How Central Banks End Crises

Fighting Crises with Secrecy⇤

Gary Gorton† Guillermo Ordonez‡

June 2017

Abstract

How does central bank lending during a crisis restore confidence? Emergencylending facilities which are opaque (in that names of borrowers are kept secret)raise the perceived average quality of bank assets in the economy, creating an in-formation externality that prevents runs. Stigma (the cost of a banks participationat the lending facility becoming public) is desirable as it keeps banks participat-ing in the lending facilities from revealing their participation. The central bank’skey policy instrument for limiting the amount lent is the haircut on bank assetsoffered as collateral at the emergency lending facility.

keywords: financial crises, information disclosure, lending facilities, runs, stigma, haircuts.

⇤We thank Huberto Ennis, Mark Gertler, Todd Keister, Pamela Labadie, Ricardo Reis, Eric Rosen-gren, Geoffrey Tootell and seminar participants at the Boston Fed, Wharton, the 2015 AEA Meetingsin Boston, the 2016 NAESM in Philadelphia, the 2016 SED Meetings in Toulouse and the “Innova-tions in Central Banking” Conference at the St. Louis Fed for comments and suggestions. This paperpreviously circulated under the title “How Central Banks End Crises.” The usual waiver of liabilityapplies.

†Yale University and NBER (e-mail: [email protected])‡University of Pennsylvania and NBER (e-mail: [email protected])

1 Introduction

How does central bank emergency lending during a financial crisis restore confi-dence? During a financial crisis central banks open new lending facilities, but theydo not reveal the identities of participating banks. Why is this and how does thiscontribute to restoring confidence? In this paper, we show that emergency lendingfacilities which are opaque (in that the names of borrowers are kept secret) raise theperceived average quality of bank assets in the economy, creating an information ex-ternality that prevents runs. Restoring confidence means successfully creating thisexternality. But the central banks policy also depends on stigma, which refers to thecost to a bank if it becomes known that the bank participated in the emergency lend-ing program. A bank may want to reveal its participation so that it is known thattheir liabilities are backed by government debt or cash. Stigma is desirable as it keepsbanks participating in the lending facilities from revealing their participation, so thatthe information externality can be implemented via the emergency lending program.The central banks key policy instrument for limiting the amount lent is the haircut onbank assets offered as collateral at the emergency lending facility.

During the financial crisis of 2007-2008 the Federal Reserve introduced a number ofnew emergency lending programs, including the Term Auction Facility, the Term Se-curities Lending Facility, and the Primary Dealers Credit Facility, which provide pub-lic funds to financial intermediaries in exchange for private assets. Also as part oftheir Emergency Economic Stabilization Act, the U.S. Treasury implemented similarprograms, such as the Troubled Asset Relief Program (TARP). These facilities werespecifically designed to hide borrowers’ identities.1 Secrecy was also integral to thespecial crisis lending programs of the Bank of England and the European CentralBank.2 Plenderleith (2012), asked by the Bank of England to review their EmergencyLending Facilities (ELA) during the financial crisis, wrote: “Was secrecy appropriatein 2008? In light of the fragility of the markets at the time . . . it was right to endeavorto keep ELA operations covert. . . in conditions of more systemic disturbance, as in

1Bernanke (2010): “. . . [because of] the competitive format of the auctions, the TAF [Term AuctionFacility] has not suffered the stigma of the conventional discount window” (p.2). Also, see Armantieret al. (2015). In the case of TARP there were fierce criticisms after their implementation by the SenateCongressional Oversight Panel about its lack of transparency. See http://www.jec.senate.gov/public/index.cfm/2009/3/03.1.09.

2For an overview see Bank for International Settlements (BIS) Committee on the Global FinancialSystem (2008).

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2008, ELA is likely to be more effective if provided covertly” (p. 70). Even before theFederal Reserve came into existence, private bank clearinghouses would also open anemergency lending facility during banking panics keeping the identities of borrowingbanks secret. See Gorton and Tallman (2016).

The secrecy of all these lending facilities has been widely criticized for hiding theidentities of weak or insolvent banks, resulting in fierce calls for transparency. Dur-ing the recent crisis, for instance, Bloomberg and Fox News sued the Fed (under theFreedom of Information Act) to obtain the identities of borrowers.3 In this paper,however, we show that lending facilities that replace “dubious private assets” with“good public bonds” in secret are indeed optimal. Secrecy creates an informationexternality by raising the average quality of assets in the banking system withoutreplacing all assets under suspicion, mitigating the desire of depositors to examinebanks’ assets, thereby avoiding a withdrawal of funds (runs hereafter) from thosebanks that are found to have less than average asset quality. A crisis here is an infor-mation event, as in Dang, Gorton, and Holmstrom (2013) and Gorton and Ordonez(2014a). Lending facilities recreate confidence and avoid inefficient examination ofbanks’ portfolios. Secrecy minimizes the social cost of resorting to lending facilities.

Our view of a bank run is in contrast to the more standard view of run as coordinationfailures that occur even if assets are solvent and focuses instead on the process of ex-amining the bank’s assets before choosing whether to withdraw funds. Our view alsocaptures both depositors seeking to withdraw from banks and repo lenders wantingto withdraw via higher haircuts on the collateral or not rolling over their loans at all.Evidence that not all banks experience runs upon examination during the recent fi-nancial crisis is provided by Perignon, Thesmar, and Vuillemey (2017) who show thatin the European market for unsecured wholesale certificates of deposit, some banksexperiences funding dry-ups while other banks did not, highlighting the relevance ofinformation for this outcome.

As these runs are motivated by suspicions about the bank’s backing collateral, theCentral Bank may lend cash against collateral using an emergency lending facility,or lend Treasury bonds against collateral as in the Term Securities Lending Facility(TSLF) to raise the perceived average quality of the backing collateral. Our argu-ments below apply in either case, but we will, for consistency, speak throughout of a

3Bloomberg L.P. v. Board of Governors of the Federal Reserve System, 649 Supp 2d 263. Fox NewsNetwork v. Board of Governors of the Federal Reserve System, 639 F. Supp 2d 388. See Karlson (2010).

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facility like the TSLF where government bonds are lent against private collateral. Ad-ditionally, we will speak of the “Central Bank”, but we also think of that as includingfiscal authorities’ facilities, as with TARP.

In our model, for simplicity, households are born with endowments already depositedin a bank which they can withdraw immediately to consume or keep in the bank. Abank is an institution with proprietary access to a productive investment opportunity,or a “project” hereafter, and relies on these deposits to invest. Each bank also has alegacy asset (hereafter simply an “asset”) that is used to back deposits. We assumethe quality of the asset is unknown unless costly (private) information is produced. Ifno information is produced, even banks with a bad asset can avoid early withdrawalsand invest in the project. The underlying problem in the economy is a scarcity of goodassets (“safe debt” to back repo, for example). When good assets are scarce, an effi-cient substitute is ignorance about which assets are good and which are bad. In thatcase, good and bad assets are pooled in an informational sense. When this poolingresults in a high enough perceived average value of banks’ portfolios, depositors donot examine their own bank’s portfolio (no run), maintain their deposits in the bankand banks efficiently invest in their projects.

A crisis happens when an (exogenous) event occurs (a fall in home prices, for in-stance) causing depositors to run, examining the bank’s portfolio and withdrawing ifthe bank has a bad asset. If this happens, banks can react by reducing the investmentscale to avoid information acquisition, or give in to the run, be examined by depos-itors and hope those depositors find out that the asset is good so it can invest at theoptimal scale. In either case, absent government intervention, aggregate consump-tion falls during a crisis.

The government’s goal is to prevent runs, which means avoiding information acquisi-tion about banks’ portfolios. How does a Central Bank end a crisis? First, the CentralBank opens an emergency lending facility (called the discount window throughout thepaper). As banks have heterogeneous collateral, which banks go to the discount win-dow depends on their private information about their asset quality and on the haircut(discount throughout the paper) on the assets they deliver as collateral at the discountwindow in exchange for government bonds.

The choice of the haircut on assets controls which banks participate in the discountwindow and by doing so determines the perceived average quality of private assetsremaining in the economy. But, banks which go to the discount window might be

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tempted to reveal that they have exchanged dubious private assets for good govern-ment bonds, so that they can maintain their deposits. But, this reveals informationabout their own asset quality which will affect them next period. If by revealing thatthey went to the borrowing facility banks are perceived as having low quality assets(because they exchanged low quality assets for high quality government backed as-sets), then they may be stigmatized. “Stigma” is the cost, in terms of future fundraising, of revealing that the bank borrowed from the emergency facility. We showhow the stigma costs arise endogenously and are necessary for the policy of opacityto work, as banks participating in lending facilities do not have incentives to revealthis information. Stigma aligns the secrecy incentives of the government with thoseof individual banks.

In contrast to the standard view that opacity prevents stigma, in our paper stigmaprevents transparency. Stigma plays an important role in sustaining secrecy, allow-ing the Central Bank to generate an information externality. If the Central Bank issuccessful, the result is a high enough perceived average value of banks’ assets in theeconomy such that runs do not occur. The threat of stigma is critical for opacity to besustainable in equilibrium. The optimal haircut is given by the point at which thereare enough banks participating at the lending facility to avoid runs on all banks, andno bank faces stigma in equilibrium.

