+πLR 1. Normal Regime: If F(cid:48)(0) < 1 + 1 1 and q aB ≥ cB−κB, then τB τL 1 0 0 0 1 1 c =aB=aB=dL,aT =0,κB=q aB−cB+κB,andtherewillbenofire-salesinthemarket 1 1 0 1 1 1 1 0 0 0 andtheassetpriceis (1−πB)(R¯−1) (1−πL)+πLR p = 1 + 1 1 1 τB τL 1 1 oranynumberlessthanorequalto p ifq aB−cB =κB,whichimpliesκB =0. 1 1 0 0 0 1 2. Fire Sale Regime with the borrower holding only the risky asset: If F(cid:48)(0) < (1−πB)(R¯−1) (1−πL)+πLR 1 + 1 1 but q aB < cB−κB, then c = aB = aB−aT = dL, aT τB τL 1 0 0 0 1 1 0 1 1 1 1 1 57

isdeterminedby c −q aB−κB F(cid:48)(aT)=q + 0 1 0 0 , 1 1 aT 1 κB =0,andtheassetpriceis p =F(cid:48)(aT). 1 1 1 3. Fire Sale Regime with the borrower at interior solution in portfolio choice: If F(cid:48)(0) > (1−πB)(R¯−1) (1−πL)+πLR 1 + 1 1 >F(cid:48)(aB)holds,thenc =aB =dL isdeterminedby τB τL 0 1 1 1 1 1 (1−πB)(R¯−1) (1−πL)+πLR F(cid:48)(aB−c )= 1 + 1 1 , 0 1 τB τL 1 1 (1−πB)(R¯−1) (1−πL)+πLR withaT =aB−c , p =F(cid:48)(aB−c )= 1 + 1 1 andκB=q aB− 1 0 1 1 0 1 τB τL 1 1 1 1 1 cB+κB. 0 0 (1−πB)(R¯−1) 4. FireSaleRegimewithmarketcollapse(Bholdsonlycash): IfF(cid:48)(aB)> 1 + 0 τB 1 (1−πL)+πLR 1 1 ,thenc =aB=dL=0,aT =aB,andκB=p aB−cB+κBwith p =F(cid:48)(aB), τL 1 1 1 1 0 1 1 0 0 0 1 0 1 soalltheassetsaresoldtoT andtherewillbenodebtcontractbetweenBandL. Proof. From Lemma 1, we know that the collateral constraint is binding, and the asset price is greaterthanthecontractprice. Nowwesolvefortheequilibriumprice p . 1 Case 1. Consider the case in which B holds a positive amount of assets, a = c > 0. The 1 1 reservationassetprice pB forBthatmakesBindifferentbetweenpurchasingtheassetandholding 1 cashis (1−πB)(R¯−1) (1−πL)+πLR pB = 1 + 1 1 . (43) 1 τB τL 1 1 Case1.1. IfBisnotsellinganyofitsassetduetobudgetconstraint,whichispossibleonlyif q aB+κ B ≥c , 1 0 0 0 then equilibrium price becomes p = pB, because otherwise T will be the only marginal buyer of 1 1 theasset,whichimpliesa =0,acontradiction. 1 Case1.2. IfBlackscashtorepaythedebtas q aB+κ B <c , 1 0 0 0 58

thentheassetpricecanbelowerthan pB andξB >0as 1 κ1 (1−πB)(R¯−1) (1−πL)+πLR pB > p = 1 + 1 1 =F(cid:48)(aT), 1 1 τB+ξB τL 1 1 κ1 1 because B liquidates some of the assets to T in order to pay the debt to L even though B prefers nottodoso. Therefore,theassetpricecouldbelowerthanthemarginalreturnoftheassetinsuch equilibrium and determined by the traditional sector’s inverse demand. B will liquidate the assets uptothenecessaryamountneededtomatchthebudgetconstraintwithκB =0, 1 aTF(cid:48)(aT)=c −q (aB−aT)−κ B 1 1 0 1 0 1 0 insuchequilibrium. Rearrangingthetermsyields c −q aB−κB F(cid:48)(aT)=q + 0 1 0 0 , (44) 1 1 aT 1 and p =F(cid:48)(aT). NotethatthereexistsauniqueaT inthiscase,becauseF(cid:48) isstrictlydecreasing. 1 1 1 Finally,thebudgetconstraintdeterminesthecashholdingsas κ B =q aB−cB+κ B, 1 1 0 0 0 thatpinsdowntheoptimaldecisionvectorofBifaT =0. IfaT >0,themarginalbuyerwillbeT, 1 1 andthepriceoftheassetwillbe p =F(cid:48)(aB−c ). (45) 1 0 1 Combining(45)with(40)yields (1−πB)(R¯−1) (1−πL)+πLR F(cid:48)(aB−c )= 1 + 1 1 . (46) 0 1 τB τL 1 1 Therefore,ifBissellingapositiveamountofasset,thetraditionalsector’sinversedemandfunction willpindowntheassetpriceaswellasthefire-saleamount. IfeitherπL ishighorτL ishigh,then 1 1 theright-handsideof(46)decreasesandaB−c increases,meaningBsellsmoreassettoT. 0 1 (1−πB)(R¯−1) (1−πL)+πLR InthecasethatF(cid:48)(aB)≥ 1 + 1 1 ,thenBsellsalltheassetholdings 0 τB τL 1 1 toT andBdoesnotborrowfromL. 59

