How Should Governments Create Liquidity?

Timothy Jackson1, George Pennacchi2

November 1, 2019

Abstract

Governments can create safe, liquid assets by issuing government debt or by insuring privatedebt, such as bank deposits. Yet this public liquidity creation is limited by the government’scapacity to raise taxes to pay its liabilities. This paper analyzes how different methods ofpublic liquidity creation affect an economy’s lending and its private liquidity. It compares abanking system with government deposit insurance to banking systems with both uninsured“broad” banks and “narrow” banks which invest only in government debt. We find that asystem with deposit insurance maximizes bank lending but leads to less efficient monitoringof borrowers. The alternative system with narrow and broad banks produces the sameamount of government liquidity but more private liquidity.

Keywords: Deposit insurance, LiquidityJEL No. E42, E44, E51, G21

∗We are grateful for valuable comments from Sebastian Di Tella, Sebastian Infante, Jing Zeng, and theparticipants of the 2018 University of Bristol Workshop on Banking and Financial Intermediation, the 2019University of South Carolina Conference on Fixed Income and Financial Institutions, the 2019 WhartonLiquidity and Financial Fragility Conference, and seminars at the University of Illinois, the Bank of Canada,and the Bank of Finland. Tim thanks the Economic and Social Research Council.

1Manchester Metropolitan University; timjackson100@hotmail.co.uk2Department of Finance, University of Illinois; gpennacc@illinois.edu

1. Introduction

This paper analyzes how the manner in which a government creates safe assets affects thestructure and performance of a country’s banking system. A government can create safe(default-free) assets by insuring bank deposits or by directly issuing government debt, suchas Treasury bills. Yet it can do so only to a limited degree. A government’s safe assetcreation is constrained by the amount of taxes that it can raise to pay its liabilities.

We compare a system where the government provides limited deposit insurance to banksthat make risky loans versus a system where the government requires that ‘narrow’ banksinvest only in the government’s directly issued debt and uninsured ‘broad’ banks makerisky loans. In both systems, safe assets may be created by the government but ‘quasi-safe’assets can also be created privately. The distinction between government safe assets andprivate quasi-safe assets is that the former are fully default-free while the latter are default-free except during a severe financial crisis or ‘catastrophe.’ Individuals are willing to pay‘liquidity’ premiums to invest in safe assets, where the premium is greater for fully-safeassets compared to quasi-safe assets. Because individuals especially value liquidity, theyaccept lower rates of return on safe assets relative to the certainty-equivalent return onrisky assets.

Our model assumes that banks can improve the returns on their risky loans by costlymonitoring of their borrowers. However, monitoring is not contractible so that the bank’sowner-manager must be given incentives to monitor efficiently. Because the bank owner haslimited liability, the bank’s choice of leverage (deposit-to-equity capital ratio) affects theincentive to monitor. A bank that limits its leverage can signal its incentive to efficientlymonitor, which can raise the bank’s firm value and potentially lower its cost of depositfunding. Because monitoring costs are assumed to vary across banks, in general somelower-cost banks may limit leverage and efficiently monitor while other higher-cost bankschoose high leverage and do not monitor. Our model also permits a bank to issue depositsthat differ by their seniority, a process referred to as ‘tranching.’ For some banks, issuingboth senior and junior deposits can be profitable because the former can be made quasi-safe.

We use the model to investigate three main regimes: a baseline, fully-private bankingsystem with no government safe assets; a scheme of fairly-priced government deposit in-surance; and a system combining narrow banks that invest only in government debt anduninsured broad banks that make risky loans. We assume the tax base available to finance agovernment’s safe assets is identical in the latter two regimes to allow for a fair comparison.

We show that the creation of safe assets that derives from the government’s uniqueability to impose future taxes can potentially improve welfare by providing more higher-quality liquidity. However, the way that a government creates liquidity has implications interms of the total amount of bank lending, the monitoring efficiency of bank lending, and

2Compared to deposit-insured banks, uninsured broad banks might be considered a type of ‘shadow’bank.

1

the amount of private liquidity creation.

Government deposit insurance is profitable for banks because deposits that are com-pletely safe, even during a catastrophe, are especially valued by savers and require thelowest risk-adjusted rate of return. However, sufficiently high levels of deposit insurance re-duce the amount of efficient monitoring in the banking system and can crowd out quasi-safedeposits.

In a system with narrow banks and uninsured broad banks, narrow banks that takedeposits and invest in government debt may, in some structures, reduce the deposits availableto broad banks that make loans. As a consequence, banking system lending can decline.However, because fewer deposits limit the leverage of broad banks, the lending that doesoccur tends to be done with more efficient monitoring and possibly greater production ofquasi-safe deposits. However, under a different narrow bank - broad bank structure, narrowbanks would not reduce deposits available to broad banks and would be equivalent to asystem of fully-private lending banks but with the benefit of additional government safeassets.

Our paper contributes to a literature on the private and public provision of safe as-sets.3 Prior research, including Gorton and Pennacchi (1990) and Dang et al. (2017), notesthat safe assets are especially valuable for making transactions due to their information-insensitivity. The ‘money-like’ feature of safe assets allows them to pay a lower rate of returncompared to riskier, less-liquid assets. Such a liquidity premium is empirically documentedby several studies including Krishnamurthy and Vissing-Jorgensen (2012), Sunderam (2015),Nagel (2016), and Kacperczyk et al. (2018).

Research shows that safe assets can be created privately via financial institutions suchas banks (e.g., Diamond (2017) and Ahnert and Perotti (2018)) or special purpose vehiclesthat invest in risky debt and issue tranched securities (e.g., DeMarzo and Duffie (1999) andDeMarzo (2005)). However, safe assets can also be created by governments in the form ofsovereign debt or by insuring privately-issued debt, such as bank deposits (e.g., Greenwoodet al. (2015), He et al. (2018), Gatev and Strahan (2006), and Pennacchi (2006)).

Most papers that analyze the co-existence of private and public safe assets concentrateon issues relating to financial stability. Research by Holmstrom and Tirole (1998), Boltonet al. (2009), and Stein (2012) present models where the provision of government safe, liquidassets can improve financial system stability relative to an economy with only private liquidassets. Yet in some models, such as Brunnermeier and Niepelt (2019), substituting public forprivate liquidity has no effect on an economy. Like us, Ahnert and Perotti (2018) examinehow government liquidity creation via directly issued debt or deposit insurance affects theprovision of private liquidity. However, their model differs because, in equilibrium, bothprivate and public liquidity are fully safe and, therefore, are perfect substitutes. They focuson deriving the equilibrium return on safe assets.

3See Gorton (2017) for an in-depth review of this literature.

2

Our paper analyzes the interaction between public and private safe assets, but our focusrelates to issues of lending efficiency and the aggregate volume of safe assets. As in Diamond(1984), banks in our model can create value by making loans and providing costly monitoringof borrowers. We study how the form of government safe assets affects banks’ monitoringefficiency and their private liquidity creation. Our paper’s main contribution is to showthat public safe assets in the form of government-insured deposits have different effects onlending efficiency and total liquidity creation compared to public safe assets in the form ofdirectly-issued government debt.

While government insurance of bank deposits is currently the dominant way that govern-ment liquidity is provided to banks, there is a long history of proposals to provide governmentliquidity in the form of narrow banks.4 More recently, proposals to create a central bankdigital currency (CBDC) envision it as a narrow bank with government debt as assets andinterest-bearing digital deposits as liabilities.5 Our model provides insights on the potentialbenefits and costs of transitioning to narrow banking systems, whether a narrow bank’sdeposits take a traditional or digital form.

The next section introduces our basic model and considers a fully-private banking sys-tem with no role for government. Section 3 examines a banking system with governmentdeposit insurance that is limited by the government’s capacity to tax agents’ future wagesin order pay insurance losses. Section 4 considers a banking system where the governmentdirectly issues debt that is held by narrow banks that operate like ‘Treasury-only’ moneymarket mutual funds. In this system, uninsured broad banks make loans. As with depositinsurance, the amount of government debt that can be issued is limited by the govern-ment’s future taxing capacity. Section 5 considers a hybrid system where narrow banks andgovernment-insured broad banks co-exist. Section 6 provides numerical illustrations of themodel’s results, and Section 7 briefly discusses the robustness of the model’s assumptions.Conclusions are given in Section 8.

2. Liquidity Creation in a Fully-Private Banking System

This section presents our basic model of a private banking system that has no role for agovernment to create safe assets. Private banks can create only quasi-safe assets, which aredefault-free except in a financial catastrophe. The following sections will consider how agovernment can create fully default-free assets due to its ability to tax individuals’ futureendowments.

Consider a single-period economy with risk-neutral agents located in a continuum ofseparate local markets who obtain utility from their end-of-period consumption. Agentsreceive initial endowments that can be transformed into end-of-period consumption using

4Pennacchi (2012) reviews the history and theory of narrow banking proposals.5See, for example, Bank for International Settlements (2018), Norges Bank (2018), and Brunnermeier

and Niepelt (2019)

3

two types of investment technologies. One is a risky investment technology that is availableto all agents. The other is a superior risky investment technology that can only be accessedthrough lending intermediaries, which we call ‘banks.’

There are two types of agents in each local market: agents who are capable of owningand managing banks and other agents who wish to save and value liquidity derived frominvesting in safe assets. We will refer to the former agents as ‘bankers’ and the latter agentsas ‘savers.’ Each banker has a fixed beginning-of-period endowment of inside equity equalto k. The bank can raise additional funds from savers in its local market in the form ofstandard, non-state-contingent debt contracts, i.e., deposits.6 We normalize the maximumamount of available local savings to 1 and the amount of total deposits actually issuedby the bank is denoted by γ ≤ 1. Therefore, a bank’s beginning-of-period assets equals(γ + k) ≤ (1 + k).7

Banks are special due to their superior lending technology that funds identical projectsin perfectly elastic supply. All projects, and therefore loans, are subject to only aggregate(macroeconomic) risk.8 We consider three end-of-period states of the world: ‘good,’ ‘bad,’and ‘catastrophe.’ The good state occurs with probability pg, in which case each loan’s end-of-period return per unit lent equals its promised return of RL. The bad state occurs withprobability pb, in which case each loan defaults but has a positive recovery value. Finally,the catastrophe state occurs with probability pc = 1−pg−pb, in which case the loan defaultsand has a zero recovery value.