A prominent branch of the literature has studied the effects of lending facilities inhelping the economy during crises. Most of this work is based on the premise that acrisis is given by an exogenous tightening in credit constraints (see Gertler and Kiy-otaki (2010) for a discussion) and lending facilities are set up to provide loans directly(increasing the role of governments as credit providers), open a discount window(mostly to improve inter-banking operations) and injecting equity (to increase the networth of banks). In our model the tightening of credit constraints is endogenous andcreated by an informational reaction in credit markets to changes in fundamentals,and then we emphasize the role of lending facilities in relaxing the informationaltightening of credit conditions.

There is also a large literature on the lender-of-last-resort summarized by Freixas, Gi-annini, Hoggarth and Soussa (1999 and 2000) and by Bignon, Flandreau, and Ugolini(2009).4 Recent historical work also includes Flandreau and Ugolini (2011) and Bignon,

4See also Flannery (1996), Freeman (1996), Rochet and Vives (2004), Castiglionesi and Wagner (2012)and Ponce and Rennert (2012).

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Flandreau, and Ugolini (2009) who document the development of the lender-of-last-resort role at the Bank of England and at the Bank of France. Unlike the existingliterature, we focus on why secrecy surrounds interventions during crises, the rolesof stigma, and a determination of how haircuts are set during a crisis.

The paper proceeds as follows. In Section 2 we specify the model, including thehouseholds’ choice to run or not to run and the banks’ choice to invest. In Section3 we describe crises and the structure of interventions that we consider. Section 4concerns the equilibrium when the economy is in a crisis and the Central Bank opensa lending facility. First, we determine the equilibrium for a fixed collateral haircutand a disclosure policy, and then we allow the Central Bank to choose the haircutlevel and the disclosure policy that maximize welfare. Section 5 concludes.

2 Model

2.1 Environment

We study a discrete-time economy composed of a Central Bank, a mass 1 of banksand a mass 1 of households in each period t. Households are short-lived (live for asingle period t), risk-neutral and indifferent between consuming at the beginning orat the end of the period. Each household is born with an endowment D of consump-tion good (numeraire) already deposited in a bank (accordingly we refer to them asdepositors), which is available for withdrawal at no cost at the beginning of the period.

Banks are also risk-neutral, but long-lived. Each bank i is matched with a singledepositor every period, and in addition to holding the household’s deposits at thebeginning of the period it holds a unit of a legacy, fixed-income, asset (in what followswe refer to it simply as the asset). With probability pit the asset is of good type, in whichcase it delivers C units of numeraire at the end of period t. With the complementaryprobability, the asset is of bad type and does not deliver any numeraire at the end ofthe period. The assets’ types are iid across periods and across banks. We refer to theex-ante probability that the bank i’s asset in period t is good, pit, as the quality of theasset held by bank i in period t.

The quality of the asset depends on an aggregate component and on an idiosyncraticcomponent, pit = pt + ⌘i. The aggregate component, pt, is time-varying and captures

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the average quality of assets in the economy. The idiosyncratic component, ⌘i, ispersistent and captures heterogeneity in the banks’ ability to manage assets, where⌘i ⇠ F [�⌘, ⌘], with E(⌘i) = 0 and ⌘ such that pi 2 [0, 1].

At the beginning of each period, a bank can finance an investment opportunity (inwhat follows we refer to it simply as the project). The project pays Amin{K,K⇤} withprobability q and 0 otherwise, where K⇤ is the maximum feasible scale of production.We make the following assumptions about the project’s payoffs

Assumption 1 Projects’ payoffs

• qA > 1. It is ex-ante optimal to finance the project up to a scale K⇤.

• D > K⇤. It is feasible to implementing the optimal investment just using deposits.

• C > K⇤. Good collateral is sufficient to cover the optimal loan size.

From this assumption, it follows immediately that the bank could use the proceedsfrom the good asset to finance the project up to K⇤, but this is not possible becausethe bank has to finance the project at the beginning of the period and these proceedsarise at the end of the period. This is why the bank requires deposits to finance theproject.

In terms of the information structure, the result of the project (whether it succeeds orfails) is known by the bank but it is neither observable nor verifiable by the depositors.With respect to the bank’s asset, no agent knows the asset type at the beginning of theperiod but there is also asymmetric information about the asset quality: while bothbanks and depositors know the average quality of assets in the economy, pt, eachbank also privately observes its own idiosyncratic component ⌘i, so the bank has abetter assessment of its own asset quality, pit.

Even though no agent knows at the beginning of the period whether a particularasset will turn out to be good or bad, depositors have a technology to privately in-vestigate the asset at the beginning of the period at a cost � in terms of numeraire.Using this technology they can determine whether the asset is worth C at the end ofthe period or not.5 We assume that information acquisition (and hence information

5Assuming that banks can also learn the bank’s asset type does not modify the main insights. SeeGorton and Ordonez (2014b) for double-sided information acquisition in a model without banks.

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itself) is private immediately after being obtained but becomes public at the end ofthe period. Still, the depositor can credibly disclose his private information immedi-ately upon acquisition if it is in his best interest to do so. This information acquisitiontechnology introduces incentives for depositors to learn about the bank’s asset typebefore deciding to keep their funds in the bank or not, taking advantage of such pri-vate information before putting their deposits at risk and before it becomes commonknowledge.

The timing during a single period t is as follows. At the beginning of the period, abank offers a contract that consists of an investment plan (the commitment to investa certain amount K in the project), an interest rate for deposits (promised repaymentR > D at the end of the period), and the collateral that secures the deposits (a fractionx of the asset to deliver to the depositor in case the bank is forced to default). Condi-tional on this contract, depositors choose whether or not to examine the bank’s assetthat is used as collateral and then decide whether or not to withdraw the deposits atthe beginning of the period. Then at the end of the period, if the depositors chose tokeep their deposits in the bank, investments payoffs are determined, the asset type isrevealed and the contract is fulfilled.

In what follows we say there is a run on a bank when depositors examine the bank’sasset at the beginning of the period, and withdraw the funds upon finding out thatthe asset is bad. In our setting we have a single depositor per bank and a run is nottriggered by a coordination failure. Our interpretation of a run is not one in which thedepositor suddenly says “give me the money!” but rather says “show me the money!”

Next, we characterize the contracts that banks choose to maximize their profits con-ditional on depositors’ participation (that is, conditional on depositors leaving theirdeposits in the bank for investment purposes). We show that by announcing theirinvestment strategies, banks can knowingly cause a run (examination of the asset be-fore the withdrawal decision) or not (no examination of the asset). In each situationthe bank will offer a repayment promise and collateral in case of default such thatthe depositors will be indifferent between withdrawing their deposits at the begin-ning of the period or not. Depending on which contract delivers higher profits inexpectation, the bank will optimally choose whether or not to induce a run and beexamined. Notice that this is not a game between banks and depositors but instead abanks’ choice of contracts to maximize profits conditional on the constraints imposedby the participation, information acquisition and withdrawal choices of depositors.

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2.2 Withdrawal Choices

We now characterize the contracts that banks’ offer to maximize their profits. As thefeasibility of these contracts depends on the bank’s investment plan, the bank canselect whether to being examined (facing a run) or not simply by committing to thecorrect investment level.

While the bank knows its own pit, households just have an expectation about theprobability that a particular bank’s asset is good, which we denote as E(pi|I), whereI is the information set of depositors at the time of deciding whether to withdraw atthe beginning of the period. In this section we focus on a single bank and a singleperiod, so we dispense with the subindex it.

2.2.1 Run

Assume the depositor examines the bank’s asset at the beginning of the period, spend-ing � of numeraire. If the asset is good, the bank can credibly guarantee that the de-positor will recover the deposits at the end of the period, even if the bank financesthe project at optimal scale K⇤ and the project fails. This is because C +D �K⇤ > D

by assumption. If the asset is bad, the bank will always default (paying less than de-posits), even if the project succeeds as it is not verifiable and the bank would rathernot repay. This implies that the depositor would always withdraw the deposits at thebeginning of the period upon information that the asset is bad.

What is the bank contract triplet (Rr, xr, Kr) (the promised repayment at the end ofthe period, the fraction of collateral that backs the loan and the investment scale) thatinduces the depositor to examine the asset and allow for investment if the asset isgood instead of just withdrawing the deposits at the beginning of the period?

Before acquiring information the depositor knows that, conditional on facing a bankthat is offering a contract that induces a run, the ex-ante probability of finding a goodasset is Er

(p) ⌘ E(p|run). As we will discuss, Er(p) will be determined in equilibrium

according to the banks that optimally choose to offer this contract. If finding out thatthe asset is good, the depositor’s expected payoffs are qRr + (1 � q)xrC � �, withthe additional constraint of a truth-telling condition Rr = xrC. If this condition wereviolated, the bank would always repay with the asset proceedings (if Rr < xrC) orwould always default (if Rr > xrC).