Onthecontrary,if (1−πB)(R¯−1) (1−πL)+πLR F(cid:48)(0)< 1 + 1 1 , τB τL 1 1 then c =aB =aB assuming c =aB, and asset price p is indeterminant as no agent is buying or 1 1 0 0 0 1 sellinginapositiveamountandanyvalueaboveF(cid:48)(0)ispossible. A.7. Proof of Proposition 3 Given the date-1 equilibrium, we can derive equilibrium at date 0. We assume F(cid:48)(a) = α for all a ≥ 0 from now on. This assumption is for tractability of the solution because otherwise the equilibriumisdeterminedbyimplicitfunctions. Theonlycasethatdisappearswiththisassumption iscase3inProposition5,whichisanintermediateequilibriumbetweenfullfiresalesandzerofire sales. Proof. A.7.1. Lender’sProblematDate0 InSection6,weshowedthatwhenthelender,L,tightensfundingliquidity,L’sreturnforeachunit of cash invested (either as cash or lending) is always τL regardless of the realization of (sL,τL), 1 1 1 becauseL alwaysholdsextracashatdate1. Takingdate1equilibriumoutcomesasgiven,L’sproblematdate0is max EL(cid:2) τ L(κ L+dL) (cid:3) 0 1 0 0 κL,dL 0 0 s.t. κ L+q dL ≤eL 0 0 0 0 withnon-negativityconstraints, κ L ≥0, dL ≥0. 0 0 Because L’s budget constraint binds at the optimum, substitute κL = eL−q dL. The first-order 0 0 0 0 conditionwithrespecttotheonlychoicevariable,dL,is 0 EL(cid:2) τ L(1−q ) (cid:3) +ξ L =0. (47) 0 1 0 d 0 By assumption 1, dL > 0 in equilibrium. Therefore, the contract price that makes L indifferent 0 60

acrossanyamountofdL is 0 q =1. (48) 0 A.7.2. Borrower’sProblematDate0 For each realization of q at date 1, the borrower, B, takes the equilibrium market outcome, in- 1 cluding B’s own optimal decisions, as given. As we have seen in Proposition 5, there are different regimesunderdifferentrealizationofq . Baccountsforthatandsolvesthefollowingoptimization 1 problem: max E 0 B(cid:2) E 1 (cid:2) C 2 B(cid:12) (cid:12)q 1 (cid:3)(cid:3) (49) c 0 ,a 0 ,κ 0 B (cid:124) (cid:123)(cid:122) (cid:125) expectedutility foreachrealizedq 1 =E 0 B(cid:2) τ 1 B(cid:0) κ 0 B+q 1 a 0 −c 0 (cid:1) + (cid:0) 1−π 1 B(q 1 ) (cid:1) a 0 (R¯−1) (cid:12) (cid:12)q 1 ≥q¯ 1 (cid:3) PrB 0 (q 1 ≥q¯ 1 ) (cid:124) (cid:123)(cid:122) (cid:125) noliquidationcase +E 0 B (cid:20)(cid:18) a 0 − c 0 − α κ 0 − B− q q 1 a 0 (cid:19) (cid:0) 1−π 1 B(q 1 ) (cid:1) (R¯−1) (cid:12) (cid:12) (cid:12) (cid:12) q 1 ≤q 1 <q¯ 1 (cid:21) PrB 0 (q 1 ≤q 1 <q¯ 1 ) 1 (cid:124) (cid:123)(cid:122) (cid:125) sellstheassettopaydebtbutnosalesduetopessimism +E 0 B(cid:2) τ 1 B(cid:0) κ 0 B−c 0 +αaB 0 (cid:1)(cid:12) (cid:12)q 1 <q 1 (cid:3) PrB 0 (q 1 <q 1 ) (cid:124) (cid:123)(cid:122) (cid:125) liquidatesalltheassets s.t. eB ≥ p aB−q c +κ B, 0 0 0 0 0 0 aB ≥c , 0 0 ˆ 1 (cid:18) 1−(1−R)π (cid:19) (cid:26) 1−(1−R)π (cid:27) π B(q )= πλ 1 τ < <τ¯ dG (π), 1 1 T q q π 0 1 1 c −κB q¯ = 0 0 , 1 a 0 (cid:124) (cid:123)(cid:122) (cid:125) cutoff forliquidityinduced firesales (1−πB(q ))(R¯−1) 1 1 α = +q ,, τB 1 1 (cid:124) (cid:123)(cid:122) (cid:125) cutoff forbelief induced firesales withnon-negativityconstraints, c ≥0,aB ≥0,κ B ≥0. 0 0 0 61

There are three different regimes for B depending on the realization of q , which corresponds to 1 eachexpectedutilityrepresentationintheaboveoptimizationproblem: 1. Ifq ≥q¯ ,thenBwillbeabletorolloverthedebtfromdate0andkeepalltheassets. 1 1 2. Ifq ≤q <q¯ ,thenBdoesnothaveenoughcashtopaythepromiseddebtamountbutstill 1 1 1 optimistic enough to hold the assets as much as possible for the given prices. Therefore, B doesliquidity-inducedfiresalesbutsellsnomorethanthenecessaryamount. 3. If q <q , then B is pessimistic about the asset payoff, so T values the asset more than B. 1 1 Therefore,therewillbebelief-inducedfiresales. BwillsellalltheassetstoT. Wefirstconfirmthatthecutoffforbelief-inducedfiresalesisabovethelevelofq thatrequires 1 full sales of assets due to liquidity. This is because B can always sell all assets and repay the debt by assumption 2, and thus, any q higher than q will be enough for B to repay the debt while 1 1 holdingapositiveamountofasset. Lemma 6. If q ≥q , B can hold a positive amount of assets c >0 while repaying the debt to L 1 1 1 infull. Proof. Recallthattheamountoffiresalesis c −κB−q a 0 0 1 0 , α−q 1 whichisdecreasinginq because 1 (cid:18) c −κB−q a (cid:19) ∂ 0 0 1 0 α−q −a (α−q )+c −κB−q a −a α+c −κB 1 = 0 1 0 0 1 0 = 0 0 0 <0, ∂q (α−q )2 (α−q )2 1 1 1 wherethelastinequalitycomesfromassumption2. Denote the Lagrangian multiplier for the collateral constraint as µ. Substituting out κB and q 0 1 usingthebindingbudgetconstraintandthecutoffequationyields 62