The banker is able to improve each loan’s recovery value in the bad state by exertingbeginning-of-period effort to monitor the borrower.9 This recovery value is denoted as d(a),where a is a banker’s beginning-of-period level of effort per unit of loan. Recovery value perunit of loan is assumed to be the following increasing and concave function of banker effort:

d(a) = RL

(1− αe−βa

)(1)

where 0 < α < 1 and β > 0. Monitoring effort is assumed to be costly in terms ofdiminishing the banker’s utility at a fixed marginal cost per unit of effort. Denote bankeri’s marginal cost of effort by ci. To generate an inelastic supply of private liquidity and

6Savers can invest only in a bank’s deposits and not its equity, though later we discuss allowing banks toissue outside equity. See footnote 15. In richer models where savers cannot verify the return on the bank’sassets or have needs to trade, they may prefer bank deposits relative to bank equity as in Diamond (1984),Townsend (1979), and Gorton and Pennacchi (1990). We also rule out inter-market lending, which maybe justified by information asymmetries across markets. As will be discussed, we assume that savers knowtheir local bank’s leverage and monitoring cost. It is plausible that such information would be unknown tonon-local agents, creating a deterrence to inter-market loans.

7As will become clear, a banker has the incentive to invest the entire amount of capital, k, in the bankbecause of its access to a superior investment technology.

8We focus on macroeconomic risk because, in general, idiosyncratic risks might be diversified awaythrough pooling as in Diamond (1984).

9We refer to this effort as monitoring, but it could also be interpreted as credit screening to determinewhich loan applicants have higher recovery value.

4

efficiently-monitored loans, we assume the economy has a continuum of local bankers whodiffer in terms of their cost of monitoring, where ci belongs to a continuous distributionhaving the range:

ci ∈ [pbβ(RL − 1), pbαβRL] ≡ [c, c]. (2)

As will be shown, this restriction on the range of monitoring costs ensures that each bank’sfirst-best effort is positive but still results in loans having a positive loss given default.Importantly, it is also assumed that each bank’s effort level, a, is not directly observable tosavers and cannot be contracted upon. This non-contractible level of effort is a potentialsource of moral hazard and inefficiency.

Savers have an alternative to depositing funds in their local bank. They have directaccess to a risky investment technology that funds projects in perfectly elastic supply andpays a return per unit investment of RR/pg only in the good state. The expected returnper unit investment in this non-intermediated investment technology, RR, is assumed to beless than RL.10

Savers are risk neutral but have an additional ‘liquidity’ demand for ‘quasi-safe’ assets.They will accept the expected return of RC < RR on an investment that is default-freein the good and bad states but not the catastrophe state. Such an investment does notdefault with probability ℘ ≡ pg + pb and defaults with probability pc = 1 − ℘, and wedefine its liquidity premium as l where RC(1 + l) = RR. This liquidity premium can beconsidered a utility bonus due to a safe asset’s value in settling transactions and for useas collateral.11 As in Stein (2012), we assume that savers’ required return on quasi-safeassets is independent of their supply. Examples of these investments might include moneymarket instruments such as A1/P1-rated commercial paper and wholesale, uninsured bankcertificates of deposit.12 Later we consider government-created assets that are default-freein all states for which savers require a return that is even lower than RC .

Because loans return zero in the catastrophe state, the best that a private bank cando is to create quasi-safe deposits. Doing so allows it to reduce its cost of funding by theliquidity premium l. The bank can augment its quasi-safe deposits in two ways. One way isby increasing the recovery value of its loans in the bad default state by efficiently monitoringits borrowers. However, for depositors to find this credible, the bank must have an incentiveto undertake this unobserved action. The bank can create this incentive by restricting itsleverage so that bank equity receives the marginal benefit from its costly monitoring effort.

The second way that a bank might increase its quasi-safe deposits is by issuing deposits

10These projects may be the same types of projects that banks fund with loans. However, savers are notable to monitor borrowers and obtain inferior expected returns relative to those received by banks.

11Gorton and Pennacchi (1990) provide a theory for why safe assets are particularly valuable for trans-actions. Several recent models, such as Stein (2012), assume that the moneyness feature of safe short-termgives it a lower required return than the certainty-equivalent return on risky assets.

12This asset class might be considered a ‘near-money’ or what Moreira and Savov (2014) refer to as‘shadow money.’

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that differ by their seniority over the recovery value of the bank’s assets. In other words,the bank can ‘tranche’ its debt by issuing both senior deposits and junior deposits (or sub-ordinated debt). Designed appropriately, the senior deposits can be made quasi-safe andsupported by the additional assets that are funded by junior deposits. We defer consider-ation of tranching until the next section. For now, we assume that the bank issues only asingle class of deposits of the amount γ.

Bankers are the only equity investors (shareholders) of their bank and have limitedliability. Since they have access to the bank’s superior investment technology, they willalways choose to invest their entire endowment, k, as capital in their bank. However,bankers’ profit-maximizing choices of leverage and effort may result in savers’ deposits beingeither quasi-safe or default-risky. We now consider the possible equilibrium behavior ofbankers and savers.

An equilibrium is defined as follows. First, a bank(er) announces that it will raise γ ≤ 1in deposits, so that its total assets equals γ + k. Second, given this choice of leverage(deposits and total assets), the bank’s promised return on deposits, RD, is set. Third, giventhis deposit rate, the bank chooses its unobserved effort level, a. An equilibrium is a choiceof γ and a that maximizes the bank’s profits and a promised deposit rate RD that satisfiesdepositors’ participation constraint given the bank’s announced γ and its equilibrium profit-maximizing choice of a.

Since bank i has monitoring cost ci, its profit maximization problem can be written as:

maxγ,a

pg [(γ + k)RL − γRD] + pb max [(γ + k)d(a)− γRD, 0]− cia(γ + k) (3)

subject to the constraint that its deposits cannot exceed 1:

γ ≤ 1 (4)

and subject to depositors’ participation constraint:

RD ≥

{RR−pb γ+kγ d(a∗)

pgif (γ + k)d(a∗) < γRC/℘

RC/℘ otherwise .(5)

Note that the expected profits given in (3) reflect the possibility of default in the bad statebut the certainty of default in the catastrophe state due to loans’ zero recovery value inthat state. Also, the depositors’ participation constraint (5) reflects either default in thebad state (the first line on the right-hand side) or no default in the bad state (the secondline on the right-hand side). In the former case, depositors’ required expected return is RR,but in the latter case it is RC since deposits are quasi-safe.

The solution to the problem is found by noting that whenever default occurs in the badstate, i.e. (γ+k)d(a∗) < γRC/℘, then from the objective function (3) we see that the bankerreceives no benefit from monitoring and the bank’s private effort choice is a∗ = al ≡ 0.

6

Deposits’ equilibrium promised return must then be RD =RR−pb γ+kγ d(al)

pg=

RR−pb γ+kγ RL(1−α)

pg

and the bank’s expected profit is

πl = pg[(γ + k)RL − γRD] = (γ + k)[pgRL + pbRL(1− α)]− γRR. (6)

Instead, whenever (γ + k)d(a∗) > γRC/℘ so that the banker obtains a return in the defaultstate, then the choice of effort is either the same corner solution al = 0 or the effort levelimplied by the first-order condition:

∂d(a)

∂a=cipb. (7)

By substituting in the functional form for d(a) from equation (1), the effort satisfying thisfirst order condition is a∗ = ah where

ah ≡ 1

βln

(βαpbRL

ci

)(8)

which results in the loan’s bad state recovery value equaling

d(ah) = RL −ciβpb

. (9)

In this case deposits are quasi safe, RD = RC/℘, and the high-effort bank’s expected profitis

πh = (γ + k)[pgRL + pbd(ah)− ciah

]− γRC (10)

where pbd(ah)− ciah > pbRL(1− α) is the expected recovery value net of monitoring costs.Thus, al = 0 or ah given by equation (8) are the only choices of effort that could possibly beprofit-maximizing for the bank. The following lemma states that a bank’s profit-maximizingchoice of effort, and its equilibrium deposit interest rate, depends on whether its initial choiceof leverage is below or above a particular threshold value which we denote as γm.

Lemma 1. If bank i chooses initial deposits less than or equal to γm(ci) ≡ k pbd(ah)−ciahpbRC/℘−(pbd(ah)−ciah)

,

then the equilibrium is one where the bank supplies first-best effort (8), RD = RC/℘, and hasprofits equal to equation (10). If it chooses initial deposits exceeding γm(ci), the equilibrium

is characterized by no bank effort, RD =RR−pb γ+kγ RL(1−α)

pg, and profits equal to equation (6).

The proof is given in Appendix A. Thus, the bank’s profit function is given by equations(6) and (10) where the switch point occurs at γ = γm. The bank’s profit maximizing choiceof γ is then the maximum of this profit function. As both profit functions, (6) and (10),are linear and increasing in γ, these maxima are at the endpoints, γ = γm and γ = 1

7

respectively.13 Bank i simply compares profits at the two levels

πl =(1 + k) [pgRL + pbd(0)]−RR (11)

πh =(γm + k)[pgRL + pbd(ah)− ciah

]− γmRC (12)

and chooses the (γ, a) combination (γm, ah) if πl ≤ πh and otherwise chooses (1, 0).

Now note from (11) and (12) that πl does not depend on the banker’s cost of effort, sinceno effort is expended. In contrast, πh is monotonically decreasing in the banker’s cost, ci.