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Before acquiring information the depositor also knows that the ex-ante probability offinding a bad asset is (1 � Er

(p)), in which case the depositor will always withdraw,just loosing the cost of examination, then obtaining D � �.

The expected payoffs for a depositor of participating in the contract are then Er(p)(Rr�

�) + (1�Er(p))(D � �), independent of Kr. The depositor would then be indifferent

between participating in the contract or withdrawing without examining the assetand consuming at the beginning of the period if

D = (1� Er(p))D + Er

(p)Rr � �.

This participation constraint pins down the interest rate

Rr = D +

�

Er(p)

, (1)

which is independent of q (the probability the project pays off).

We have characterized the contract that maximizes the bank’s payoffs under runsas a triplet, (Rr, xr, Kr), where Rr is given by equation (1), xr =

RrC

and Kr = K⇤.Now we can compute the bank’s ex-ante expected profits given this contract. As thebank knows its asset is good with probability p, and that it will suffer a withdrawalin case its asset is bad, its ex-ante expected profits are p(D + qAK⇤ �K⇤ � Rr) + pC.Substituting Rr in equilibrium, the period expected net profits (net of the asset expectedvalue pC) from inducing a run are:

Er(⇡|p, Er(p)) = max{pK⇤

(qA� 1)� p

Er(p)

�, 0}. (2)

These expected profits depend not only on the probability that the bank’s asset isgood, p, but also on the average quality of the assets of all other banks inducing arun. This implies that there is cross-subsidization among banks that face a run: bankswith p > Er

(p) end up paying more to compensate depositors for the informationcosts in expectation, as p

Er(p) > 1. The opposite happens for banks with p < Er(p).

2.2.2 No-Run

Another possible contract is one where the depositor does no produce information. Inthis case, the depositor expects to obtain Rnr in case of repayment. And in case of de-

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fault, the non-invested deposits plus a fraction xnr of the collateral (of expected valueEnr

(p)C, where Enr(p) ⌘ E(p|no run) is the expected quality of assets among banks

that offer a contract that does not trigger examination), this is D �Knr + xnrEnr(p)C.

By the truth-telling constraint, these two promises should be the same, otherwisethe bank will always repay or default, whatever is the minimum. This implies thatxnr =

Rnr�D+KnrEnr(p)C .

The depositor will be indifferent between participating in the contract or not as longas D = Rnr. This is a feasible contract if xnr =

KnrEnr(p)C 1. This gives the first

constraint on investment from this contract,

Knr Enr(p)C. (3)

The second constraint on investment comes from guaranteeing that the depositordoes not have an incentive to deviate and examine the bank’s asset privately at thebeginning of the period. Depositors want to deviate because they can maintain de-posits at an expected gain if they know the asset received as collateral is good andwithdraw if the asset is bad. Depositors want to deviate if the expected gains fromacquiring information, evaluated at Rnr (and then at xnr), are greater than the gainsfrom not acquiring information. This is

(1� Enr(p))D + Enr

(p)⇥qRR

nr + (1� q)[D �Knr + xnrC]

⇤� � > D.

Substituting in the definitions of Rnr (and then xnr), there is no incentive to privatelydeviate and acquire information about the bank’s asset if

(1� Enr(p)) (1� q)Knr �.

Intuitively, by acquiring information the depositor only keeps his deposits in the bankif the asset is good, which happens with probability Enr

(p). If there is default, whichoccurs with probability (1 � q), the depositor gets xnrC for a unit of asset that wasobtained at a price Enr

(p)xnrC = Knr, making a net gain of (1� Enr(p)) xnrC =

(1� Enr(p)) Knr

Enr(p) with probability Enr(p)(1 � q). In other words, when deviating

and examining the asset, the depositor withdraws at the beginning of the period ifthe asset is bad and keeps the funds at the bank if the asset is good.

The condition that guarantees that depositors do not want to deviate and produce

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information about the bank’s asset can then be expressed in terms of the size of theproject, Knr,

Knr <�

(1� q) (1� Enr(p))

. (4)

If the bank scales back the project, then the depositor has less of an incentive to ac-quire information about the bank’s asset.

Imposing constraints (3) and (4), the investment size that is consistent with a no-runequilibrium is

Knr(Enr(p)) = min

⇢K⇤,

�

(1� q) (1� Enr(p))

, Enr(p)C

�, (5)

and the bank’s expected profits, net of the asset’s expected value pC, are

E(⇡|Enr(p)) = Knr(E

nr(p))(qA� 1). (6)

2.3 Investment Choices

By choosing the investment level, a bank can choose between facing a run (exami-nation of its assets) or not. The bank then selects the option (and related contract)that delivers the highest profits in expectation. The optimal decision of how much toinvest in the project is then isomorphic to the bank announcing an investment strat-egy that either finances the project at optimal scale (triggering a run) or at a restrictedscale (avoiding a run).

All the banks’ decisions must be consistent in aggregate. Even though banks are notfundamentally linked to each other, the choices of banks who choose to induce a runor not will affect the inference of depositors about the quality of the asset that a bankholds as a function of the contract it offers. As we have shown, the expected profits ofa bank suffering a run (equation 2) depends both on p and Er

(p) while the expectedprofits of a bank without a run (equation 6) depends on Enr

(p).

Since Er(⇡) increases in p while Enr(⇡) is independent of p, conditional on the strate-gies of all other banks, if a bank with asset of quality p announces an investment planthat induces a run, then all p0 > p will also. Similarly, if a bank with an asset of qualityp announces an investment plan that avoids a run, then all p0 < p will also. Intuitively,a bank with a high quality asset (high p so the bank believes its asset is very likely

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to be good) is more willing to open the asset to examination and to face the lotterythat a run represents. This is why banks with better assets would be more likely toannounce investment plans that induce examination.

This implies that the optimal investment strategy is given by a cutoff rule underwhich all banks with p < p⇤ restrict their investments to avoid runs and all bankswith p > p⇤ invest at the optimal scale and open themselves to examination of theasset (a run), where p⇤ is the asset quality that makes a bank indifferent between arun or not.

Er(⇡|p⇤, Er(p)) = Enr(⇡|Enr

(p)),

where Er(p) = E(p|p > p⇤) and Enr

(p) = E(p|p < p⇤).

More formally, allowing for corner solutions in which all banks either face a run ornot, the equilibrium cutoff is such that

p⇤ =

8>>><

>>>:

p+ ⌘ if Er(⇡|p+ ⌘, p+ ⌘) < Enr(⇡|p)

p⇤ s.t. Er(⇡|p⇤, E(p|p > p⇤)) = Enr(⇡|E(p|p < p⇤)).

p� ⌘ if Er(⇡|p� ⌘, p) > Enr(⇡|p� ⌘)

(7)

Remark on off-equilibrium beliefs: When the threshold p⇤ is a corner (either p⇤ =

p � ⌘ or p⇤ = p + ⌘), the expected quality of the asset of a bank following a strategythat is off equilibrium is not well-defined. More formally, if p⇤ = p + ⌘ no bank facesa run and then Er

(p) is not well-defined as it is an off-equilibrium strategy. The sameis the case for Enr

(p) if p⇤ = p� ⌘ and all banks are expected to face a run.

Following the Cho and Kreps (1987) criterion, we assume that if a bank follows astrategy that is not supposed to be followed in equilibrium, depositors believe thebank holds an asset that maximizes its incentives to deviate from the expected strat-egy. This is, if p⇤ = p+⌘ and a depositor observes a bank investing in a large project soas to induce a run, then the depositor believes that the bank has the highest availablequality asset, Er

(p) = p + ⌘. Similarly, if p⇤ = p � ⌘ and a depositor observes a bankinvesting in a project that discourages examination of the asset, then the householdbelieves that the bank has the lowest available quality asset, Enr

(p) = p� ⌘.

Remark on multiple cutoff values: As both Er(⇡|p⇤, E(p|p > p⇤)) and Enr

(⇡|E(p|p <

p⇤)) increase with p⇤ there may be multiple p⇤ in equilibrium. In what follows we will

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focus on the largest p⇤ as this represents the best equilibrium, the one that guaranteesthe highest sustainable output.

The next Proposition shows that in the best equilibrium the threshold p⇤ increaseswith the average quality of assets in the economy, p. This is, the higher the averagequality of assets in the economy the fewer banks face runs and asset examination.

Proposition 1 In the best equilibrium, the threshold p⇤ is increasing in p. There exist beliefspH > pL such that if p > pH no bank faces a run and if p < pL all banks face runs.

Proof

We first characterize pH . The maximum profits that a bank with the highest assetquality p+ ⌘ expects when inducing a run, conditional on no other bank facing a run(that is, when Er

(p) = p+ ⌘), are

Er(⇡|p+ ⌘, p+ ⌘) = (p+ ⌘)K⇤(qA� 1)� �.

while the expected profits when not inducing a run, conditional on no other bankfacing a run, are

Enr(⇡|p) = Knr(p)(qA� 1)

where, from equation (5)

Knr(p) = min

⇢K⇤,

�

(1� q)(1� p), pC

�.