ˆ 1/τ L 0 B = (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:2) τ 1 B(cid:0) eB 0 +q 0 c 0 −p 0 a 0 +q 1 a 0 −c 0 (cid:1) + (cid:0) 1−π 1 B(q 1 ) (cid:1) a 0 (R−1) (cid:3) dH(q 1 ) a 0 (50) ˆ (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:18) c −(eB+q c −p a )−q a (cid:19) + a 0 a − 0 0 0 0 0 0 1 0 (cid:0) 1−π B(q ) (cid:1) (R−1)dH(q ) 0 α−q 1 1 1 q 1 ˆ 1 q + 1 τ B(cid:0) eB+q c −p a −c +αaB(cid:1) dH(q ) 1 0 0 0 0 0 0 0 1 R/τ +µ(a −c )+ξ c +ξ a +ξ (eB+q c −p a ), 0 0 c 0 0 a 0 0 κB 0 0 0 0 0 0 where H(·) is B’s subjective distribution function of q . Then, we can apply the Leibniz integral 1 rule for derivatives of the Lagrangian function. The first-order conditions of B’s optimization problemare FOCforc : 0 1−q (cid:20) (cid:18) (1−q )c −eB+p a (cid:19) − 0 τ B eB+q c −p a + 0 0 0 0 0 a −c (51) a 1 0 0 0 0 0 a 0 0 0 0 (cid:18) (cid:18) (1−q )c −eB+p a (cid:19)(cid:19) (cid:21) + 1−π B 0 0 0 0 0 a (R−1) 1 a 0 0 ˆ 1/τ − (1−q 0 )c 0 −eB 0 +p 0 a 0 τ 1 B(1−q 0 )dH(q 1 ) a 0  (1−q )c −eB+p a  c −(eB+q c −p a )− 0 0 0 0 0 a + 1− a q 0    a 0 − 0 0 0 0 (1 0 − 0 q )c −eB+p a a 0 0    0 α− 0 0 0 0 0 a 0 (cid:18) (cid:18) (1−q )c −eB+p a (cid:19)(cid:19) × 1−π B 0 0 0 0 0 (R−1) 1 a 0 ˆ (1−q 0 )c 0 −eB 0 +p 0 a 0 − a 0 (1−q 0 ) (cid:0) 1−π B(q ) (cid:1) (R−1)dH(q ) α−q 1 1 1 q 1 ˆ 1 q − 1 τ B(1−q )dH(q )−µ+ξ +ξ q =0, 1 0 1 c 0 κB 0 R/τ 0 63

FOCfora : 0 p a − (cid:0) (1−q )c −eB+p a (cid:1)(cid:20) (cid:18) (1−q )c −eB+p a (cid:19) − 0 0 0 0 0 0 0 τ B eB+q c −p a + 0 0 0 0 0 a −c a2 1 0 0 0 0 0 a 0 0 0 0 (52) (cid:18) (cid:18) (1−q )c −eB+p a (cid:19)(cid:19) (cid:21) + 1−π B 0 0 0 0 0 a (R−1) 1 a 0 0 ˆ 1/τ + (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:2) τ 1 B(q 1 −p 0 )+ (cid:0) 1−π 1 B(q 1 ) (cid:1) (R−1) (cid:3) dH(q 1 ) a 0  (1−q )c −eB+p a  + p 0 a 0 − (cid:0) (1−q 0 a ) 2 c 0 −eB 0 +p 0 a 0 (cid:1)    a 0 − c 0 −(eB 0 +q 0 c 0 − (1 p 0 − a 0 q ) ) − c −eB+ 0 p 0 a a 0 0 0 0 a 0    0 α− 0 0 0 0 0 a 0 (cid:18) (cid:18) (1−q )c −eB+p a (cid:19)(cid:19) × 1−π B 0 0 0 0 0 (R−1) 1 a 0 ˆ (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:18) (cid:19) p −q + a 0 1− 0 1 (cid:0) 1−π B(q ) (cid:1) (R−1)dH(q ) α−q 1 1 1 q 1 ˆ 1 q + 1 τ B(α−p )dH(q )+µ+ξ −ξ p =0, 1 0 1 a 0 κB 0 R/τ 0 64

A.7.3. EquilibriumTradeandPrices FOCforc ,(51),canbefurthersimplifiedas 0 1−q (cid:18) (cid:18) (1−q )c −eB+p a (cid:19)(cid:19) − 0 1−π B 0 0 0 0 0 a (R−1) a 1 a 0 0 0 ˆ ˆ 1/τ q − (1−q 0 )c 0 −eB 0 +p 0 a 0 τ 1 B(1−q 0 )dH(q 1 )− 1 τ 1 B(1−q 0 )dH(q 1 ) R/τ a 0 1−q (cid:18) (cid:18) (1−q )c −eB+p a (cid:19)(cid:19) + 0 1−π B 0 0 0 0 0 a (R−1) a 1 a 0 0 0 ˆ (1−q 0 )c 0 −eB 0 +p 0 a 0 − a 0 (1−q 0 ) (cid:0) 1−π B(q ) (cid:1) (R−1)dH(q )−µ+ξ +ξ q q α−q 1 1 1 1 c 0 κ 0 B 0 ˆ 1 ˆ 1/τ q =− (1−q 0 )c 0 −eB 0 +p 0 a 0 τ 1 B(1−q 0 )dH(q 1 )− 1 τ 1 B(1−q 0 )dH(q 1 ) 0 a 0 ˆ (1−q 0 )c 0 −eB 0 +p 0 a 0 − a 0 (1−q 0 ) (cid:0) 1−π B(q ) (cid:1) (R−1)dH(q )−µ+ξ +ξ q q α−q 1 1 1 1 c 0 κ 0 B 0 1 =−µ+ξ +ξ =0, c 0 κB 0 andthesecondtolastinequalityholdsbecauseofq =1,whichisfrom(48). 0 If B does not borrow at all c = 0, then the collateral constraint should be binding, implying 0 a = c = 0. If B is borrowing a positive amount as c > 0, then ξ = 0 and there can be two 0 0 0 c 0 differentcases. Considerthefirstcaseinwhichξ positive,implyingκB =0and µ =ξ q >0. κB 0 κB 0 0 0 Thus,thecollateralconstraintisalsobindingandc =a . Inthesecondcaseinwhichξ iszero, 0 0 κB 0 µ should also be zero, implying that the collateral constraint is not binding and B purchases some assetswithoutleverageasa >c . 0 0 The result holds because of the linearity of B’s utility. If B holds a positive amount of cash, B should be indifferent between holding more or less cash. However, if the return of purchasing moreassetsbyborrowingmorefromLatdate0atthecostoflessamountofcashatdate1exceeds thecashreturn,Bborrowswithfullcapacityandspendsallthecashtopurchasetheasset. 65