14

Assuming thatπh(c) < πl < πh(c), (13)

then there exists a unique threshold value c∗ such that πl = πh(c∗). It satisfies the implicitequation:

c∗ =πlpb(RL −RC/℘)(

1β

+ ah(c∗))

(πl − pgkRC/℘)(14)

where ah(c∗) = 1β

ln(βαpbRL

c∗

). This logic leads to the following proposition whose proof is

in Appendix A.

Proposition 1. If bank i’s cost of monitoring is ci < c∗, it chooses leverage equal toγm(ci), provides first best effort of ah(ci), sets RD = RC/℘, and has profits equal toequation (12). Instead, if its cost is ci > c∗, it chooses γ = 1, provides no effort, sets

RD = RR−pb(1+k)RL(1−α)pg

, and has profits equal to equation (11).

Thus, only low monitoring cost banks, defined as having ci < c∗, limit leverage, providefirst-best effort, and create quasi-safe deposits.

Our results to this point assume that a bank issues only one class of deposits. The nextsection considers whether a bank might choose to issue both senior and junior deposits.Importantly, we show that this process of tranching deposits allows even high-cost, no-effortbanks to issue quasi-safe deposits.

2.1. Tranching

We continue to assume that a bank issues standard deposit contracts to savers but nowallow these deposits to differ in terms of their seniority. Thus, consider a bank that offersboth senior deposits and junior deposits (or subordinated debt). Suppose that the bankrestricts the amount of its senior deposits, γs, such that it has sufficient loan recovery valuein the bad default state to pay off senior depositors in full. Intuitively, the bank may have an

13We rule out parameter values for which profit is decreasing in γ since that implies that banks chooseto issue no deposits.

14See the proof in Appendix A.

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incentive to do so because it ensures that senior deposits are quasi-safe and their promisedreturn is relatively low at RC/℘. In addition, let the amount of junior deposits be γj andtheir promised return be RD,j. For a given effort level, a, the amount of senior, quasi-safedebt that the bank could issue, γs, satisfies:

(γs + γj + k)d(a) ≥ γsRC/℘ (15)

or, equivalently, senior leverage must be below a maximum, γs, which is increasing in effort,a, and other forms of bank funding, γj + k,

γs ≤ γs ≡ (γj + k)d(a)

RC/℘− d(a). (16)

The bank’s profit maximization problem is now characterized as:

maxγj ,γs,a

pg[(γs + γj + k)RL − γsRC/℘− γjRD,j

](17)

+pb max[(γs + γj + k)d(a)− γsRC/℘− γjRD,j, 0

]− cia(γs + γj + k)

subject to the constraint (15) that senior deposits are quasi-safe and subject to juniordepositors’ participation constraint:

RD,j ≥

RR−

pbγj

[(γs+γj+k)d(a∗)−γsRC/℘]pg

if (γs + γj + k)d(a∗)− γsRC/℘ < γjRC/℘

RC/℘ otherwise(18)

To solve this problem, let us start by considering the bank’s incentive to exert effort. Notethat bank equity benefits from effort only if it receives the marginal profit from effort in thebad default state. That can only occur when

(γs + γj + k)d(a)− γsRC/℘− γjRD,j > 0. (19)

But if inequality (19) holds, then junior deposits are also quasi-safe, can be paid a depositinterest rate of RD = RC/℘, and are no different from senior deposits. Consequently, abank that monitors at first best, a∗ = ah, cannot benefit from issuing more than one classof (senior) deposits. Thus, if a bank has a relatively low monitoring cost so that its profit-maximizing effort is first-best, then its effort, leverage, and profits are {ah, γm, πh}, the sameas when it does not tranche.

Instead, consider a bank that has a relatively high monitoring cost such that its profit-maximizing effort is zero and inequality (19) does not hold. Without tranching, Proposition1 shows that this bank would choose maximum leverage, γ = 1, and all deposits sufferlosses in the bad default state. Consequently, the bank’s per unit expected cost of depositfunding is RR. Tranching now allows the bank to reduce part of its deposit funding costs.Since quasi-safe senior deposits have the lower per unit expected cost of RC < RR while theexpected cost of junior deposits is unchanged at RR, the bank has the incentive to issue themaximum amount of quasi-safe senior deposits, γs = γs. As a result, junior deposits receive

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nothing in the bad state so that their promised return is RD,j = RRpg

.15

Since a bank that chooses no effort, al = 0, profits by maximizing total leverage, γ = 1,even when it does not tranche its deposits, its profit-maximizing amount of junior deposits isγj = 1− γs. Thus, tranching does not affect this bank’s equilibrium effort or total leverage,but allows it to issue quasi-safe deposits of

γs =(1− γs + k)RL(1− α)

RC/℘−RL(1− α)=

(1 + k)RL(1− α)

RC/℘. (20)

Doing so yields increased profits as a result of the reduced funding cost of senior deposits.Denoting the no-effort bank’s profits under tranching as πlT , we have

πlT =pg

[(1 + k)RL − γs

RC

℘− (1− γs)RR

pg

]=(1 + k) [pgRL + pbRL(1− α)]− γsRC − (1− γs)RR

=(1 + k) [pgRL + (pb + ℘l)RL(1− α)]−RR, (21)

where recall that the liquidity premium, l, is defined by RR = (1 + l)RC . Compared toprofits without tranching in equation (11), we see that tranching raises the no-effort bank’sprofits by the liquidity premium that it saves on its quasi-safe senior deposits:

πlT − πl =℘l(1 + k)RL(1− α) = lγsRC ≥ 0.

This increase in no-effort profits alters the monitoring cost threshold above which bankschoose no effort. By equating πlT in equation (21) to πh in equation (12), the critical valueof ci is now:

cT =πlTpb(RL −RC/℘)(

1β

+ ah(cT ))

(πlT − pgkRC/℘). (22)

Since πlT > πl and pgkRC/℘ > 0, we see that cT < c∗. Intuitively, tranching deposits makeschoosing no effort relatively more profitable than choosing high effort. Hence, there is alower cost threshold at which banks choose no effort. We summarize these results in thefollowing proposition:

Proposition 2. If bank i’s cost of monitoring is ci < cT , it does not benefit from tranchingand its leverage, effort, deposit rate, and profits are the same as without tranching. Instead,if its cost is ci > cT , it exerts no effort and issues senior deposits of γs and junior depositsof 1− γs where γs is given by equation (20). The deposit rates of senior and junior depositsare RC/℘ and RR/pg, respectively, and the bank’s profits are given by equation (21). Since,

15The fact that in equilibrium junior deposits receive nothing in the bad state makes their payoff similarthat of the banker’s inside equity, k. In this sense, junior deposits might be considered similar to outsideequity. However, the effort incentive of the banker depends his/her inside equity, so issuing more juniordeposits is not equivalent to the banker increasing inside equity.

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cT < c∗, more banks choose no effort when tranching is possible than when it is not.

It is interesting to note that tranching allows creation of quasi-safe deposits by high-cost, no-effort banks, and furthermore the amount that they create can exceed that of thelow-cost, high-effort banks. That is, there are parameter values such that γs > γm(ci) for arange of ci < cT . It may seem ironic that high-cost banks, which do not monitor and haveriskier loan portfolios, are able to create more quasi-safe deposits. Of course they do so bya form of financial engineering that entails issuing different seniority deposits that avoidsthe need to signal the incentive to monitor by having low total leverage. The next sectionconsiders the aggregate amount of quasi-safe deposits produced by a private, uninsuredbanking industry.

2.2. Quasi-Safe Deposit Production

The provision of quasi-safe liquidity may be considered a key measure of banking systemperformance.16 Let f(ci) be the economy’s density of bankers with monitoring cost ci.Then assuming banks are permitted to tranche deposits, the aggregate quantity of quasi-safe deposits produced by banks that choose first-best effort is∫ cT

c

γm(ci)f(ci)dci, (23)

where each bank that exerts first best effort issues quasi-safe deposits equal to γm(ci) =

kpbd(ah(ci))−ciah(ci)

pbRC/℘−[pbd(ah(ci))−ciah(ci)]and where cT is given by equation (22).

Each no-effort bank issues quasi-safe, senior deposits equal to γs given by equation (20).Thus, denoting total quasi-safe deposit production under tranching as ScT , it satisfies

ScT =

∫ cT

c

γm(ci)f(ci)dci + γs∫ c

cTf(ci)dci. (24)

Assuming a uniform density for monitoring costs, equation (24) simplifies to

ScT =1

c− c

[∫ cT

c

γm(ci)dci + γs(c− cT )

]. (25)

Since quasi-safe deposits are vulnerable to losses in the catastrophe state, there is thepotential for a government with taxing authority to improve welfare by producing assetsthat are safe in all future states, including severe crises. The next section considers thispossibility by way of government deposit insurance.

16Berger and Bouwman (2009) provide several measures of liquidity creation by U.S. banks.

11

3. Liquidity Creation with Government Deposit Insurance

As a prelude to analyzing deposit insurance, we discuss why a government’s ability to taxsources of income that may not be available to private creditors can give it an advantage inproducing safe assets.

3.1. Taxes as a Source of Public Liquidity

A government might raise revenue from sources of income that may be difficult to pledgeunder private contracts. Future revenue from human capital, i.e., wage income, may be anexample. Payment from an individual’s future wages may be difficult to collect under privatecontracts due to bankruptcy laws that limit protection of private creditors. In contrast, agovernment typically is in a stronger position to collect taxes that are owed.