There is always a p large enough such that K⇤ < �(1�q)(1�p) and K⇤ < pC such that

Knr(p) = K⇤. Then no bank would rather face a run and have a positive probability ofnot being able to invest, given that it can invest at optimal scale without examination.For all p > pH , all banks invest without runs, where pH is defined by Er(⇡|pH

+⌘, pH+

⌘) = Enr(⇡|pH), or

(pH+ ⌘)K⇤

(qA� 1)� � =

�

(1� q)(1� pH)

(qA� 1).

In this region, p⇤ = p+ ⌘, which trivially increases one for one with p.

We now characterize pL. The maximum expected profits that a bank with the lowestasset quality p � ⌘ can obtain when avoiding a run when all other banks face runs

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(that is, Enr(p) = p� ⌘) are

Enr(⇡|p� ⌘) = Knr(p� ⌘)(qA� 1)

where, from equation (5)

Knr(p� ⌘) = min

⇢K⇤,

�

(1� q)(1� (p� ⌘)), (p� ⌘)C

�.

The expected profits when the bank induces a run, conditional on all other banksinducing a run, are

Er(⇡|p� ⌘, p) = (p� ⌘)

K⇤

(qA� 1)� �

p

�.

Defining pL by the point at which Er(⇡|pL � ⌘, pL) = Enr(⇡|pL � ⌘), such that

(pL � ⌘)

K⇤

(qA� 1)� �

pL

�>

�

(1� q)(1� (pL � ⌘))(qA� 1),

then when p < pL all banks invest such that there is examination of their assets. Inthis region, p⇤ = p� ⌘, which also trivially increases one for one with p.

In the best equilibrium and by monotonicity, in the intermediate region of p thethreshold p⇤ also increases with p. Q.E.D.

3 Crises and Interventions

In the previous section we have characterized the contracts and investment levels ofall banks in the economy as a function of the aggregate pt in each period t. In thissection we restrict attention to two periods (t={1,2}), the first in which the economysuffers a crisis and the second in which the economy gets back to normal. How theeconomy fares during the crisis in the first period will determine output in the secondperiod.

The crisis in the first period comes from a shock to the economy as follows. First,to capture an information crisis, we assume that pt=1 = pL < pL, such that all banksface runs. Second, to capture the turbulent nature of crises, we assume that absent

14

interventions, there are potential information leakages; with probability " the bank’sasset type is revealed.

In the second period the economy goes back to normal times, which we capture byassuming that pt=2 = pH > pH such that there are no runs on any bank.

When the economy is in normal times it achieves the maximum potential consump-tion: households consume D and all banks invest at the optimal scale, producing anadditional amount K⇤

(qA� 1) of numeraire to consume. Consumption is then

WN = D + pHC +K⇤(qA� 1).

During crises, however, absent government intervention, all banks face runs, depos-itors examine the banks’ assets and only a fraction pL of banks retain their deposits,at an informational cost �, while the remaining fraction (1 � pL) of banks face with-drawals at the beginning of the period and are not able to finance their projects. Inthis case, consumption is

WC = D + pLC + pLK⇤(qA� 1)� �.

In crises, absent government intervention, consumption is clearly lower than con-sumption in normal times and the assumption of information leakages become irrel-evant as there is information acquisition about all banks anyway.

3.1 Lending Facilities

We focus next on analyzing a particular set of lending facilities that can be used by aCentral Bank to relax the incentives to acquire information in the economy and boostinvestment during the crisis in the first period. This is the timing of these facilities.

1. The Central Bank opens a discount window where it will exchange B governmentbonds (hereafter ”bonds” for short) per unit of asset, to be paid in numeraire atthe end of the period. It also announces whether it will reveal the identities ofbanks participating at the discount window (a policy of transparency) or whetherthese identities will be secret (a policy of opacity). The Central Bank chooses both

15

B and its disclosure strategy and can commit to its announced policy.6

2. Banks choose whether to go to the discount window or not. Even if the CentralBank announces a policy of opacity, still a bank may choose to reveal its partici-pation to its depositors. In that case, it becomes public knowledge that the bankhas bonds in its portfolio. But, this will result in an endogenously determinedstigma cost, denoted �, which we will characterize later in equilibrium.7

3. At the end of the first period, discount window participants with successfulprojects repay deposits using the proceeds from production and repurchasetheir assets back from the Central Bank. Failing banks lose their bonds to de-positors, who redeem them. Successful banks that did not borrow from thediscount window, repay their deposits with the proceeds from production andretain their assets. Unsuccessful banks default and hand over their assets to thedepositors, who consume them at the end of the period.

4. The Central Bank can liquidate the assets left in its possession by defaultingbanks but only imperfectly. The Central Bank can only extract a fraction � ofthe value of the banks’ asset in its possession at the end of the period. Thenthe numeraire generated by the asset in possession of the Central Bank plusdistortionary taxes (transfers at a social cost � per unit of numeraire) are used toredeem the bonds.

Our focus will be on whether the optimal policy of the Central Bank is one of trans-parency or one of opacity. This decision of the Central Bank will take into account thestrategies of banks and depositors.8 Importantly, the Central Bank does not produceinformation about the assets it receives through the discount window.

Step 2 is the critical step for banks if the Central Bank chooses the opacity policy. Ifneither the Central Bank nor the bank reveals participation at the window, the de-posit is backed by a portfolio with uncertain composition. Depositors only know that

6Borrowing a Treasury bond, posting assets as collateral, corresponds to using the Fed’s Term Se-curities Lending Facility; see Hrung and Seligman (2011). But, bonds could also be thought of as cashor reserves.

7See Armantier et al. (2015), Anbil (2017), Ennis and Weinberg (2010) and Furfine (2003) for otherways to model stigma costs.

8We are interested in the optimal disclosure policy within the realm of a lending facility interven-tion. The intervention corresponding to the solution of an optimal mechanism is outside the scope ofthis paper, but certainly an interesting problem for further research.

16

a fraction of banks participated at the discount window in equilibrium. But, underopacity, discount window borrowers may still wish to reveal that they borrowed sothat they can show that their portfolio consists of bonds guaranteed by the govern-ment instead of an asset of uncertain type. But, this makes them vulnerable to stigma.We will show how this stigma is determined and how it keeps borrowing banks fromrevealing that they borrowed so that the pooling of collateral that the Central Bankseeks can be accomplished.

Step 4 is also important because it concerns the costs of intervention. As the Cen-tral Bank is less efficient than private agents at extracting value from assets, the ex-tra resources that are needed to redeem bonds come from distortionary taxation ortransfers. With no costs there would be no trade-off faced by the Central Bank indetermining the optimal disclosure policy.

4 The Roles of Opacity and Stigma in Fighting Crises

We solve the Central Bank’s problem in two steps. First, we compute the equilibriumand welfare in the economy under opacity (Op) and then under transparency (Tr), as afunction of the bonds B that the Central Bank exchanges per unit of asset through thediscount window. Then we allow the Central Bank to choose the disclosure policy,{Op, Tr}, and the optimal B⇤ that maximizes welfare in equilibrium.

We will discuss policy in terms of this discount. Define

B ⌘ epC,

such that the Central Bank choosing ep implicitly chooses how many bonds B to offerper unit of asset with quality p. Then the discount for bank with asset of quality p isgiven by (p � ep)C. For convenience, and based on this one-to-one mapping betweenep and B, we will discuss comparative statics in terms of ep.9

In computing welfare the Central Bank has to take into account two equilibrium ob-jects, the fraction of banks that participate in the discount window, y(ep|{Op, Tr}) andthe probability of runs, �(ep|{Op, Tr}).

9We can also define the haircut by the ratio 1 � BpC = 1 � ep

p . When we refer to discount, it can alsobe interpreted as the haircut of government bonds.

17

4.1 Ending a Crisis with Opacity

4.1.1 Runs under Opacity

In a crisis, in the absence of interventions, all banks will choose to face a run. Since weare focusing on the conditions under a which a Central Bank can steer these decisions,prevent runs and improve investment, we now characterize the no-run contract in thepresence of the interventions described above.

In any equilibrium in which the Central Bank successfully maintains the anonymityof participating banks, banks make the same investment decisions regardless of theirparticipation at the discount window as long as they do not face runs. Still the banks’expected payoffs differ according to whether they go to the discount window or not.The cost of no participation is given by the probability that information about thebank’s asset type leaks and that the bank suffers a withdrawal of funds in the casethat the asset is revealed to be bad. This cost decreases in expectation with the qualityof the bank’s asset p. The cost of participation is the discount, (p � ep)C imposed bythe government, which increases in expectation with the quality of the bank’s asset p.

As the cost of participation increases with p and the cost of not participating decreaseswith p, banks tend to participate more when they have lower p. In other words, if abank with asset quality p borrows from the discount window, then all other bankswith p0 < p will do the same. In contrast, if a bank with asset quality p does notborrow from the discount window, then no bank with p0 > p will borrow. Hence,there is a threshold p⇤w in equilibrium such that all banks p < p⇤w participate andall banks p > p⇤w do not. We can redefine the fraction of banks participating at thediscount window as

y(p⇤w) = Pr(p < p⇤w).