FOCfora ,(52),canbefurthersimplifiedas 0 p a − (cid:0) (1−q )c −eB+p a (cid:1)(cid:20)(cid:18) (cid:18) (1−q )c −eB+p a (cid:19)(cid:19) (cid:21) − 0 0 0 0 0 0 0 1−π B 0 0 0 0 0 a (R−1) a2 1 a 0 ˆ 0 0 1/τ + (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:2) τ 1 B(q 1 −p 0 )+ (cid:0) 1−π 1 B(q 1 ) (cid:1) (R−1) (cid:3) dH(q 1 ) a 0 p a − (cid:0) (1−q )c −eB+p a (cid:1)(cid:18) (cid:18) (1−q )c −eB+p a (cid:19)(cid:19) + 0 0 0 0 0 0 0 1−π B 0 0 0 0 0 a (R−1) a2 1 a 0 0 0 ˆ (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:18) (cid:19) + a 0 α−p 0 (cid:0) 1−π B(q ) (cid:1) (R−1)dH(q ) α−q 1 1 1 q 1 ˆ 1 q + 1 τ B(α−p )dH(q )+µ+ξ −ξ p 1 0 1 a 0 κB 0 R/τ 0 ˆ 1/τ = (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:2) τ 1 B(q 1 −p 0 )+ (cid:0) 1−π 1 B(q 1 ) (cid:1) (R−1) (cid:3) dH(q 1 ) a 0 ˆ (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:18) (cid:19) + a 0 p 0 −α (cid:0) 1−π B(q ) (cid:1) (R−1)dH(q ) q −α 1 1 1 q 1 ˆ 1 q + 1 τ B(α−p )dH(q )+µ+ξ −ξ p =0. 1 0 1 a 0 κB 0 R/τ 0 Wefinallyderivethedate-0equilibriumallocationandprices. FromtheFOCsforc anda , 0 0 ˆ 1/τ (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:2) τ 1 B(q 1 −p 0 )+ (cid:0) 1−π 1 B(q 1 ) (cid:1) (R−1) (cid:3) dH(q 1 ) a 0 ˆ (1−q 0 )c 0 −eB 0 +p 0 a 0 (cid:18) (cid:19) + a 0 p 0 −α (cid:0) 1−π B(q ) (cid:1) (R−1)dH(q ) q −α 1 1 1 q 1 ˆ 1 q + 1 τ B(α−p )dH(q )+ξ +ξ −ξ (p −1)=0. 1 0 1 c 0 a 0 κB 0 R/τ 0 Fromtheborrower’soptimalityconditionundera =c >0,Bholdszeroamountofcash,κB=0. 0 0 0 66

Underthiscase,B’sreturnfromthisleveragedassetpurchasebecomes (cid:34)ˆ 1/τ ξ = (cid:2) τ B(q −p )+ (cid:0) 1−π B(q ) (cid:1) (R−1) (cid:3) dH(q ) (53) κB 1 1 0 1 1 1 0 1 ˆ + 1(cid:18) p 0 −α (cid:19) (cid:0) 1−π B(q ) (cid:1) (R−1)dH(q ) q −α 1 1 1 q 1 ˆ 1 (cid:35) q + 1 τ B(α−p )dH(q ) /(p −1). 1 0 1 0 R/τ It is sufficient to show that ξ >τB, so B prefers to purchase the asset with leverage to holding κB 1 0 cash. IfBusesupallthecashendowmentstopurchasetheassetwithleverage,theassetpriceis A+eB p = 0. 0 A The price should be higher than T’s willingness-to-pay for the asset, which is also above the debt payment amount at date 1, so B is able to repay the debt in full if B wants to liquidate the assets. Therefore, A+eB p = 0 >α ≥1. (54) 0 A Fromassumption1, eB τ B <A(EB[R]−1) 0 1 0 A EB[R]−1 ⇒τ B < (EB[R]−1)= 0 , (55) 1 eB 0 p −1 0 0 wherethelastequalitycomesfrom(54). Hence,ifthemarginalreturnofpurchasingtheassetwith EB[R]−1 leverage,ξ ,isgreaterthanorequalto 0 ,thena =c =AandκB=0istheequilibrium κ 0 B p 0 −1 0 0 0 portfolioofB. Thedifferencebetweenthetworeturnsis ˆ 1/τ ξ −(EB[R]−1)= (cid:2) π B(q )(1−R)+τ B(q −p ) (cid:3) dH(q ) (56) κB 0 1 1 1 1 0 1 0 1ˆ 1(cid:20) (cid:18) p −q (cid:19) (cid:21) + π B(q )(1−R)− 0 1 (cid:0) 1−π B(q ) (cid:1) (R−1) dH(q ) 1 1 α−q 1 1 1 q 1 ˆ 1 q + 1 (cid:2) π B(q )(1−R)+τ B(α−p )−(1−π B(q ))(R−1) (cid:3) dH(q ). 1 1 1 0 1 1 1 R/τ 67