We model a government’s taxing authority by assuming that savers receive riskless, end-of-period wage income equal to ω per unit of beginning-of-period savings.17 These wagesare not pledgeable under private contracts, but a government is able to tax a proportiont ≤ 1.18 This taxable proportion might be determined by moral hazard considerations: ifthe tax rate is too high, individuals may have an incentive to evade taxes.19

The government’s taxing capacity gives it the ability to create assets that are default-free, even in the catastrophe state. Examples might include Treasury debt, such as Treasurybills, and government-insured debt or deposits. These assets’ moneyness and liquidity areassumed to be particularly attractive to savers such that perfectly safe assets’ requiredreturn equals RF where RF < RC < RR. Written in terms of liquidity premia, we haveRR = (1 + l)RC = (1 + l)(1 + lf )RF . Thus, increasing safety is associated with higherliquidity premia that reduce the returns required by savers.

3.2. Deposit Insurance

Suppose that the government offers fairly-priced deposit insurance to private banks thatpermits insured deposits to be default-free in all end-of-period states, including both the‘bad’ and ‘catastrophe’ states when loans default. Since deposit insurance is backed bythe government’s power to tax savers’ end-of-period wages, the amount of these taxes is

17Our results are not sensitive to the assumption that wage income is riskless. What matters is that thereis some strictly positive minimum level of wages that can be taxed even in the catastrophe state.

18If wages could be pledged under private contracts, individuals could underwrite insurance against abank’s default on its deposits; that is, savers could provide credit protection. Or, alternatively, individualscould issue riskless debt at the beginning of the period backed by their future wages. Both actions couldincrease the amount of safe assets.

19Alternatively one might assume a direct cost of raising taxes that is increasing and convex in the taxrate. Such a cost would capture the social costs of an inflation tax. For simplicity, our model makes nodistinction between real and nominal quantities, making it most applicable to a zero inflation economy.

12

state-contingent. Yet for deposit insurance to be fully credible, it must insure deposits incase of catastrophe when each bank’s assets are worthless. This worst-state scenario limitsinsured deposits to equal the government’s maximum revenue that can be raised in taxesat the end of the period.

We assume that the insurer limits its maximum liability by restricting the quantity ofdeposits that it will insure at each bank. Specifically, insurance is limited to small ‘retail’deposits that are assumed to equal a proportion γr of total savings in each banking market.Consequently, the maximum promised end-of-period payment to insured depositors is γrRF

for each bank.

Because our intent is to study the potential amount of safe assets that a government cancreate with deposit insurance, we do not consider the well-known distortions due to insurancemis-pricing.20 Rather, we assume the government assesses a fair insurance premium payableby the bank at the end of the period, where φ is the premium per promised payment oninsured deposits.21 Specifically, if a bank chooses to issue the maximum amount of insureddeposits, then its total promised payment to insured depositors and the deposit insurer isγrRF (1 + φ). Because deposit insurance is fairly priced, the premium φ will vary acrossbanks based on their default risk.

Not all banks may choose to issue the maximum amount of insured deposits. A bank’sequilibrium choice can depend on its individual cost of monitoring, ci. We begin by taking asgiven the government deposit insurance limit of γr for each banking market and determiningan individual bank’s profit-maximizing choice of insured deposits. Then we aggregate overall banking markets to determine the maximum level of γr that can be supported by thegovernment’s taxing power.

3.2.1. High-Cost Banks

Recall that in the absence of deposit insurance, banks with relatively high monitoring costschoose maximum leverage (γ = 1), zero effort (al = 0), and tranche their deposits such thatthe promised payment on senior deposits equals the bank’s asset return in the bad state(γs given by equation (20)). In equilibrium, these banks’ expected cost of quasi-safe seniordeposits is RC and their expected cost of junior deposits is RR.

For the sake of both simplicity and realism, we assume that the maximum amount ofinsured deposits, γr, is such that γrRF ≥ γsRC/℘ = (1 + k)RL(1− α). In other words, the

20Extensive empirical evidence suggests deposit insurance is typically under-priced. See, for example,Pennacchi (2010). In principle, a government might over-price insurance in order to capture the liquiditypremium from its creation of fully-safe assets. Since governments do not appear to adopt such a practice,we leave this possibility for future research.

21Assuming a promised end-of-period payment makes the premium analogous to a credit spread on unin-sured debt. We assume the premium is distributed as a lump sum payment to savers in states whereaggregate premiums exceed aggregate insurance losses.

13

maximum promised payment on insured deposits exceeds the no-effort bank’s asset returnin the bad state. Thus, if this bank issues the maximum amount of insured deposits, ithas no incentive to issue senior deposits. Moreover, we will show that when the bank ischarged a fair deposit insurance premium, its expected cost of insured deposits is RF andits expected cost of uninsured deposits is RR. Since the bank’s insured deposits exceeds thesenior deposits for the uninsured case, its (uninsured) deposits that exceed the insurancelimit are less than junior deposits for the uninsured case. Therefore, the bank is strictlymore profitable under deposit insurance and, indeed, has an incentive to issue the maximumamount of insured deposits and total deposits.

To see this, consider the fair deposit insurance premium. We continue to use the su-perscript ‘l’ to denote the no-effort bank’s quantities. Now when deposit insurance is fairlypriced, the expected deposit insurance premium equals expected deposit insurance losses.In the ‘bad’ state, losses equal the shortfall between insured deposits owed per bank, γrRF ,and the recovery rate. As in actual practice, the insurer has the same seniority (bankruptcyclaimant status) as uninsured depositors. Thus the insurer only receives the proportion γr

of the recovery value, (γl + k)d(al) = (1 + k)RL(1 − α). In the catastrophe state, whichoccurs with probability pc = (1− pg− pb), no assets are recovered. The premium is set suchthat:

pgφlγrRF = pb[γ

rRF − γr(1 + k)RL(1− α)] + pcγrRF . (26)

Solving for the insurance premium, one can see that it is independent of the proportion ofinsured deposits:

φl =(1− pg)RF − pb(1 + k)RL(1− α)

pgRF

. (27)

As bank funding costs are linear in γr, banks will either insure up to the limit or notat all. However, since RF < RR, the funding cost of insured deposits is lower than thatof uninsured deposits. Defining the promised return on uninsured deposits as RD,u, it isstraightforward to show that

(1 + φl)RF ≤ RD,u (28)

where the uninsured deposit return is the same as under the no insurance, no tranchingcase:22

RD,u =RR − pb(1 + k)RL(1− α)

pg. (29)

The intuition is that even when a bank pays a fair premium that covers the deposit insurer’sexpected loss, there is still a benefit that accrues to the bank because depositors requirea lower interest rate when deposits are fully default-free and liquid. The no-effort bank’sprofit with deposit insurance, denoted as πlDI , equals

πlDI =pg{(1 + k)RL − γr(1 + φl)RF − (1− γr)RD,u} (30)

=(1 + k)[pgRL + pbRL(1− α)]− {γrRF + (1− γr)RR}

22See Proposition 1.

14

Comparing the bank’s profit relative to the case of uninsured, tranched deposits, we obtain:

πlDI − πlT =γr(RR −RF )− lγsRC

=l(γr − γs)RC + γrlfRF > 0. (31)

Insurance allows banks to reduce their funding costs and is increasing in the liquidity premia,l and lf , and the deposit insurance limit γr.

3.2.2. Least-Cost banks

There may be banks with the lowest monitoring costs that choose total leverage, γ, satisfyingγr ≤ γ ≤ γmDI,L, where γmDI,L(ci) is a maximum level of total leverage that gives these leastcost banks an incentive to exert high effort, ah(ci). In other words, these least-cost bankslimit their leverage, but this limited leverage is still above the deposit insurance limit. Thesebanks would not default on their insured or uninsured deposits in the bad state, so thatinsurance only needs to cover the catastrophe state. Consequently, these banks would paya reduced premium per insured deposit, φh, in both good and bad states that satisfies:

℘φhγrRF = (1− ℘)γrRF (32)

This premium is equal to the simple ratio of the probability that the catastrophe occurs tothe probability that it does not

φh =1− ℘℘

=pc

1− pc. (33)

In the absence of deposit insurance, these high-effort banks paid RD = RC/℘ on all depositsin both the good and bad states. They continue to pay that amount on their currentlyuninsured deposits that equal γ − γr. However, on their γr of insured deposits they pay(1 + φh)RF = RF/℘ in both the good and bad states. Hence, they obtain a per depositsavings of (RC −RF )/℘ on their insured deposits, which reflects the savings of the liquiditypremium lfRF/℘.

The lower promised payment on insured deposits allows these banks to increase theirtotal leverage beyond the amount γm(ci) which was their maximum in the absence of depositinsurance. Appendix A shows that their new maximum leverage under deposit insurance is

γmDI,L =pbγ

r(RC −RF )/℘+ k(pbd(ah)− ciah)pbRC/℘− [pbd(ah)− ciah]

,

=γm(ci) + γrpb [RC −RF ] /℘

pbRC/℘− [pbd(ah)− ciah]. (34)

With this higher leverage, the bank’s profits are now

πhDI,L = (γmDI,L + k)[pgRL + pbd(ah)− ciah]− γmDI,LRC + γr(RC −RF ). (35)

15

Comparing the profit in equation (35) to that of no insurance case, we obtain

πhDI,L − πh =(γmDI,L − γm)(pgRL + pbd(ah)− ciah −RC) + γr(RC −RF ) > 0 (36)

3.2.3. Moderate-Cost Banks

There may be banks with moderately-low monitoring costs that choose total leverage, γ,satisfying γ ≤ γmDI,M ≤ γr, where γmDI,M(ci) is the maximum total leverage that gives thesemoderately-cost banks an incentive to exert high effort, ah(ci). Hence, these banks maximizeprofits by limiting leverage to a level that is below the deposit insurance limit. Thereforethey issue only insured deposits so that the expected cost of all of their deposits falls fromRC to RF . The lower deposit rate and premium of RF/℘ compared to RC/℘ allows themto raise maximum leverage relative to the no deposit insurance case:

γmDI,M = kpbd(ah)− ciah

ciah + pb[RF/℘− d(ah)]> γm(ci). (37)

One can see that γmDI,M takes the exact same form as γm except that the smaller valueRF replaces RC in the denominator, making it larger than γm. Given this higher leverage,profits for these moderately-cost banks are

πhDI,M = (γmDI,M + k)[pgRL + pbd(ah)− ciah]− γmDI,MRF , (38)

which is, of course, greater than profits in the no-insurance case:

πhDI,M − πh =(γmDI,M − γm)[pgRL + pbd(ah)− ciah −RF ] + γmlfRF > 0 (39)

3.3. Cost Threshold for Effort under Deposit Insurance

The maximum leverage levels for moderate-cost and least-cost banks are defined by beingon either side of the insurance limit, γr; that is γmDI,M ≤ γr ≤ γmDI,L. Since, by definition,moderate-cost banks have higher screening costs than the least-cost banks, the cost thresholdbetween high effort and no effort is defined as cDI that sets the profits associated withmoderate-cost banks under insurance equal to that of insured no-effort banks:

πhDI,M(cDI) = πlDI(γr). (40)

We can now state the following proposition:

Proposition 3. If deposit insurance is sufficiently generous such that γr > γr∗, thencDI < cT ; that is, more banks make no effort to monitor compared to the case of no depositinsurance. Moreover, in this case total leverage and lending is greater when deposits areinsured.