The expected asset quality of banks participating at the discount windows is

Ew(p) = E(p|p < p⇤w)

and the expected asset quality of banks not participating is

Enw(p) = E(p|p > p⇤w).

Depositors know that with probability y(p⇤w) a particular bank has borrowed from the

18

discount window and obtained B bonds. Then its portfolio expected value is

y(p⇤w)B + (1� y(p⇤w))Enw(p)C.

Given this expectation, depositors are indifferent between withdrawing or not with-drawing at the beginning of a period when

D = qRRnr + (1� q)RD

nr,

where RDnr is now given by the expected return in case the bank defaults,

RDnr = D �Knr + xnr[yB + (1� y)Enw

(p)C],

where xnr is again the fraction of the portfolio promised to the depositor in case ofdefault conditional on no run.

In equilibrium the bank will promise in expectation the same in case of repayment ordefault (for the same reasons as in the previous section), so D = RR

nr = RDnr. We can

then obtain the fraction of assets in the portfolio that go to the depositors in case ofdefault,

xnr = min

⇢Knr

yB + (1� y)Enw(p)C

, 1

�.

Now we can compute the incentive of a depositor to privately acquire informationabout the portfolio of the bank. At a cost � the depositor can privately learn whetherthe bank has bonds or the asset in its portfolio, and in case the where the bank has anasset, whether the asset is of good or bad type.

The benefits of acquiring information are as follows: with probability y(p⇤w) the bankhas bonds and the depositor that examines the portfolio does not withdraw his de-posits because he finds the bank holds valuable collateral, getting a payoff of:

qRRnr + (1� q)[D �Knr + xnrB]� �.

With probability (1� y)(1�Enw(p)) the bank has a bad asset and the depositor with-

draws at the beginning of the period, getting a payoff of D � �. Finally, with proba-bility (1 � y)Enw

(p) the bank has a good asset, and the depositor keeps his deposits

19

in the bank, getting a payoff of

qRRnr + (1� q)[D �Knr + xnrC]� �.

As RRnr = D, and adding the previous payoffs weighted by the respective probabili-

ties, there are no incentives to acquire information as long as:

D + y(1� q)[xnrB �Knr] + (1� y)Enw(p)(1� q)[xnrC �Knr]� � D.

Rearranging

(1� q)xnr(yB + (1� y)Enw(p)C)� (1� q)[y + (1� y)Enw

(p)]Knr �.

Since xnr(yB + (1� y)Enw(p)C) = Knr, there is no information acquisition as long as

Knr(Enw(p)) �

(1� q)(1� y)(1� Enw(p))

. (8)

Based on this condition we can obtain �(ep), the probability of a run in which depos-itors privately acquire information about a bank’s portfolio before choosing whetherto withdraw

�(ep) =

8>>><

>>>:

0 if K < �(1�q)(1�y(ep))(1�Enw(p|ep))

[0, 1] if K =

�(1�q)(1�y(ep))(1�Enw(p|ep))

1 if K > �(1�q)(1�y(ep))(1�Enw(p|ep)) .

(9)

The next Proposition is trivial from comparing the condition for no information acqui-sition in the absence of intervention (equation 4) and in the presence of intervention(equation 8). It is also straightforward to check that condition (8) is more likely tohold when y is higher.

Proposition 2 Runs are less likely with intervention when there are many banks participat-ing at the discount window (i.e., high p⇤w and then high y).

4.1.2 Discount Window Borrowing under Opacity

The previous discussion focused on the depositors’ incentives to examine a bank’sportfolio conditional on the fraction of banks participating at the discount window,

20

y. Now we solve for this fraction y in equilibrium as a function of ep.

First, defineL(p, ep) ⌘ pK⇤

(qA� 1)� p

Enw(p|ep)�

as the “relatively (L)ow” bank’s expected payoffs when when the bank does not par-ticipate at the discount window and faces a run.

Second, defineH(K(ep)) ⌘ K(ep)(qA� 1)

as the “relatively (H)igh” bank’s expected payoffs when the bank does not face a runand is able to invest K(ep).

Finally, defined(p, ep) ⌘ (p� ep)C

as the bank’s discount when borrowing from the discount window.

In this setting define stigma, �(p, ep), to be the cost in terms of a higher probabilityof a run in the second period coming from information that the bank participatedat the discount window in the first period. Even though we will derive this valueendogenously later, intuitively it captures that if a bank’s participation in the discountwindow is revealed, future depositors with access to this information will know thatthe quality of the bank’s asset is lower than average, as p < p⇤w.10

Having defined all these elements, we can compare the payoffs of a bank p fromparticipating at the discount window or not. If the discount is ep, the expected payoffsof a bank p that does not borrow from the discount window are

Enw(⇡|p, ep) = �(ep)L(p, ep) + (1� �(ep))[(1� ")H(K) + "L(p, ep)] (10)

while its expected payoffs from borrowing in the discount window are

Ew(⇡|p, ep) = �(ep)[H(K)� d(p, ep)� �(p, ep)] + (1� �(ep))[H(K)� d(p, ep)]. (11)

As can be seen, these payoffs depend both on the quality of the bank’s asset and onthe discount from participating in the discount window.

10There is in principle a symmetric positive stigma, a benefit in terms of reducing bank runs fromthe revelation that a firm has not participated in the discount window, then having asset with qualityabove the average. We do not introduce any notation for this, as later we show it is zero.

21

The next four lemmas characterize the optimal depositors’ run (examination) strate-gies and banks’ participation strategies, as a function of ep, under opacity.

Lemma 1 Very low discount region.

There exists a cutoff eph < pL+⌘ such that, for all ep 2 [eph, pL+⌘) (“very low discount region”),no depositor runs (that is, �(ep) = 0) and all banks borrow from the discount window (that is,y(ep) = 1).

Proof Assume first that the Central Bank chooses a discount given by ep = pL + ⌘.In this case there is no discount for the bank with the highest asset quality, this isd(pL + ⌘, ep) = 0 and there is indeed a subsidy for all other banks (the next analysistrvially applies then for all discounts such that ep > pL + ⌘). Compare equations (10)and (11) for p = pL + ⌘. It is optimal for the bank with the highest quality, p = pL + ⌘,to borrow from the discount window, and then it is also the optimal strategy for allother banks to also borrow. This implies y = 1 and there is no stigma (i.e., � = 0),confirming that this is indeed the best sustainable equilibrium.11 Hence, for ep � pL+⌘,a fraction y(ep) = 1 of banks participate, from equation (9) �(ep) = 0 and from equation(8) K(ep) = K⇤.

For lower levels of ep, this is still an equilibrium as long as the bank with the highestasset quality finds it optimal to participate, and then all other banks do as well. Thecritical level eph is determined by the point at which the bank with the highest qualityis indifferent between participating at the discount window or not,

H(K⇤)� d(pL + ⌘, eph) = (1� ")H(K⇤

) + "L(pL + ⌘, eph),

Using the definitions of L(p, ep), H(K(ep)) and d(p, ep),

eph = (pL + ⌘)� "

C[(1� pL � ⌘)K⇤

(qA� 1) + �] .

Q.E.D.11Notice that this is only one possible equilibrium. If everybody believes that some banks did not

borrow from the discount window, � > 0, it may be indeed optimal for those banks not to borrow fromthe window. This shows how endogenous stigma may induce equilibrium multiplicity and may gen-erate self-confirming collapses in the use of discount windows. Here we focus on the best equilibriumbased on intervention, and show its limitations.

22

This region shows that there are always levels of discount low enough (ep large enough)to induce all banks to participate in the discount window and, based on this outcomeno depositor would examine the bank’s portfolio, maintaining their deposits avail-able for investment.

Lemma 2 Low discount region.

There exists a cutoff epm < eph such that, for all ep 2 [epm, eph) (“low discount region”), no de-positor runs (that is, �(ep) = 0), not all banks participate (this is, y(ep) < 1) and participationdeclines with the level of discount (that is, y(ep) increases with ep).

Proof Assume first the extreme case in which ep = eph. From the previous proposition,y(eph) = 1 and �(eph) = 0. For ep = eph � ✏ (from the definition of eph),

H(K⇤)� d(pL + ⌘, eph � ✏) < (1� ")H(K⇤

) + "L(pL + ⌘, eph � ✏),

and then banks with asset quality pL + ⌘ strictly prefer to not participate at the dis-count window. This implies that y(eph � ✏) ⌘ Pr(p < p⇤w(eph � ✏)) < 1 where p⇤w(eph � ✏)

is given by the indifference condition

H(K⇤)� d(p⇤w, eph � ✏) = (1� ")H(K⇤

) + "L(p⇤w, eph � ✏),

ord(p⇤w, eph � ✏) = "[H(K⇤

)� L(p⇤w, eph � ✏)],

where p⇤w declines monotonically as we reduce ep. Notice that this construction relieson the conjecture that �(eph � ✏) = 0, but for relatively low ✏ this is the case as long asy(eph � ✏) and Enw

(p|eph � ✏) are such that

K⇤ <�

(1� q)(1� y)(1� Enw(p))

.