(1−πB(q))(R−1) We will show that this difference is positive. Because 1 +q is increasing in q τB 1 (1−πB(q ))(R−1) andα = 1 1 +q ≥1, τB 1 1 (1−πB(q))(R−1) α−q< 1 τB 1 foranyq∈[q ,1]. Thus,wehave 1 ˆ 1(cid:20) (cid:18) p −q (cid:19) (cid:21) π B(q )(1−R)− 0 1 (cid:0) 1−π B(q ) (cid:1) (R−1) dH(q ) 1 1 α−q 1 1 1 q 1 ˆ 1 (cid:34) (cid:32) (cid:33) (cid:35) 1 p −q > π B(q )(1−R)− τ B 0 1 (cid:0) 1−π B(q ) (cid:1) (R−1) dH(q ) 1 1 1 (cid:0) 1−πB(q ) (cid:1) (R−1) 1 1 1 q 1 1 ˆ 1 1 = (cid:2) π B(q )(1−R)+τ B(q −p ) (cid:3) dH(q ). (57) 1 1 1 1 0 1 q 1 (1−πB(q ))(R−1) Again,becauseα = 1 1 +q and(1−πB(q))isincreasinginq, τB 1 1 1 ˆ q 1 (cid:2) π B(q )(1−R)+τ B(α−p )−(1−π B(q ))(R−1) (cid:3) dH(q ) 1 1 1 0 1 1 1 R/τ ˆ (cid:34) (cid:32) (cid:33) (cid:35) q (1−πB(q ))(R−1) > 1 π B(q )(1−R)+τ B 1 1 +q −p −(1−π B(q ))(R−1) dH(q ) 1 1 1 τB 1 0 1 1 1 ˆ R/τ 1 q > 1 (cid:2) π B(q )(1−R)+τ B(q −p ) (cid:3) dH(q ). (58) 1 1 1 1 0 1 R/τ Combining(56),(57),and(58)implies ˆ 1/τ ξ −(EB[R]−1)> (cid:2) π B(q )(1−R)+τ B(q −p ) (cid:3) dH(q ) κB 0 1 1 1 1 0 1 0 1ˆ 1 + (cid:2) π B(q )(1−R)+τ B(q −p ) (cid:3) dH(q ) 1 1 1 1 0 1 q ˆ 1 q + 1 (cid:2) π B(q )(1−R)+τ B(q −p ) (cid:3) dH(q ) 1 1 1 1 0 1 ˆ R/τ = (cid:2) π B(q )(1−R)+τ Bq −τ Bp (cid:3) dH(q ) 1 1 1 1 1 0 1 =π (1−R)+τ BEB[q ]−τ Bp . (59) 0 1 0 1 1 0 68

(1−πL)+πLR Recallthatq = 1 1 from(2),whichimplies 1 τL 1 EB[q ]=EB (cid:20) (1−π 1 L)+π 1 LR (cid:21) =EB(cid:2) (1−π L)+π LR (cid:3) EB (cid:20) 1 (cid:21) = (cid:2) (1−π )+π0R (cid:3) EB (cid:20) 1 (cid:21) , 0 1 0 τL 0 1 1 0 τL 0 1 0 τL 1 1 1 where the last two equalities hold because of the independence between τL and sL and the law of 1 1 iteratedexpectations,respectively. Because f(x)=1/x isaconvexfunction, (cid:20) (cid:21) 1 1 1 EB > = 0 τL EB (cid:2) τL (cid:3) τL 1 0 1 0 byJensen’sinequality. Finally,from(55),EB[R]>τBp . 0 1 0 Therefore,(59)becomes ξ −(EB[R]−1)>π (1−R)+τ BEB[q ]−τ Bp κB 0 0 1 0 1 1 0 0 >π (1−R)+(1−π )+π R−(1−π )R−π R=0. 0 0 0 0 0 Therefore,thereturnfrominvestingintheassetwithleverageξ >isgreaterthanthereturnfrom κB 0 cashholdingsτB. Finally,thisalsoshowsthattheequilibriumwithκB=0istheonlyequilibrium. 1 0 IfκB >0,thenthepriceoftheasset p willbeevenlower,increasingtheassetreturnevenhigher. 0 0 A.8. Proof of Lemma 2 Proof. We show how learning amplifies misallocation during a fire sale for any q (i.e. shock 1 pair (πL,τL)), relative to a benchmark in which beliefs don’t change—that is, agents have private 1 1 information but do not update their beliefs in response to new information. Note that in the Fire SaleRegime,wehave (cid:32) EB[R]−EB(cid:2) Rd(cid:3) EL(cid:2) Rd(cid:3)(cid:33) aB =aB−F(cid:48)−1 1 1 2 + 1 2 (60) 1 0 τB τL 1 1 p =F(cid:48)(aT) 1 1 Since F(cid:48)(·) is monotonically decreasing, F(cid:48)−1(·) is also monotonically decreasing. Therefore, EL[Rd] holdingtheborrower’sbeliefsconstantforamoment,alowerq = 1 2 impliesthataB islower. 1 τL 1 Note that the spread EB[R]−EB(cid:2) Rd(cid:3) = (cid:0) 1−πB(cid:1)(cid:0) R−1 (cid:1) is decreas 1 ing in πB (since R > 1 by 1 1 2 1 1 69

assumption). Alowerq impliesthatπB ishigher(Bismorepessimistic)byProposition1,which 1 1 alsomakesaB lower. Thus,alowerq leadstoaloweraB intwoways: thedirecteffectofalower 1 1 1 q onaB andtheeffectoflowerq onaB throughgreaterpessimismandalowerspread. 1 1 1 1 SincealargerdecreaseinfundingliquidityleadstoloweraB,thisalsoimpliesthataT ishigher 1 1 andhence p islower. Hence,thefiresaleismoresevereintheFireSaleRegimewhenthefunding 1 illiquidity is more severe. Hence, pessimism/the information externality is associated with greater misallocation/worsefiresalesduringafiresale. A.9. Date-1 Equilibrium under Common Information Benchmark IntheNormalRegimeundertheCommonInformationBenchmarkwehave aT,ξ L,ξ B,ξ B,ξ L,ξ B =0 1 d 1 c 1 a 1 κ1 κ1 E (cid:2) Rd(f ) (cid:3) q = 1 2 1 1 τL 1 E [R]−E (cid:2) Rd(f ) (cid:3) p = 1 1 2 1 +q 1 τB 1 1 aB =cB =dL =aB 1 1 1 0 κ B =q aB−cB+κ B 1 1 0 0 0 κ L =κ L+dL−q aB 1 0 0 1 0 κ T =κ T −p aT 1 0 1 1 CB =τ B κ B+aBR−aBRd(f ) 2 1 1 1 1 2 1 CL =τ L κ L+RddL 2 1 1 2 1 CT =κ T +F(aT) 2 1 1 µ B = (cid:0) τ B+ξ B(cid:1) p −E [R]−ξ B 1 1 κ1 1 1 a 1 70