Proof: See Appendix A which also gives the value of γr∗ in equation (82).

16

An implication of this proposition is that while greater deposit insurance coverage createsmore fully-safe assets, it comes at a cost of less-efficient monitoring by the banking industry.Note that since cT < c∗, sufficiently generous deposit insurance reduces bank effort relativeto the no deposit insurance case whether or not banks tranche their deposits.

The second result of the proposition, namely that total leverage and lending is greaterunder deposit insurance, follows from two of our prior results. First, we showed that forany given level of monitoring cost, high-effort least-cost banks and high-effort moderate-costbanks choose higher leverage (and enjoy higher profits) under deposit insurance. Second,since no-effort banks choose maximum leverage of γ = 1 and there are more no-effort banksunder deposit insurance when γr > γr∗, then leverage is always greater in equilibrium forany given bank’s level of monitoring cost. Consequently, the greater deposit cost savingfrom sufficiently generous deposit insurance expands total lending, even though a greaterproportion of banks lend inefficiently.

3.4. The Maximum Level of Deposit Insurance

If the catastrophe state occurs, banks’ assets equal zero and the deposit insurer’s liabilityequals the total of all insured deposits. Deposit insurance will be credible ex ante onlyif tax revenue can cover these catastrophic insurance claims. Recall that credibility isachieved by limiting insurance so that the maximum promised end-of-period payment toinsured depositors is γrRF for each bank. However, we know that moderate-cost banks donot insure up to the limit, and this fact needs to be taken into account when setting themaximum insurance level, γr, that in equilibrium is consistent with the deposit insurer’stotal liability being no greater than the government’s taxing capacity.

Define the maximum liability of the deposit insurer per unit of initial savings as L(γr).It must equal the government’s taxing capacity per initial savings:

L(γr) = tω. (41)

It is assumed that the tax base tω is large enough such the equilibrium value of γr impliesthat no-effort banks have no desire to tranche their deposits. In other words, as was assumedearlier γrRF ≥ γsRC/℘. Therefore, no-effort banks’ uninsured deposits will then equal1− γr.

Banks with the least and highest monitoring costs will issue the maximum insured de-posits, but banks with moderate monitoring costs will choose to restrict their leverage toγmDI,M < γr. Let cm be the cost threshold which sets γmDI,M = γr and distinguishes aleast-cost bank from a moderate-cost bank. The total liability of the insurer is

L(γr) =

∫ cm

c

γrRF · f(ci)dci +

∫ cDI

cmγmDI,M(ci)RF · f(ci)dci +

∫ c

cDIγrRF · f(ci)dci, (42)

17

or, if we assume a uniform distribution for ci ∈ (c, c),

L(γr) =1

c− c

[∫ cDI

cmγmDI,M(ci)dci + γr(cm − c + c− cDI)

]RF . (43)

Equating the right-hand sides of equations (42) and (41) and using equation (40) determinesthe equilibrium values of cDI and γr. Appendix B sets out the strategy for computing thesolution.

3.5. Aggregate Liquidity under Deposit Insurance

Government deposit insurance provides catastrophe-proof, safe deposits in quantity SfDI =L(γr)/RF = tω/RF . The quantity of ‘quasi-safe’ assets, ScDI , produced by least-cost banksis

ScDI =

∫ cm

c

(γmDI,L(ci)− γr)f(ci)dci, (44)

or with uniform costs,

ScDI =1

c− c

[∫ cm

c

γmDI,L(ci)dci − γr(cm − c)

]. (45)

Having analyzed deposit insurance, the next section considers an alternative means by whichgovernments can create safe assets.

4. Liquidity Creation with Government Debt and Narrow Banking

Rather than insuring deposits, suppose the government utilizes its taxing authority to offerdefault-free Treasury securities that pay the fully risk-free return per unit investment ofRF .23 Treasury securities could be sold directly to savers, but to enhance their liquidity itmight be more realistic to think that these securities are sold to financial institutions thatuse them to back deposit-like accounts. We refer to these institutions as ‘narrow banks.’Narrow banks are assumed to have a mutual fund structure, hold only Treasury securitiesas assets, and issue accounts that are proportional ownership interests in the securities. Inother words, they operate exactly like actual ‘Treasury-only’ money market mutual funds.

These narrow banks are assumed to be uniformly distributed across the economy’s bank-ing markets. Since, by law, they must hold only government securities, the maximum

23Our model assumes a homogeneous class of government debt while, in reality, government debt variesin maturity. Longer-maturity government debt may be subject to short-run interest rate risk. Infante(2017) analyzes how repurchase agreements backed by long-term Treasuries can substitute for short-termTreasuries, such as Treasury bills, to meet the demand for liquid, money-like assets.

18

amount of deposits that they can issue per unit savings is

γn =tω

RF

(46)

where the parametric condition tω < RF is assumed, so that γn < 1. We assume that eachnarrow bank in each market issues the maximum amount of deposit accounts, γn, becausesavers have a (slight) preference for completely-safe assets, even at the lower required returnof RF .

Since each market continues to have a banker with a superior lending technology, unin-sured ‘broad’ banks that make loans will operate in each market. We assume that thesebroad banks function similar to the private banks of our baseline model analyzed in Sec-tion 2. In particular, they issue deposits to savers that may be tranched. Importantly, theamount of deposits that these broad banks can raise from savers depends critically on howthe government uses its revenue from selling its Treasury securities to narrow banks.

One possibility is to assume that the revenue raised per unit savings in each market, γn,is invested by the government in the publicly-available risky investment technology whichreturns RR/pg per unit investment in the good state. When the good state occurs, thegovernment returns the amount γnRR/pg in a lump sum to savers at the end of the period.Under this assumption, the maximum amount of deposits per market that is available to fundbroad banks would be 1− γn. We refer to this assumption as the ‘Government Investment’assumption.

Another possibility is to assume that the government’s revenue from Treasury sales,γn, is instantly rebated in a lump sum to savers at the beginning of the period. Underthis assumption, the deposits issued by narrow banks have no net effect on each market’ssavings that can be tapped by broad banks. Consequently, the maximum amount of depositsavailable to broad banks is 1.24 We refer to this assumption as the ‘Government Rebate’assumption.

The next section considers the equilibrium under the Government Investment assump-tion. Following that, we discuss the equilibrium under the Government Rebate assumption.

4.1. Equilibrium with Narrow Banks and Government Investment

With a government investing its Treasury revenue in the risky technology, the maximumamount of savings that an uninsured broad bank can attract for deposits is γd ≡ 1− γn < 1rather than γ = 1. Thus, the effect of narrow banks taking market share is to constrain

24An equivalent assumption is that the government does not rebate the revenue to savers but offers todeposit it in broad banks. In markets where broad banks choose deposits of γ < 1, then the governmentinvests its residual revenue in the risky investment technology. Any deposit and investment returns receivedby the government at the end of the period is returned to savers in a lump sum.

19

the leverage and total size of broad banks. This market-induced leverage limit changes theincentives for expending effort by broad banks because profits in the no-effort case are less.With tranching permitted, the analysis is similar to that in Section 2.1. Only no effortbanks have an incentive to tranche their deposits. They issue ‘quasi-safe’ deposits at rate

RC/℘ up to the reduced limit γsNB = (γd+k)RL(1−α)RC/℘

. Their profits are

πlNB =pg[(γd + k)RL − γsNBRC/℘− (γd − γsNB)RR/pg

]=(γd + k) [pgRL + pbRL(1− α)]− γsNBRc − (γd − γsNB)RR

=(γd + k) [pgRL + (℘(1 + l)− pg)RL(1− α)]− γdRR < πlT . (47)

High effort banks limit leverage to the same level, γm(ci) = k pbd(ah)−ciahRC/℘−(pbd(ah)−ciah)

as before

unless γm(ci) > γd, in which case they choose γd.

4.1.1. Threshold for Effort under Government Investment

Define πh(ci) as the high effort profit of banker i given in equation (10) when γi =min[γm(ci), γ

d]. Then assuming πh(c) < πlNB, define cn as the critical value of c suchthat a bank’s profits are equal when it provides high versus no effort. Its profits satisfyπh(cn) = πlNB, which implies

cn =πlNBpb(RL −RC/℘)

( 1β

+ ah(cn)(πlNB − pgkRC/℘). (48)

Suppose that the model parameters are such that γm(cn) ≤ γd so that the bank which isindifferent between high effort and no effort limits its leverage under high effort to less thanγd. Since πlNB < πlT , then using the same logic as in Proposition 2 leads to cn > cT . In otherwords, when deposits are limited to γ < γd, a greater proportion of broad banks choose higheffort compared to the proportion choosing high effort when uninsured banks have accessto deposits of γ < 1.