Define by p⇤w the threshold such that y(p⇤w) is the fraction of banks borrowing fromthe discount window and E

nw(p|p⇤w) is the expected quality of the non-participating

banks’ assets, such that depositors are indifferent between running or not when thebank invests K⇤, i.e.,

K⇤=

�

(1� q)(1� y)(1� Enw(p))

.

23

The bank with the marginal asset quality p⇤w(epm) is determined by

H(K⇤)� d(p⇤w, epm) = (1� ")H(K⇤

) + "L(p⇤w, epm),

such that

y = Pr(p < p⇤w(epm)) and Enw(p) = E(p|p > p⇤w(epm)).

Finally, the threshold p⇤w is well-defined, as both y and Enw(p) monotonically increase

in p⇤w, which monotonically decreases in ep. Q.E.D.

Intuitively, when the discount is low (ep is large), many banks choose to borrow at thediscount window because the cost in terms of exchanging assets for bonds at a lowdiscount more than compensates for the risk of a run and information about the assetbeing revealed. Given this, depositors do not have incentives to run and examine thebank’s portfolio.

The next lemma characterizes the equilibrium for an intermediate discount region.

Lemma 3 Intermediate discount region.

There exists a cutoff epl < epm such that, for all ep 2 [epl, epm) (“intermediate discount region”),depositors run with positive probability but not always (that is, �(ep) 2 (0, 1)) and a constantfraction y of banks go to the discount window (that is, y(ep) = y(epm)).

Proof

Assume first the extreme case where ep = epm. From the previous lemma, y(epm) = y

and �(eph) = 0. For ep = epm � ✏, the bank that is indifferent about borrowing from thediscount window has slightly lower quality, p⇤w(epm � ✏) < p⇤w(epm). Then y(epm � ✏) < y

and Enw(p|epm � ✏) < E

nw(p), and there are incentives to run if investment is K⇤, as

there are relatively few participants at the discount windows (low y) and the assetsof those not participating at the discount windows are worse in expectation (lowEnw

(p)). Formally,

K⇤ >�

(1� q)(1� y(epm � ✏))(1� Enw(p|epm � ✏))

.

One possibility for banks to prevent runs, �(ep) = 0, is to scale back the investment in

24

the project to K(p⇤w) < K⇤, to avoid information acquisition. The size of the invest-ment K, however, also determines y, as p⇤w(epm � ✏) is pinned down by the condition

d(p⇤w, epm � ✏) = "[H(K(p⇤w))� L(p⇤w, epm � ✏)].

A lower K relaxes the constraint and reduces the incentive to run, but at the sametime reduces p⇤w for a given ep, increasing the incentive to run. Intuitively, for a givendiscount, a reduction in the gains of borrowing from the discount window (fromlower H(K)) reduces the quality of the marginal bank which is indifferent betweenborrowing or not, i.e., reducing p⇤w further.

If H(K(p⇤w)) declines faster than L(p⇤w), then no participant will go to the discountwindow if, at the lowest possible p, which is pL � ⌘,

H(K(pL � ⌘))� L(pL � ⌘, pL) < 0

which we have assumed in the definition of a crisis (pL is low enough such thatall banks would rather face examination than restricting their investments to avoidruns). In words, it is not in the best interests of banks to discourage runs by reducingthe size of their investments in the project, which is in contrast to what happens in theabsence of intervention. Our result here comes from the endogenous participation ofbanks at the discount window. By reducing K, the effect of a lower y in inducinginformation acquisition is stronger than the effect of a lower K in discouraging infor-mation acquisition, thus increasing on net the incentives for depositors to examine abank’s asset as K declines.

Under these conditions, the equilibrium should involve either the discount windowsustaining an investment of K⇤ (when a fraction y of banks borrows from the discountwindow) or no participation in the discount window at all, which replicates the al-location without intervention. To maintain the fraction y constant in this region as epdeclines, the marginal bank with asset quality p⇤w(epm) should be indifferent betweenborrowing from the discount window or not. This is achievable only if depositorschoose to run with some probability and examine the portfolio of banks as ep declines,as this increases the incentives to have bonds in the portfolio.

As the fraction of banks participating at the discount windows, y, is constant, depos-itors are indifferent between running or not and �(ep) > 0 is an equilibrium. To deter-mine �(ep) in equilibrium we next discuss the determination of endogenous stigma.

25

With positive information acquisition (�(ep) > 0) there is stigma when the depositordiscovers a bank’s participation at the discount window. The reason there is stigmais that those banks borrowing from the discount window are the ones with relativelylow asset quality (relatively low ⌘i). Once a bank is stigmatized, it may face with-drawals during normal times in the second period. To be more precise about theendogeneity of stigma, once back in normal times, the bank will face a run wheninvesting at the optimal scale of production if

K⇤ >�

(1� q)(1� Ew(p))

,

and the bank will not suffer a run in the second period based on an indifferencecondition that pins down p⇤2 in the second period where

Er(⇡|p⇤2, Er(p|p < p⇤w)) = Er(⇡|Enr

(p|p < p⇤w)).

We denote by Kw(p, ep) the investment size that a bank with an asset of quality p

can obtain in the second period conditional on it having been revealed that the bankborrowed from the discount window in the first period.

Then, stigma is given by

�(p, ep) = [K⇤ �Kw(p, ep)](qA� 1),

where � is an increasing function of the discount (a decreasing function of ep). As thediscount increases, p⇤w decreases, y(p⇤w) decreases and Ew

(p) decreases. This leads toa decline in Kw and then an increase in stigma from going to the discount window.

Given y, to maintain the investment size K⇤ without triggering information, the in-difference of the marginal bank p⇤w pins down the probability the depositor runs. Thisis Enw

(⇡|p⇤w) = Ew(⇡|p⇤w), which implies

�L(p⇤w, ep) + (1� �)[(1� ")H(K⇤) + "L(p⇤w, ep)] = [H(K⇤

)� d(p⇤w, ep)]� ��(p⇤w, ep)

and then�(ep) = d(p⇤w, ep)� "[H(K⇤

)� L(p⇤w, ep)](1� ")[H(K⇤

)� L(p⇤w, ep)]� �(p⇤w, ep). (12)

Finally, depositors randomize between running and not given the bank investing K⇤

in the project, and a bank with asset quality p⇤w is indifferent between borrowing from

26

the discount window or not. Q.E.D.

In the intermediate discount range the equilibrium cannot involve pure strategies bydepositors. Since participation at the discount window when depositors do not run islow, depositors have incentives to run. In contrast, if depositors run, banks have moreincentives to borrow from the discount window, which discourages runs. Depositorshave to be indifferent between running or not. As the discount increases in this range,banks incentives to borrow from the discount window have to be compensated for byan increase in the probability of runs.

Finally, the next lemma characterizes the case with large discount.

Lemma 4 High discount region.

There exists a cutoff epl > 0 such that, for all ep 2 (0, epl] (“high discount region”), banks donot borrow from the discount window (that is y(ep) = 0) and depositors always run (that is,�(ep) = 1).

Proof

From equation (12), �(p⇤w) = 1 for d(p⇤w, ep) � H(K⇤)� L(p⇤w, ep)� �(p⇤w, ep). Define epl to

be the discount that solves this condition with equality. Evaluating Enw(⇡) and E(⇡)

at �(ep) = 1 given a project of size K⇤ and at the threshold p⇤w a bank with asset p⇤wgoes to the discount window whenever

H(K⇤)� d(p⇤w, ep)� �(p⇤w, ep) > L(p⇤w, ep)

which is never the case in this region by the previous condition. Then y(ep) = 0 andthen it is indeed optimal for depositors to run, that is �(ep) = 1. Q.E.D.

Intuitively, when the discount is high (ep is low), no bank chooses to borrow from thediscount window, even when depositors are running. Given this reaction, depositorsalways run. The economy generates the same consumption as in the case of a crisiswithout intervention.

The equilibrium strategies derived in Lemmas 1-4 are illustrated in Figure 1. On thehorizontal axis we show the average discount d(pL, ep), the red solid function showsthe fraction of depositors who run, �(ep), and the black dashed function shows thefraction of banks that borrow from the discount window, y(ep). We use the average

27

discount instead of ep as it is more intuitive to think of the discount as the cost ofparticipation. The strategies in the “very low discount region” [0, d(pL, eph)] are shownin Lemma 1, in the “low discount region” [d(pL, eph), d(pL, epm)] are shown in Lemma2, in the “intermediate discount region” [d(pL, epm), d(pL, epl)] in Lemma 3 and in the“high discount region” [d(pL, epl), C] in Lemma 4.