µ T =F(cid:48)(0)−p 1 1 π λ (cid:0) εL =sL−R (cid:1) π :=π L =Pr (cid:0) R=R|sL,I (cid:1) = 0 ε 1 1 1 1 1 0 (1−π )λ (cid:0) εL =sL−R (cid:1) +π λ (cid:0) εL =sL−R (cid:1) 0 ε 1 1 0 ε 1 1 π B =π 1 1 IntheFireSaleRegimeundertheCommonInformationBenchmark,wehave µ T,ξ B,ξ B,ξ L,ξ L,ξ B =0 1 c 1 a 1 d 1 κ1 κ1 E (cid:2) Rd(f ) (cid:3) q = 1 2 1 1 τL 1 p =F(cid:48)(aT) 1 1 (cid:32) E [R]−E (cid:2) Rd(f ) (cid:3) E (cid:2) Rd(f ) (cid:3)(cid:33) aB =aB−F(cid:48)−1 1 1 2 1 + 1 2 1 1 0 τB τL 1 1 aT =aB−aB 1 0 1 cB =dL =aB 1 1 1 κ B =q aB−cB+κ B−p (cid:0) aB−aB(cid:1) 1 1 1 0 0 1 1 0 κ L =dL−q dL+κ L 1 0 1 1 0 κ T =κ T −p aT 1 0 1 1 CB =τ B κ B+aBR−aBRd(f ) 2 1 1 1 1 2 1 CL =τ L κ L+RddL 2 1 1 2 1 71

CT =κ T +F(aT) 2 1 1 µ B =τ Bp −E [R] 1 1 1 1 π λ (cid:0) εL =sL−R (cid:1) π :=π L =Pr (cid:0) R=R|sL,I (cid:1) = 0 ε 1 1 1 1 1 0 (1−π )λ (cid:0) εL =sL−R (cid:1) +π λ (cid:0) εL =sL−R (cid:1) 0 ε 1 1 0 ε 1 1 π B =π 1 1 A.10. Proof of Lemma 3 HerewecharacterizethethresholdbetweentheNormalandCrisisRegimesintheCommonInformationBenchmark. Proof. FrontierinBaselineEconomy Recalltheequilibriumassetprice. EB[R]−EB(cid:2) Rd(f ) (cid:3) p = 1 1 2 1 +q . 1 τB 1 1 The threshold price is defined by asset price at threshold of Normal and Fire Sale Regimes. Thisthresholdassetpriceis pˆ =F(cid:48)(0)satisfying 1 EB[R]−EB(cid:2) Rd(f ) (cid:3) 1 1 2 1 +qˆ =F(cid:48)(0) τB 1 1 (cid:0) 1−πB(qˆ ) (cid:1)(cid:0) R− f (cid:1) qˆ + 1 1 1 =F(cid:48)(0) 1 τB 1 (cid:0) 1−π B(qˆ ) (cid:1)(cid:0) R− f (cid:1) +τ Bqˆ =τ BF(cid:48)(0). (61) 1 1 1 1 1 1 Thisdefinesthethresholdvalueqˆ . Giventhis,thesetof (cid:0) τL,πL(cid:1) consistentwithqˆ isgivenby 1 1 1 1 (cid:0) 1−πL(cid:1) f +πLR 1 1 1 =qˆ . τL 1 1 SolveforπL : 1 f −τLqˆ 1−τLqˆ π L = 1 1 1 = 1 1 . 1 f −R 1−R 1 This defines the curve of the frontier partitioning the state space into the Normal and Crisis 72

Regimes. The y-intercept of the curve (when τL = 0, though never occurs) is 1 > 1, while the x- 1 1−R intercept(whenπL =0)is 1 >0. 1 qˆ 1 Theslopeofthecurveisnegativeandconstant: dπL −qˆ 1 = 1 <0. dτL 1−R 1 FrontierinCommonInformationBenchmark WhenπB =πL,thethresholdinbenchmarkisdefinedby 1 1 (cid:0) 1−π L(cid:1)(cid:0) R− f (cid:1) +τ Bqˆ =τ BF(cid:48)(0) 1 1 1 1 1 i.e. (cid:0) R− f (cid:1) − (cid:20) (cid:0) R− f (cid:1) + τ 1 B (f −R) (cid:21) π L+ τ 1 B f =τ BF(cid:48)(0) (62) 1 1 τL 1 1 τL 1 1 1 1 We now trace out the frontier of all (cid:0) τL,πL(cid:1) such that this is satisfied. First, we derive the 1 1 x-interceptofthefrontierbysupposingthatπL =0: 1 (cid:0) R− f (cid:1) + τ 1 B f =τ BF(cid:48)(0) 1 τL 1 1 1 (cid:0) R− f (cid:1) τ L+τ Bf =τ L τ BF(cid:48)(0) 1 1 1 1 1 1 τ Bf =τ L(cid:0) τ BF(cid:48)(0)−R+ f (cid:1) 1 1 1 1 1 τBf τ L = 1 1 . 1 τBF(cid:48)(0)−R+ f 1 1 Claim1. Thisx-interceptisgreaterthanthex-interceptintheBaselinecase. Proof. We want to show that the x-intercept of the frontier in the Common Info benchmark is largerthanthatintheBaselineeconomy,i.e. τBf f 1 1 > 1 τBF(cid:48)(0)−R+ f qˆ 1 1 1 τBF(cid:48)(0)− (cid:0) R− f (cid:1) qˆ > 1 1 1 τB 1 whereqˆ isdefinedby(61)intheBaselinecase. Recallthatqˆ isdefinedby 1 1 73