4.1.2. Liquidity Provision under Government Investment

Fully safe deposits are now produced only by narrow banks. Since they are limited by thegovernment’s taxing authority, the maximum that can be produced is

SfNB = γn. (49)

Note that, in aggregate, the maximum fully-safe deposits that can be produced under narrowbanking is exactly the same as under deposit insurance. This is because under depositinsurance, the government needs to have enough taxes to pay off the entire amount ofinsured deposits in the catastrophe state. Similarly under narrow banking, the governmentrequires enough taxes to pay off narrow banks in all states. In both cases the government’stax capacity, tω, limits the maximum level of fully-safe deposits.

20

However, quasi-safe asset production differs under narrow banking and deposit insurance.Under narrow banking, one source of quasi-safe deposits comes from high-effort broad bankswhich provide ∫ cn

c

min [γm(ci), γd]f(ci)dci. (50)

The second source is the senior deposits of no-effort banks who add

γsNB

∫ c

cnf(ci)dci. (51)

Together, total quasi-safe deposits equal

ScNB =

∫ cn

c

min [γm(ci), γd]f(ci)dci + γsNB

∫ c

cnf(ci)dci. (52)

If a uniform distribution of costs is assumed, we have

ScNB =1

c− c

[∫ cn

c

min [γm(ci), γd]dci + γsNB(c− cn)

]. (53)

Recall that under deposit insurance only least cost banks produce quasi-safe deposits, givenby ScDI in equation (44). Moreover, under some parametric assumptions least cost banksdo not exist and only moderately-low cost banks exert first-best effort. Consequently, it ishighly likely that quasi-safe deposits are greater under narrow banking with GovernmentInvestment compared to deposit insurance. As a result, total liquidity creation is alsogreater.

4.2. Equilibrium with Narrow Banks and Government Rebate

If a government’s beginning-of-period proceeds from Treasury sales are instantly rebated tosavers, the constraint on deposits raised by broad banks is γ ≤ 1. Therefore, the equilibriumfor broad banks is exactly the same as the fully-private banking system analyzed in Section2. Thus, all of the results in that section apply to broad banks, but now the banking systemproduces the maximum amount of fully-safe deposits issued by narrow banks.

Comparing this narrow bank with Government Rebate system to the deposit insurancesystem analyzed in Section 3, we can immediately draw the following conclusions. First,the amount of fully-safe deposits are exactly the same under the two systems, since they arelimited by the same government constraint on tax capacity. Second, quasi-safe deposits areunambiguously greater under narrow banking relative to deposit insurance. This followsbecause under deposit insurance only high-effort, least-cost banks produce quasi-safe de-posits. In contrast, under narrow banking, quasi-safe deposits of broad banks comprise thetranched senior deposits of no-effort banks and the total deposits of all high-effort banks.Consequently, total public and private liquidity creation is greater under narrow bankingversus deposit insurance.

21

Third, when deposit insurance is sufficiently generous such that γr > γr∗, then a higherproportion of broad banks exert effort and lend efficiently compared to banks with depositinsurance. This result follows from Proposition 3 which compares deposit insurance to anuninsured banking system, which now applies to broad banks. A fourth conclusion alsofollows from Proposition 3: a system with sufficiently generous deposit insurance leads togreater leverage and total lending compared to a system of narrow and bank broad banks.

Why does a system of narrow banking with a government rebate produce the sameamount of fully-safe deposits as under a system of deposit insurance but far more quasi-safedeposits? The intuition is as follows. In terms of liquidity creation, a government’s powerto tax allows it to create assets that are fully default-free in all future states, even the catas-trophe state. An uninsured bank, on the other hand, can create quasi-safe assets that aredefault-free in all states except for the catastrophe state. By insuring this bank’s deposits,a government starts with a financial structure that is already capable of producing quasi-safe assets and adds safety in only one additional state, the catastrophe state. Hence, thegovernment’s special ability to create safety via taxation adds a small margin of additionalsafety to deposits that are already quasi-safe.

Thus, creation of fully-safe assets via deposit insurance is inefficient relative to narrowbanking with a government rebate. By issuing Treasuries purchased by narrow banks,a government creates fully default-free deposits from scratch. By not layering on alreadyexisting banks, the government does not extinguish the existing quasi-safe deposits producedby the private banking system. If there is value to quasi-safe deposits, the system withnarrow banking and a government rebate would be preferred along this dimension.

5. Hybrid Systems

5.1. Analysis of a Hybrid Model

A system of narrow banks with fully-uninsured broad banks may seem unrealistic if onebelieves that a minimum level of deposit insurance is required to prevent depositor runs onbanks that make loans. Hence, this section analyzes a system where narrow banks operatetogether with deposit-insured banks. Such a ‘Hybrid’ system is denoted with the subscript‘H’. Let narrow banks of size γnH < γn operate in each market while the government alsooffers a limited amount of deposit insurance, γrH , for each market’s loan-making ‘broad’banks.

For simplicity, assume that the government’s proceeds from issuing debt to narrow banksare rebated back to savers at the initial date. Thus, under this Government Rebate systemthe total amount of savings available to each insured broad bank remains at 1. Equatingthe government’s total liability in this hybrid regime to tax capacity yields the maximumlevel of deposit insurance that can be credibly offered to each bank, γrH , as the solution to

L(γrH) +RFγnH = tω, (54)

22

where L(·) captures the total liability of the insurer in the catastrophe state given in equation(42). Assuming the government’s tax capacity is unchanged, this hybrid system with narrowbanks must lead to less deposit insurance coverage relative to the system with only deposit-insured banks and no narrow banks; that is, γrH < γr.

Using a system with only government deposit-insured banks as the benchmark, considerthe effects of introducing narrow banks. Equation (54) implies that an increase in narrowbanking has the same effect as lowering tax capacity. In response to the reduction inmaximum insurance coverage, high-cost, no-effort banks will issue more uninsured depositsat the higher funding cost of RR, which leads to a reduction in their profits. If depositinsurance is sufficiently restricted, γrHRF < γsRC/℘ = (1 + k)RL(1 − α), no-effort bankswill also issue quasi-safe senior deposits as well as insured and junior deposits. These bankscontinue to lever up to the maximum limit of 1. In this case, the insurer does not lose in thebad state and charges a premium of φh = 1−℘

℘. These no-effort banks will receive expected

profits of

πlDI,T =pg{(1 + k)RL − γrH(1 + φh)RF − (γs − γrH)RC/℘− (1− γs)RR/pg}. (55)

Least-cost banks, which limit leverage but are able to issue insured deposits plus quasi-safedeposits in excess of the insurance limit, will now be limited to fewer insured deposits andexpand their quasi-safe deposits. Since the cost of quasi-safe deposits, RC , exceeds thatof fully-safe insured deposits, RF , these banks also experience lower profits. The increasedfunding cost affects these banks’ ability to credibly exert effort, γmDI,L,H < γmDI,L, thus thereis a reduction in leverage for this class of banks.

Moderate-cost banks restrict their deposits to below the insurance limit and, therefore,will not be affected by the reduction in coverage. However, some banks will switch frommoderate-cost to least-cost banks, cmH > cm, and issue both insured and quasi-safe uninsureddeposits. As there is no reduction in moderate-cost bank profits, the threshold for exertingeffort will increase, cDIH > cDI , as some banks switch from no-effort to high-effort due tothe reduction in profits for no-effort banks. The hybrid system thus has better averagemonitoring efficiency relative to the pure deposit insurance system.

The total quantity of fully-safe deposits is determined by the government’s tax capacityand is independent of the choice of banking system, SfH = SfDI = tω/RF . However, thequantity of quasi-safe deposits increases due to the increased issuance by least-cost banksand also by high-cost banks if deposit insurance is sufficiently restricted,

ScH =

∫ cmH

c(min[γd, γmDI,L,H(ci)]− γrH)f(ci)dci +

∫ c

cDIH

max[0, γs − γrH ]f(ci)dci > ScDI . (56)

Since least-cost banks have lower leverage under a hybrid system, and some no-effort banksswitch to lower leveraged moderate-cost banks, total economy-wide lending will decrease.

Thus, a hybrid system can be seen as a blend of a pure narrow banking system and apure insured-deposit system. Relative to a system with only government deposit insurance,

23

a hybrid system has less lending but better average monitoring efficiency. It also has greatertotal liquidity provision in the form of fully-safe narrow bank deposits, fully-safe insureddeposits, plus quasi-safe uninsured deposits.

Which system is welfare-improving depends on how lending, monitoring efficiency, andsafe asset production are weighted. In a richer model with decreasing marginal social benefitsto both greater aggregate lending and safe asset production, the hybrid system may providean interior solution for the overall social optimum.

5.2. Combining Narrow and Broad Banks

Instead of completely independent institutions, narrow and broad banks could operate asseparate subsidiaries under a single bank holding company. A ring-fenced narrow banksubsidiary would issue fully-safe deposits that are backed by government debt. Anothersubsidiary makes loans and issues fully-safe insured deposits, uninsured deposits, and equitycapital. This structure may have merit if there are economies of scope in activities suchas administering payments for narrow and broad bank deposits. However, the emergenceof payments mechanisms provided by non-banks such as PayPal, mobile wallets, and peer-to-peer payments systems demonstrate that banks do not hold a monopoly on settlingpayments. Therefore, it is unclear whether such a holding company structure provides asubstantial efficiency gain relative to specialized narrow and broad banks.

5.3. Central Bank Digital Currencies

Many central banks have assets that are largely government debt and have liabilities thatare mostly reserve notes and deposits. Consequently, these central banks are structured notunlike narrow banks. Recently, the similarity has become even closer as some central banks,such as the U.S. Federal Reserve, began paying competitive interest on reserve deposits.

Proposals to create a central bank digital reserve deposit or currency (CBDC) would fol-low a similar narrow bank structure (Bank for International Settlements (2018)). If reservedeposits were available to the general public, then our hybrid model would be applicable topredicting the consequences of an introduction of CBDCs.