Figure 1: Equilibrium Strategies under Opacity

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�

y(ep) �(ep)

d(pL, ep)d(eph) d(epm) d(epl)

Very Low

Discount

Region

Low

Discount

Region

Intermediate

Discount

Region

High

Discount

Region

Role of Stigma in Sustaining Opacity: It is straightforward to check that no bankwould like to deviate from the opaque policy of the Central Bank in terms of disclos-ing its own participation, or lack thereof, at the discount window. If revealing theirown participation banks have to pay the stigma cost without getting any benefit (thebest the bank can accomplish by disclosing participation in the discount window isto prevent examination and invest K⇤, which is what happens by not revealing par-ticipation). Similarly, banks not borrowing from the discount window do not wantto reveal their lack of participation, otherwise they have a higher chance of sufferinga run (depositors will always try to examine the bank’s portfolio once they know for

28

sure that they hold assets instead of bonds).

4.2 Ending a Crisis with Transparency

When the Central Bank discloses information about the identity of banks participat-ing at the discount window, the information acquisition strategy of depositors is con-ditional on this information. More specifically, when depositors know a bank has bor-rowed from the discount window, they never run on it, as they know the bank usesgovernment bonds as collateral. Then �(ep) = 0 for all ep, conditional on participationat the discount window, and Ew

(⇡) = H(K⇤) � d(p, ep) � �(p, ep). In contrast, when

depositors know a bank has not participated at the discount window, they alwaysrun on it, as they know the bank has an asset in its portfolio. Then �(ep) = 1 for all ep,conditional on no participation in the discount window, and Enw(⇡|p, ep) = L(p, ep).

This implies that the borrower p⇤w that is indifferent about borrowing from the dis-count window, when the discount is given by ep, is determined by

H(K⇤)� d(p⇤w, ep)� �(p, ep) = L(p⇤w, ep).

Notice that this is the same condition that determines p⇤w(epl) in Lemma 4. Still thisdoes not imply that policies of opacity and transparency coincide at epl. While theequilibrium with opacity is characterized by only some banks participating in the dis-count window, y < 1, the equilibrium with transparency have all banks participating.The same discount induces more banks participating under transparency than underopacity because transparency eliminates the strategic incentive of banks to “hide”.This is, under opacity, a bank has incentives not to borrow from the lending facilityso not to pay a discount and still investing K⇤ because no bank is examined.

Lemma 5 In the best equilibrium under transparency, all banks borrow from the discountwindow y(ep) = 1 for ep 2 [epT , p+ ⌘] such that epT < eph, and y(ep) = 0 for ep < epT .

Proof Define epT byH(K⇤

)� d(pL + ⌘, epT ) = L(pL + ⌘, epT )

For all ep > epT

H(K⇤)� d(pL + ⌘, ep) > L(pL + ⌘, ep)

29

and the bank with asset of quality pL + ⌘ strictly prefers to borrow from the discountwindow. Notice that as all banks participate, �(p, epT ) = 0 for all p.

Recall the condition that pins down eph in Lemma 1 is

H(K⇤)� d(pL + ⌘, eph) = (1� ")H(K⇤

) + "L(pL + ⌘, eph).

Then, as the right-hand side is larger than in the condition above that pins down epT ,the discount has to be smaller and epT < eph. Q.E.D.

Intuitively, when all banks participate under a policy of transparency, the alternativeof not participating results in suffering a run for sure. This is in stark contrast to theopacity policy in which, if all banks participate, the alternative of not participatinghas only a slight chance " of an information leakage and a run.

There is no parameter restriction that prevents epT from being smaller than epl. In such acase, for all discount levels epT < ep < epl, there is full participation under transparencyand no participation under opacity. The reason is that at epl depositors always runconditional on there being y banks participating. If all banks were participating, de-positors would not run and then the alternative gains from not participating underopacity are very large, inducing individuals to deviate, making this equilibrium un-sustainable. This is not the case under transparency in which the alternative to notparticipating always leads to runs, turning the deviation unprofitable.

4.3 Opacity or Transparency?

Given the equilibrium strategies for each ep under both opacity and transparency, wecan compute the total production (or welfare in our setting) for each ep under eachdisclosure policy. As banks keep the net gains from production, welfare is an aggre-gation of the payoffs of all banks with different quality p, given by

W (ep) =

Z

p

[Iw[H +B � pC] + (1� Iw)[�L(p) + (1� �)((1� ")H + "L(p))]] dF (p)

+

Z

p

Iw[q(pC � B) + (1� q)(�pC � B)]dF (p)

+

Z

p

[Iw[q(H � ��(p))� (1� q)�(1� �)pC] + (1� Iw)H] dF (p),

30

where Iw is an indicator function that takes the value 1 if the bank participates at thediscount window and 0 otherwise.

Taking integrals and rewriting the expression,

W (ep) = y(H +B � Ew(p)C) + (1� y)[�bL+ (1� �)((1� ")H + "bL)]

+y[q(Ew(p)C � B) + (1� q)(�Ew

(p)C � B)]

+y[q(H � �b�)� (1� q)�(1� �)Ew(p)C] + (1� y)H,

where H ⌘ H(K⇤), bL ⌘

Rp|nw

L(p, ep)dp and b� ⌘R

p�(p, ep)dp.

The first two terms (the first line) represent the welfare of banks in the crisis period.A fraction y of banks borrow from the discount window leading to a production ofH and exchanging asset for bonds at an average discount of B � Ew

(p)C. A fraction1� y of banks do not participate and their investments lead to a production level thatdepends on whether they suffered a run or if there was an informational leak. Thisfirst line of the welfare function can be rewritten as

H � y(Ew(p)C � B)� (1� y)("+ �(1� "))(H � bL).

The third term (the second line) represents the welfare for the government. From thefraction y of banks borrowing from the discount window, a fraction q has their assetseized, which delivers Ew

(p)C in expectation while a fraction 1 � q defaults and theasset has to be liquidated, recovering just �Ew

(p)C. Still in both cases the governmenthas to repay B for the bonds. This second line of the welfare function can be rewrittenas

y(Ew(p)C � B)� y(1� q)(1� �)Ew

(p)C.

The last term (the third line) captures investments in the second period (no discount).Those banks that did not borrow from the discount window and those that did with-out their participation being revealed can borrow without triggering a run in the sec-ond period (then leading to production level H). Those banks that participated andtheir participation was revealed (because they were examined) can potentially suffera run in the second period because of stigma (captured by L). Finally, the govern-ment has to repay (facing inefficiency costs 1 � � and distortionary taxation costs �)the bonds that could not be covered by liquidating assets in the previous period. The

31

third line can then be rewritten as

H � y[q�b�+ (1� q)�(1� �)Ew(p)C].

Adding (and canceling) terms, total welfare is

W (ep) = 2H � (1� y)("+ �(1� "))(H � bL)� y[(1� q)(1 + �)(1� �)Ew(p)C � q�b�].

Since the unconstrained welfare is 2H in both periods, we can denote the distortionfrom the crisis as

Dist(ep) = (1� y)("+ �(1� "))(H � bL) + y[(1� q)(1 + �)(1� �)Ew(p)C + q�b�]. (13)

The first component shows the distortion that comes from lower output from banksthat did not borrow from the discount window, either due to information leaks or be-cause they suffered a run. The second component shows the costs of the distortionarytaxation that is needed to cover deposits from defaulting banks, and that cannot becovered by liquidating the asset. Finally, the third component shows the lower pro-duction in the second period that arises from stigma – banks who were discoveredborrowing from the discount window and then were revealed to have collateral ofrelatively lower quality – being more likely to suffer a run and produce less in thesecond period.

Now, we can compare the welfare levels of distortions for each ep under opacity andunder transparency, and follow the Ramsey approach of choosing the one that maxi-mizes welfare. The next Proposition characterizes the optimal policy

Proposition 3 When the distortionary costs of taxation (1+�) and public liquidation (1��)of assets are large relative to the leakage probability ("),

(1 + �)(1� �)

"� (1� pL)K⇤

(qA� 1) + �

(1� q)pLC,

a discount rate of epm under opacity dominates transparency.

Proof Here we compare the distortions for different levels of discount ep (in the dif-ferent regions that we characterized in the previous section) for different disclosurepolicies.

32

We consider first the parametric case for which epT < epl.

Under transparency, all banks participate when ep � epT . This implies that y = 1, � = 0

and Ew(p) = pL. Distortions in this range are then

Dist(ep|Tr) = (1 + �)(1� q)(1� �)pLC.

In contrast, for all ep < epT , the discount window is not sustainable and distortions arethe same as without intervention, H � L ⌘ (1� pL)K⇤

(qA� 1) + �.

Under opacity, distortions also depend on the equilibrium regions. In the “very low”discount region all banks participate, then y = 1 and � = 0, and distortions are thesame as under transparency (in the region ep � epT above).

In the “low” discount region, y < 1 but � = 0, then from equation (13)

Dist(ep|Op) = (1� y)(H � bL)"+ y(1 + �)(1� q)(1� �)Ew(p|Op)C.

Since yEw(p|Op) < pL and H � L > H � bL, the sufficient condition for the distortion

under opacity is lower than the distortion under transparency is

(1 + �)(1� q)(1� �)pLC > "(H � L)

which is the condition in the Proposition. In words, even though a fraction " of non-participating banks produce less, there are fewer banks that participate and need tobe covered by distortionary taxation in case they default.