(cid:0) 1−πB(qˆ ) (cid:1)(cid:0) R− f (cid:1) qˆ =F(cid:48)(0)− 1 1 1 1 τB 1 Sowewanttoshowthat (cid:0) 1−πB(qˆ ) (cid:1)(cid:0) R− f (cid:1) τBF(cid:48)(0)− (cid:0) R− f (cid:1) F(cid:48)(0)− 1 1 1 > 1 1 . τB τB 1 1 Theinequalityisequivalentto (cid:0) 1−πB(qˆ ) (cid:1)(cid:0) R− f (cid:1) (cid:0) R− f (cid:1) F(cid:48)(0)− 1 1 1 >F(cid:48)(0)− 1 τB τB 1 1 (cid:0) 1−π B(qˆ ) (cid:1)(cid:0) R− f (cid:1) < (cid:0) R− f (cid:1) 1 1 1 1 (cid:0) 1−π B(qˆ ) (cid:1) <1, 1 1 whichholdsbecauseπB >0. 1 Hence, we have that the x-intercept is larger in the Common Info Benchmark compared to the Baseline case. (Note that this automatically implies that the x-intercept is positive, since it’s obviouslypositiveintheBaselinecase.) What is the y-intercept of the frontier in the Common Info Benchmark? (i.e. what happens to πL asτL approacheszero?): Thefrontierequation(62)canberearrangedas 1 1 (cid:0) R− f (cid:1) − (cid:20) (cid:0) R− f (cid:1) + τ 1 B (f −R) (cid:21) π L+ τ 1 B f =τ BF(cid:48)(0) 1 1 τL 1 1 τL 1 1 1 1 − (cid:20) (cid:0) R− f (cid:1) + τ 1 B (f −R) (cid:21) π L =τ BF(cid:48)(0)− τ 1 B f − (cid:0) R− f (cid:1) 1 τL 1 1 1 τL 1 1 1 1 τ 1 B f + (cid:0) R− f (cid:1) −τBF(cid:48)(0) π L = τ 1 L 1 1 1 1 (cid:0) (cid:1) τB R− f + 1 (f −R) 1 τL 1 1 τBf +τL(cid:0) R− f (cid:1) −τLτBF(cid:48)(0) π L = 1 1 1 1 1 1 1 τL (cid:0) R− f (cid:1) +τB(f −R) 1 1 1 1 τBf +τL(cid:0) R− f −τBF(cid:48)(0) (cid:1) π L = 1 1 1 1 1 . 1 τL (cid:0) R− f (cid:1) +τB(f −R) 1 1 1 1 They-intercept(whenτL =0)is f 1 . Hence,thisisthesamey-interceptasintheBaselinecase. 1 f −R 1 TheslopeofthisfrontierisdefinedbytakingthederivativeofπL withrespecttoτL: 1 1 74

(cid:18) (cid:19) ∂ τ 1 Bf 1 +τ 1 L(R−f 1 −τ 1 BF(cid:48)(0)) ∂π 1 L = τ 1 L(R−f 1 )+τ 1 B(f 1 −R) ∂τL ∂τL 1 1 (cid:0) R− f −τBF(cid:48)(0) (cid:1)(cid:0) τL(cid:0) R− f (cid:1) +τB(f −R) (cid:1) − (cid:0) R− f (cid:1)(cid:0) τBf +τL(cid:0) R− f −τBF(cid:48)(0) (cid:1)(cid:1) = 1 1 1 1 1 1 1 1 1 1 1 1 (cid:0) τL (cid:0) R− f (cid:1) +τB(f −R) (cid:1)2 1 1 1 1 (cid:0) R− f −τBF(cid:48)(0) (cid:1)(cid:0) τB(f −R) (cid:1) − (cid:0) R− f (cid:1)(cid:0) τBf (cid:1) = 1 1 1 1 1 1 1 <0, (cid:0) τL (cid:0) R− f (cid:1) +τB(f −R) (cid:1)2 1 1 1 1 which holds because (cid:0) R− f −τBF(cid:48)(0) (cid:1) < (cid:0) R− f (cid:1) and τBf > τB(f −R). Denote Ψ ≡ 1 1 1 1 1 1 1 (cid:0) R− f −τBF(cid:48)(0) (cid:1)(cid:0) τB(f −R) (cid:1) − (cid:0) R− f (cid:1)(cid:0) τBf (cid:1) . Also,thesecondderivativebecomes 1 1 1 1 1 1 1 ∂2πL −2 (cid:0) R− f (cid:1)(cid:0) τL(cid:0) R− f (cid:1) +τB(f −R) (cid:1) Ψ 1 = 1 1 1 1 1 >0, ∂ (cid:0) τL (cid:1)2 (cid:0) τL (cid:0) R− f (cid:1) +τB(f −R) (cid:1)4 1 1 1 1 1 wherethelastinequalityholdsbyR> f >RandΨ<0. Therefore,thefrontierisstrictlyconvex 1 withanegativeslope(inourrelevantdomain). A.11. Proof of Condition for Intersection of the Frontiers Using the results from the proof of Lemma 3 in Appendix A.10, we derive the necessary and sufficient condition for the existence of the intersection of the frontiers in the Baseline Economy andtheCommonInformationBenchmark. Proof. In the Baseline Economy, the slope is always dπ 1 L = −qˆ 1 < 0. In the Common Info dτ 1 L f 1 −R Benchmark,theslopeis (cid:0) R− f −τBF(cid:48)(0) (cid:1)(cid:0) τB(f −R) (cid:1) − (cid:0) R− f (cid:1)(cid:0) τBf (cid:1) 1 1 1 1 1 1 1 (cid:0) τL (cid:0) R− f (cid:1) +τB(f −R) (cid:1)2 1 1 1 1 whichisincreasingasshowninLemma3. Therefore,oncethefrontierintheCommonInformation Benchmark is at the same point (τL∗ ,πL∗ ) or above that point as (τL∗ ,πL) with πL ≥πL∗ and has 1 1 1 1 1 1 the same slope as the frontier in the Baseline Case at τL∗ , the two frontiers would never meet for 1 anyτL >τL∗ (seeFigure6foragraphicalexample). 1 1 First, suppose that τˆL ≤τ˜L. We claim that the two frontiers will never meet for πL ≤1. Since 1 1 1 1 thetwofrontiershavethesamey-intercept, ,theaverageslopeofthefrontierintheCommon 1−R Information Benchmark within the interval [0,τˆL] has to be greater than that of the frontier in the 1 BaselineCasewithinthesameinterval. Becauseofstrictconvexity,theslopeofthefrontierinthe 75