One method of making CBDCs available to the public would be to simply allow individ-uals and institutions to open accounts directly with the central bank. However, this wouldinvolve the central bank providing depository services that it may not be willing, or in themost efficient position, to undertake. A second method would be for narrow banks thatspecialize in settling payments to hold CBDC reserve deposits as assets and issue matchingdigital deposit accounts to their customers.25 A third method would be to allow such nar-

25A recent proposal for a financial institution known as The Narrow Bank (TNB) would operate in asimilar manner but have conventional reserves as assets and offer conventional deposits to institutionalcustomers.

24

row banks to operate as a separate subsidiary of a bank holding company that also offersconventional deposits through another broad bank loan-making subsidiary as described inthe previous section.

6. Numerical Illustrations

This section provides numerical comparisons of aggregate performance, or loosely-speaking‘welfare,’ measures for the three main previously-discussed government regimes: depositinsurance, narrow banks with government investment, and narrow banks with governmentrebates.

6.1. Welfare Measures

We assume that banks are able to tranche deposits and that the density of bank monitor-ing costs is distributed uniform. Using the fully-private banking system as a baseline, weconsider the following measures of social welfare, set out explicitly in Appendix C.

1. Quasi-safe deposit production and its aggregate surplus.2. Total bank loans and the surplus from financing projects.3. Total monitoring effort exerted by banks.4. Bank profits.

The surplus from issuing quasi-safe deposits for regime r is equal to the quantity producedmultiplied by their liquidity premium, (RR −RC) = lRC :

SurplusSr = (RR −RC)Scr . (57)

Loan surplus is defined as the expected loan revenue less the costs of monitoring and fundingat RR. That is, a bank with monitoring cost, effort and total leverage {ci, ai, γi} has surplusequal to (γi + k)[pgRL + pbd(ai) − ciai] − γiRR. In the fully-private bank system, this issimply equal to aggregate profits, ΠT , less the reduction in funding costs due to the liquiditypremium from issuing quasi-safe deposits:

SurplusLT =ΠT − SurplusST . (58)

This is also the loan surplus for the case of narrow banking with a government rebate sincebroad banks operate the same as banks in the fully-private system. For the case of narrowbanking with government investment, loan surplus is

SurplusLNB,I =ΠNB,I − SurplusSNB,I . (59)

Under deposit insurance, some loans are funded by fully-safe deposits that generate extrasurplus for banks

SurplusLDI =ΠDI − SurplusSDI − (RR −RF )SfDI . (60)

25

Note that deposit insurance and narrow banking produce equal amounts of safe deposits,so there is no need for comparing this aggregate welfare measure.

6.2. Calibration

Table 1 reports the benchmark parameter values used in our illustrations. We set the fully-safe return at 1.02, the expected quasi-safe return at 1.03, and the expected risky return at1.04. These rates imply a liquidity premium of 100 basis points for each increase in safety.The promised loan return is assumed to be 1.10, and the probabilities of the good, bad,and catastrophe states are 90%, 9%, and 1%, respectively. The amount of banker capitalper unit of market savings, k, is set at 10%. The recovery rate with no effort is assumedto be 50%, implying a value of α = 0.5. The coefficient on bank effort, β, is set to 1. Themaximum tax rate t equals 40% of end-of-period endowment of ω = 1.5.26

Table 1: Parameter Values

Parameter ValueRisk-free return RF 1.02Safe liquidity premium l 0.01Quasi-safe liquidity premium lf 0.01Promised loan return RL 1.10Probability of good state pg 0.90Probability of bad state pb 0.09Bank capital per savings k 0.10Loan loss parameter α 0.50Monitoring effort parameter β 1.00Tax limit t 0.40Endowment ω 1.50

Table 2 reports particular deposit limits implied by the parameters in Table 1. Themaximum level of insured deposits, γr, is 0.63 and the maximum level of deposits availableto broad banks under narrow banking with government investment, γd, is 0.41. Recall thatunder government investment, narrow banks’ deposits ‘crowd out’ the deposits availableto broad banks. Narrow banking with a government rebate avoids this problem. Theproportion of banks exerting high effort under the regime of narrow banking with governmentrebate is 0.49. In addition, Proposition 3 holds for these parameter values, so that underdeposit insurance a smaller 0.35 proportion of banks exert high effort.27 Narrow bankingwith government investment has the largest proportion of (broad) banks exerting high effort,

26The choice of taxation is made to arrive at reasonable values of maximum insured deposits, γr. Notethat FDIC data from 2017 Quarter 3 imply a ratio of insured domestic U.S. deposits to total bank debt of0.78.

27The restriction on insurance, γr = 0.63 is greater than the threshold, γr∗ = 0.39.

26

0.73. This occurs because less deposit availability makes choosing the highest leverage andno effort less profitable.

Table 2: Implied Deposit Limits and Effort Levels

ParameterDeposit

InsuranceNarrow Bankingwith Investment

Narrow Bankingwith Rebate

Tax-limit on leverage γr, γd 0.63 0.41 1.00Proportion of high effort banks c∗

c−c 0.35 0.73 0.49

Figure 1 illustrates the structure of the banking system under each regime.28 Bankswith relatively low monitoring costs, ci, find it optimal to exert high effort and have profitsshown in black. Banks with relatively high monitoring costs exert no effort, with profits inred. Under deposit insurance, for this baseline calibration, only moderately-low cost banksexist. Their profits are illustrated by the dashed black line.

Note that no-effort banks profit the most under deposit insurance because the fully-safeliquidity premium makes their cost of deposit funding the least. No-effort broad banksunder the narrow bank system with government investment make the least profit becausetheir leverage, 1− γn, is less due to crowding out from narrow banks.

Figure 1: Individual Bank Profits Under Each Regime.

28Of course for narrow banking systems, these graphs refer to only broad banks. Narrow banks extendno loans and issue no quasi-safe deposits. They invest only in Treasury debt and issue fully-safe deposits,making zero profit.

27

Table 3 compares various measures of welfare under the three government regimes. Thesemeasures are averages per loan-making bank. As discussed earlier, for this calibration theonly high-effort banks that exist under deposit insurance are those with moderately-lowcosts. Since only least-cost banks would issue quasi-safe deposits, there are no quasi-safedeposits under this deposit insurance regime. Under narrow banking regimes, quasi-safedeposits are issued by broad banks, and since there is more deposit availability under narrowbanking with a government rebate, this regime produces the most.

Due to the low cost of insured deposits and less incentives to limit leverage, the tablealso shows that insured banks make the most loans. Among narrow bank regimes, broadbanks make more loans when there is a government rebate due to the greater availabilityof deposits. One also sees that monitoring effort is highest under narrow banking withgovernment investment. Yet with a rebate, effort is still substantially higher than underdeposit insurance. The last line of Table 3 aggregates the bank profits that were illustratedin Figure 1 and shows that average bank profits are greatest under deposit insurance.

Table 3: Welfare Comparisons

Measure of SocialWelfare

DepositInsurance

Narrow Bankingwith Investment

Narrow Bankingwith Rebate

Quasi-safe deposits Sc 0.000 0.241 0.494Liquidity surplus SurplusS 0.000 0.002 0.005Loans L 1.016 0.699 0.900Loan surplus SurplusL 0.128 0.111 0.115Monitoring effort A 0.182 0.506 0.340Profit per bank Π 0.116 0.108 0.110

6.3. Comparative Statics

This section considers how the aggregate production of quasi-safe deposits, loans, and bankprofits are affected by variation in key parameters. We begin by adjusting the liquiditypremium on quasi-safe deposits which allows banks to fund loans at less than the expectedreturn RR, which is held constant. In Figure 2 the premium on quasi-safe deposits, l, variesby plus or minus 100 basis points while holding the premium on fully-safe deposits, lf ,constant. Equivalently, RC changes while changing RF by the same amount.29 The resultsshow that increasing the quasi-safe liquidity premium leads broad banks to issue morequasi-safe deposits and more loans. Profits rise, but the proportion of high-effort banksfalls. Apparently, the higher liquidity premium gives a relative advantage to high leverageand tranching of deposits, rather than high effort. Under deposit insurance, insured banksissue no quasi-safe deposits but the reduction in RF leads to greater profits and slightly lesshigh effort banks.

29The results are very similar if we vary RC while holding RF constant.

28

Figure 2: Varying the Quasi-safe Liquidity Premium, l, Holding lf Constant

0 0.005 0.01 0.015 0.020

0.1

0.2

0.3

0.4

0.5

0.6

0 0.005 0.01 0.015 0.020.2

0.4

0.6

0.8

1

1.2

0 0.005 0.01 0.015 0.020.106

0.108

0.11

0.112

0.114

0.116

0.118

0.12

0.122

0 0.005 0.01 0.015 0.020

0.2

0.4

0.6

0.8

1

Figure 3 demonstrates that an increase in banker capital, k, raises average bank profitsand the proportion of high-effort banks in all three regimes. However, broad banks’ quasi-safe deposit and loan production is not monotonic in capital. Ceteris paribus, more capitalincreases total assets available to banks, but as greater capital makes high effort relativelymore attractive, a greater proportion of banks limit their leverage such that quasi-safedeposit and loan production can decline.

29

Figure 3: Varying Bank Capital, k

0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.05 0.1 0.15 0.2 0.250.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.05 0.1 0.15 0.2 0.250.05

0.1

0.15

0.2

0.25

0.3

0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 4 shows that increasing the loss rate reduces profits and increases the relativebenefit of high-effort monitoring.30 Since collateral for senior tranched deposits of no-effortbroad banks declines and more broad banks limit leverage, the result is that quasi-safedeposits and loans decline.

30Note that for values of α exceeding 0.55, all broad banks find it optimal to exert high effort; that is,cn = c.