In the “intermediate” discount region, y = y, but � > 0 and increases with the dis-count. From equation (13) it is clear that in this region the distortion increases with �

and then with the discount. While the fraction of banks participating is fixed, thereare more runs and then more banks not participating end up producing less, both inthe first period (less deposits) and the second period (more stigma).

Finally, in the “high” discount region, the discount window is not sustainable underopacity, so distortions are the same as without intervention, H � L.

Summarizing, as the discount increases, distortions under opacity are fixed in thevery low discount region, decrease in the low discount region, increase in the in-termediate discount region and reach the maximum in the high discount region. In

33

contrast, distortions under transparency are fixed whenever the discount window issustainable, as either all banks participate or neither does. This implies the optimaldiscount is epm under opacity.

Consider finally the parametric case for which epT � epl. In this case the discountwindow under opacity collapses at lower discount levels than under transparency.Still the optimal policy is a discount of epm under opacity because epT < eph, as shownin Lemma 5. Q.E.D.

Figure 2 displays this proposition graphicall, showing welfare under both disclosurepolicies (dashed red for transparency and solid black for opacity) for all discountrates, using a set of parameters for the illustration such that d(pL, epl) < 0.4 < d(pL, epT ).

Figure 2: Welfare Comparison

2 / 2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.417

17.2

17.4

17.6

17.8

18

18.2

18.4

18.6

18.8

��

��

��

��

��

��

��

��

��

��

��

�

��

��

��

��

��

��

��

��

��

��

��

�

��

��

��

��

��

��

��

��

��

��

��

�

Welfare Opacity

Welfare Transparency

d(pL, ep)d(eph) d(epm) d(epl)

Very Low

Discount

Region

Low

Discount

Region

Intermediate

Discount

Region

High

Discount

Region

Notice that the welfare implemented with a transparent intervention is the same forall discounts in the range of this illustration, as we have assumed the discount thatinduces banks not to participate in a transparent lending facility is larger than 0.4.This level of welfare under transparency is naturally lower than the unconstrained

34

first best because the government has to rely on distortionary taxation to meet thedeposits of insolvent banks.

Under opacity, welfare depends on the discount. In the very low discount region allbanks participate at the discount window, which implies that welfare is the same asthat obtained under transparency. In the low discount region, some banks prefer totake advantage of pooling and not participate. In this region welfare increases withthe discount as no depositor runs, and then the government needs to rely less ondistortionary taxation. The sufficient condition for welfare to increase in this region(which is the one displayed in Proposition 3 and assumed in the figure) guaranteesthat the distortionary costs of interventions are larger than the costs of some banksfacing a run because there is a leak, with probability ".

In the intermediate discount region, only a fraction of banks (y) borrow from the dis-count window but more and more depositors run as the discount increases. Thisreduces welfare because the level of distortionary taxation is lower than under trans-parency but more and more banks face runs and produce relatively less in the cri-sis period and face stigma that reduces their production in the second period. Inthis region, once the discount level is so large that the “stigma” effect dominates the“less distortion” effect, then welfare under opacity is lower than welfare under trans-parency. Finally, in the high discount region, the level of the discount is so high thatthere is no participation in equilibrium under opacity, with welfare reaching a nointervention level.

As is clear, the policy that maximizes welfare is imposing an average discount ofd(epm) under opacity. At this discount, and without disclosure, the participation ofbanks in the discount window (which creates distortionary costs) is minimized, with-out triggering any runs.

Remarks on Securitization and Deposit Insurance In our setting, if banks were ableto sign contracts that perfectly eliminate the idiosyncratic risk of individual portfoliosthen no depositor would have an incentive to acquire information about a bank’s as-set. In other words, if idiosyncratic risk were eliminated by pooling all projects in theeconomy, there are no runs and no role for intervention. Indeed, with diversificationnot only would there be no crisis but all banks would be able to invest at the optimalscale.

Even though we have assumed that banks cannot diversify their individual portfolio

35

risk, there could be in principle two institutions that allow for such diversification:securitization (sustained by private contracts) and deposit insurance (imposed by publicregulation).

In the case of securitization, a bank can sign a contract at the beginning of the period,selling shares of its own asset and buying shares of the assets of other banks, elimi-nating the idiosyncratic risk as the value of its portfolio would be deterministic andequal to pLC. This contract discourages depositors in the bank from acquiring infor-mation about its portfolio, which is now irrelevant for the probabilities of recoveringthe deposit. There are no runs and no crisis. These private contracts are difficult tosustain, however. Banks with high ⌘ subsidize banks with low ⌘ and may not haveincentives to enter into these contracts as a way to signal their high ⌘. Studying thesustainability of these contracts is interesting to understand the effects of securitiza-tion as a stabilizing innovation, but it is outside the scope of this paper.

Assuming securitization is not feasible, the government may have incentives to im-pose diversification in the form of deposit insurance. In the standard view of bankruns, under which they are triggered by a collective action problem, deposit insur-ance prevents panics and then it is not used in equilibrium. In our setting a run isnot driven by lack of coordination among depositors but instead by individual incen-tives to investigate the bank’s portfolio, withdrawing the funds if it is found that theportfolio is of low value. The government can prevent the examination of a bank’sportfolio by forcing banks to pay a premium, ex-post, in case their assets are goodand to receive insurance in case their assets are bad. As this cross-subsidization isself-financed inside the system, no taxation is used in equilibrium.

One interpretation of what happened during the recent financial crisis is that somebanks (commercial) were under deposit insurance (and nothing happened to them).Some others (shadow) were using securitization. Securitization may be a fragile con-tract. In particular if adverse selection concerns are present among banks, these con-tracts may not be sustainable and the same problem analyzed in the paper develops.Again, this is a subject that requires more research.

Comparison with the Bagehot’s Rule The classic rule for a central bank to follow ina crisis is Walter Bagehot’s (1873) rule that the central bank should lend freely, at ahigh rate, and on good collateral. In the recent financial crisis, Ben Bernanke, MervynKing and Mario Draghi, the respective heads of the Federal Reserve System, the Bankof England, and the European Central Bank, reported that they followed Bagehot’s

36

advice; see Bernanke (2014a and 2014b), King (2010) and Draghi (2013). But, in fact,there was more to their responses to the crisis. All three central banks also engagedin anonymous or secret lending to banks.

Indeed, it is not obvious why Bagehot’s advice would work to restore confidence, orwould be expected to work. It worked because of secrecy. Bagehot did not mentionsecrecy because “. . . a key feature of the British [banking] system, its in-built protec-tive device for anonymity was overlooked [by Bagehot]” (Capie (2007), p. 313). Capieexplains that in England geographically between the country banks and the Bank ofEngland was a ring of discount houses. Also, see Capie (2002). If a country bankneeded money during a crisis it could borrow from its discount house, which in turnmight borrow from the Bank of England. In this way, the identities of the actual endborrowers was not publicly known.12

While the identities of discount window borrowers were not publicly known, theBank of England knew those identities and, in particular, the identities of the non-bank borrowers in the crisis of 1866, the Overend-Gurney Panic. See Flandreau andUgolini (2011)) on the importance of non-bank borrowers (from the shadow bankingsystem). They argue that access to the Bank of England’s discount facility meant thatthese non-bank borrowers faced increased monitoring by the Central Bank.

5 Conclusions

A financial crisis occurs when some public information causes depositors to worryabout the collateral backing their deposits such that they want to produce informationabout the bank’s portfolio. This is a bank run. Secret public lending during a financialcrisis is important to avoid runs and asset examination. Bernanke (2009): “Releasingthe names of [the borrowing] institutions in real-time, in the midst of the financialcrisis, would have undermined the effectiveness of the emergency lending and theconfidence of investors and borrowers ” (p. 1). Recreating confidence means raisingthe perceived average value of collateral in the economy so that it is not profitable toproduce information about banks. The government can achieve this by exchangingbonds (or cash) for lower quality assets. Interestingly, there is no need to replace all

12King (1936) provides more discussion on the industrial organization of British banking in the 19th

century. Also see Pressnell (1956). The Bank of England did not always get along with the discounthouses, and there is a complicated history to their interaction. See, e.g., Flandreau and Ugolini (2011).

37

the bad assets in the economy to discourage information acquisition. There is an in-formational externality in the use of opacity by pooling assets. Opacity was adoptednot only by governments to deal with crises but also by private bank clearinghousesin the U.S. prior to the Federal Reserve System. Clearinghouse banks pooled theirassets so that deposits were claims on all the assets not just one bank’s assets.

The Central Bank can choose the haircut optimally to determine the optimal amountof bond collateral that is put into the economy. The ability to adopt a policy of opacitydepends on the threat of stigma, to realign its opacity incentives with those of thebanks. Stigma is costly for borrowers, as their participation reveals their holding ofworse assets in expectation. Opacity does not try to avoid stigma but stigma is crucialto avoid transparency. Stigma is not observed in equilibrium, not because opacity butbecause participation on lending facilities imply paying a discount rate.

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How Central Banks End Crises

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