Common Information Benchmark at τˆL has to be greater than that of the frontier in the Baseline 1 Caseaswell. Thus,thetwofrontierswillnevermeetforthestateswithπL ≤1. 1 NowsupposethatthetwofrontiersnevermeetforthestateswithπL≤1. WeclaimthatτˆL≤τ˜L 1 1 1 shouldholdandprovethisbycontradiction. Supposethecontrary,τˆL>τ˜L. Becausethex-intercept 1 1 in the Common Information Benchmark is greater than the x-intercept in the Baseline case, the frontier in the Baseline case continuously goes from πL = 1 to πL = 0 within an interval that is 1 1 contained in the interval between τ˜L and the x-intercept in the Common Information Benchmark. 1 Sincethefrontieriscontinuouslydecreasingandconvex,thereexistsapointthatthetwofrontiers meetbytheintermediatevaluetheorem. A.12. Proof of Proposition 4 Proof. ProofofPart(A):EffectsofmisinformationontheallocationintheNormalRegime BycomparingaNormalRegimeequilibriuminbaselinewiththeCommonInformationBenchmark,wecanseethatinbothequilibria,aB =aB. SointheNormalRegime,beliefshavenoeffect 1 0 on the allocation of the risky asset. What they have is an effect on the allocation of cash between the lender vs the borrower, via q , and the asset price p (which itself doesn’t matter in this 1 1 regime beyond ensuring aB =aB, since pecuniary externality doesn’t lead to misallocation here): 1 0 EL[Rd(f )] κB =q aB−cB+κB. In both versions, q = 1 2 1 determiend by beliefs of the lender, which 1 1 0 0 0 1 τL 1 is equivalent to that of the borrower in the common information benchmark. Hence, tighter q is 1 met with cash holdings in this regime, but belief disagreements (learning mechanism) don’t affect tighterq . 1 EB[R]−EB[Rd(f )] Beliefdisagreementdoesaffect p (since p = 1 1 2 1 +q ),butthishasnoallocative 1 1 τB 1 1 consequencesinthisregime. Thus,beliefdisagreementonlyaffectequilibriumallocation(ofasset or cash) at date 1 only to the extent that they affect allocation of asset. (Beliefs in general affect allocationofcash,butthisisalwaysdeterminedbyq whichispinneddownbythelender’sbelief. 1 So,theborrower’sbeliefmattersonlytotheextentthatitaffectsdesiredaB). 1 Proof of Part (B): Effect of misinformation on the severity of fire sale when Condition 1 holds As we outlined for the Normal Regime, belief disagreement affects the equilibrium allocation (ofassetorcash)atdate1onlytotheextentthattheyaffectallocationofasset. (Thisisbecausein boththebaselinecaseandthecommoninfobenchmarket,q ispinneddownbythelender’sbeliefs 1 EL[Rd(f )] in equilibrium q = 1 2 1 ). So at the margin, belief disagreements can affect the equilibrium 1 τL 1 allocation only to the extent that it affects aB and/or p . Recall that in the Fire Sale Regime, we 1 1 have(60) 76

(cid:32) EB[R]−EB(cid:2) Rd(f ) (cid:3) EL(cid:2) Rd(f ) (cid:3)(cid:33) aB =aB−F(cid:48)−1 1 1 2 1 + 1 2 1 1 0 τB τL 1 1 whileforthebenchmarkwithcommoninformation,wehave (cid:32) E [R]−E (cid:2) Rd(f ) (cid:3) EL(cid:2) Rd(f ) (cid:3)(cid:33) aB =aB−F(cid:48)−1 1 1 2 1 + 1 2 1 (63) 1 0 τB τL 1 1 Since F(cid:48)−1(·) is monotonically decreasing, aB is lower in the baseline case (i.e. the identi- 1 fication problem makes the fire sales more severe) only if the spread EB[R]−EB(cid:2) Rd(f ) (cid:3) = 1 1 2 1 (cid:0) 1−πB(cid:1)(cid:0) R− f (cid:1) is lower. This occurs iff πB > πL. Whether this is true in equilibrium or not 1 1 1 1 depends on the actual realization of πL,τL given the observed q . So, for a given q , the identifi- 1 1 1 1 cationproblemwillamplifythefiresalesrelativetothecommoninfobenchmark(loweraB)when 1 πL islowandτL ishigh;whileitwilldampenthefiresalesrelativetothecommoninfobenchmark 1 1 (higheraB)whenπL ishighandτL islow. 1 1 1 Theneteffectofthesetwoforcesdeterminestheoveralleffectofmisinformationontheseverity offiresales. Proof of Part (C): Effect of misinformation on the severity of fire sales when Condition 1 doesnothold AsweestablishedintheproofofPart(B),aBislowerinthebaselinecaseifandonlyifπB>πL. 1 1 1 As we show in the the characterization of optimism and pessimism at date 1, if Condition 1 does nothold,thenπB >πL (exceptforknife-edgecases,inwhichπB =πL). 1 1 1 1 A.13. Figures for Lemma 5 Figure 6 illustrates the bisection of the state space into two regimes at date 1 under the case when Condition1doesnothold. Itplotsthisbisectionforthebaselinecaseinwhichthelender’sliquidity shock τL and beliefs πL are private information, and under the Common Information Benchmark 1 1 case in which this information is directly observable by the borrower. The equilibrium is in the Fire Sale Regime if and only the equilibrium value of p is below a threshold value pˆ . (For the 1 1 baseline case, this corresponds to a threshold value of q .) The solid curve in the figure plots the 1 combinations of the states (cid:0) τL,πL(cid:1) consistent the threshold pˆ , based on the lender’s optimality 1 1 1 condition for dL, and denotes the frontier between the two regimes. The dashed curve plots the 1 samefrontierintheCommonInformationBenchmarkinwhichtheborrowerdirectlyobservesthe lender’s private information, and hence πB =πL all along this curve. For both cases, the region to 1 1 thesouthwestofthesecurvesistheNormalRegime,whilethenortheastistheFireSaleRegime. ThedottedcurveinthefiguredemarcatestheregionofthestatespaceinwhichπB=πL onthe 1 1 77

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