30

Figure 4: Varying the Default Loss Rate, α

0.4 0.45 0.5 0.55 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.4 0.45 0.5 0.55 0.60.2

0.4

0.6

0.8

1

1.2

0.4 0.45 0.5 0.55 0.60.105

0.11

0.115

0.12

0.125

0.13

0.4 0.45 0.5 0.55 0.60

0.2

0.4

0.6

0.8

1

Figure 5 demonstrates how a hybrid system of both narrow banking and deposit insur-ance provides results that are intermediate to either regime. The solid line assumes that thegovernment invests the funds issued by narrow banks and the dashed line assumes a gov-ernment rebate. The pure deposit insurance and pure narrow banking regimes are denotedby an asterisk at the end-points of each line, when γnH = 0 or γnH = tω/RF respectively. Asthe relative size of narrow banks in the economy increases, loan provision and bank profitsfall, but average monitoring efficiency and quasi-safe asset provision rise.

31

Figure 5: Varying the Size of Narrow Banks in the Hybrid System, γnH

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.50.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.50.108

0.11

0.112

0.114

0.116

0.118

0 0.1 0.2 0.3 0.4 0.50.1

0.2

0.3

0.4

0.5

0.6

7. Robustness of the Model’s Results

This section discusses the robustness of our results to reasonable changes in the model’sassumptions. One modification might allow for direct costs of taxation that are an increasingand convex function of the revenue raised. The expected value of such direct costs would belower under deposit insurance because the average amount of required taxes are less thanunder narrow banking. However, the uncertainty of state-dependent taxes under depositinsurance might be unattractive to consumers, particularly since taxes are highest in thecatastrophe state. Hence, there may be a trade off between a higher average cost of taxesversus the predictability of taxes.

Another assumption of our model is that the liquidity premia for fully-safe and quasi-safe deposits are fixed. A richer general equilibrium model might lead to liquidity premiathat decline with the economy’s supply of safe deposits. Since our model implies that theprovision of fully-safe deposits is the same under deposit insurance and narrow banking, the

32

direct effects of fully-safe deposit supply on liquidity premia would not differ across the tworegimes. However, since the supply of quasi-safe deposits tends to be greater under narrowbanking, the equilibrium quasi-safe liquidity premium would tend to be lower. Moreover,if quasi-safe and fully-safe deposits have substitutability, the equilibrium fully-safe liquiditypremium might also be lower under narrow banking.

The assumption that a bank’s equity capital is fixed is clearly a strong one. Yet inter-preting this capital as “inside” equity, it might be reasonable to believe that it is relativelyinelastic if it is linked to the capital that can be contributed by the bank’s manager. Wecould allow the bank to issue “outside” equity to consumers, but as discussed earlier, itwould play the same role as junior deposits in the existing model.

Our model’s assumption that bank loans are worthless in the catastrophe state is stark.Instead, one might expect that loans have a positive minimum recovery value, even if acatastrophe occurs. If that were the case, then our model with zero recovery over-estimatesa government’s deposit insurance losses. Thus, for a given tax capacity, a government couldcreate more fully-safe deposits under deposit insurance if there was a positive recovery valuein the catastrophe state.

However, if there was a strictly positive minimum recovery value to bank loans, theneven uninsured broad banks could issue some fully-safe deposits. They could do so bytranching deposits, where the most senior, fully-safe deposits would be limited to the mini-mum recovery value of the bank’s loans. As a result, when the maximum fully safe depositsof narrow banks and broad banks are combined, they would equal the same amount offully-safe deposits created by a deposit insurance system. Therefore, assuming a positiveminimum recovery value does not overturn our result that the amount of fully-safe depositsis independent of the government regime.

In our model’s system of narrow banking, broad banks produce quasi-safe deposits which,in aggregate, exceed the amount of quasi-safe deposits that are produced by a system ofgovernment deposit insurance. Under government deposit insurance, only least-cost, high-effort banks produce quasi-safe deposits. As discussed earlier, the low production of quasi-safe deposits occurs because government deposit insurance crowds out what would havebeen quasi-safe deposits in the absence of deposit insurance. However, in principle onemight imagine a slightly different system where a government offers two types of depositinsurance coverage: fully-safe and quasi-safe. In this case, more quasi-safe deposits couldbe produced under government deposit insurance.

Such as system would work as follows. As in our model, a government provides asimilar level of fully-safe deposit insurance limited to some level γr. In addition, it wouldoffer supplemental, quasi-safe deposit insurance that covers losses in all states except thecatastrophe state. A government is able to offer such supplemental quasi-safe insurancebecause banks’ positive recovery value in the bad state provides unused taxing capacityrelative to the catastrophe state where there is zero recovery. This bad state taxing capacitywould allow for limited, supplemental quasi-safe deposit insurance. While such a system

33

could replace some risky, uninsured deposits with quasi-safe insured deposits, it might beimplausible to think that a government would offer two classes of insurance where for oneclass it fails to provide protection for some depositors in some states. Hence, it is not clearthat this two-class deposit insurance system would be politically feasible and that it shouldbe taken seriously.

A related issue is whether quasi-safe deposits are truly a social good. Our analysisimplicitly assumed that the greater private liquidity under narrow banking compared togovernment deposit insurance was a net social benefit. However, in richer models thatincorporate additional frictions, a competitive private banking system might inefficientlyover-issue quasi-safe deposits (Gersbach (1998)), creating negative externalities such as fire-sale costs when a crisis occurs (Stein (2012)). When deposits are not fully safe, coordinationfailures can lead to inefficient bank run equilibria as in Diamond and Dybvig (1983) andGoldstein and Pauzner (2005).

While our model neglects these adverse consequences of quasi-safe deposits, it also doesnot account for potential costs of government liquidity. It was assumed that a governmentalways respects its limit on tax capacity so that its debt and bank deposit guaranteesare fully-safe. But as Reinhart and Rogoff (2009) document, history provides numerousexamples of government defaults. Even without default, government liquidity in the formof deposit insurance may create inefficiencies due to bank risk-shifting.31 Mitigating thismoral hazard may require costly bank regulation. So while we acknowledge that our modelmisses potential costs of private liquidity, it also neglects other costs of government liquidity.A more complete modeling of these costs is needed to provide a definitive answer to thequestion of how governments should create liquidity.

8. Conclusions

This paper considers constraints on a financial system’s liquidity. The amount of liquiddeposits that can be created by private, uninsured banks is limited by their assets’ recoveryvalues in bad states of nature. Recovery values can be enhanced by a bank limiting itsleverage, thereby instilling incentives to efficiently monitor borrowers. Alternatively, bankscan maximize leverage and not monitor borrowers but create liquid senior deposits usingthe extra collateral provided by junior deposits (subordinated debt).

The amount of liquidity provided by a government is limited by its future taxing ca-pacity since ultimately taxes are need to cover the government’s liabilities. Importantly,the method that a government uses to create its liquid assets has consequences for privateliquidity creation. Government liquidity created via bank deposit insurance crowds out pri-vate liquidity. It also reduces bank monitoring incentives but maximizes bank lending due

31Our model assumes deposit insurance is risk-sensitive and fairly priced. In practice, deposit insurancetends to be risk-insensitive and under-priced, which can worsen risk-shifting incentives.

34

to the government liquidity premium that minimizes banks’ cost of funding.

In contrast, a system by which governments create liquidity by issuing debt held bynarrow banks allows narrow banks to create public liquidity while uninsured broad bankscreate private liquidity. Such a system avoids the crowding out of private liquidity whilemaintaining broad banks’ better incentives to monitor borrowers. Yet since broad banks’cost of funding is higher, they do not match the quantity of lending made by government-insured banks.

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37

A. Proofs

A.1. Proof of Lemma 1

For some values of leverage, γ, the effort choice that is profit-maximizing becomes apparentsince the banker will either always get a return in the default state or it will never get areturn in the default state, independent of whether it monitors at al = 0 or ah. For example,suppose the bank’s choice of leverage (deposits) is sufficiently low such that γ < γlow where

(γlow + k)d(a = 0) = γlowRC/℘ (61)

or

γlow =kRL(1− α)

RC/℘−RL(1− α)(62)

Since in this case deposits are default-free even if the bank made no effort, bank equityreceives the entire benefit from effort. Consequently, given the parametric restriction in (2),the profit-maximizing choice of effort is ah and RD = RC/℘. For this case of γ < γlow, thebank’s expected profit is (10). In contrast, suppose that the bank’s choice of leverage isvery high such that γ > γhigh where

(γhigh + k)d(ah) = γhighRC/℘ (63)

γhigh = kβpbRL − c

c− βpb(RL −RC/℘)(64)

In this case even if effort was at the first-best level ah, bank equity would receive none ofthe benefit from monitoring effort. Therefore, because effort is not contractible, the profit-

maximizing effort level is al = 0 and RD =RR−pb γ+kγ d(al)

pg=

RR−pb γ+kγ RL(1−α)

pg. For this case

of γ > γhigh, the bank’s expected profit is (6).

Now for γlow < γ < γhigh, we can determine whether effort al = 0 or ah given in (8) ischosen by comparing the difference in the bank’s expected profits under these two choices.Hence, for a bank’s given choice of γ and depositors’ given choice of RD, the difference inthe bank’s expected profits from effort level ah versus al is

pb max[(γ + k)d(ah)− γRD, 0]− cah(γ + k)− pb max[(γ + k)d(al)− γRD, 0] (65)

Now note that since d(al) = RL(1− α) < RC/℘ ≤ RD, we have

(γ + k)d(al)− γRD = kRL(1− α)− γ (RD −RL(1− α)) (66)

which is always negative for γ > γlow. Using this result and substituting in for ah, thedifference in expected profit becomes

pb max

[(γ + k)

[RL −

c

βpb

]− γRD, 0

]− c(γ + k)

βln

(βαpbRL

c

)(67)

38

The rise in expected profit from effort in (67) is weakly decreasing in RD. Moreover, forany γ ≤ γhigh, deposits are default-free when the bank chooses the first-best effort levelah. Therefore, for γ ≤ γhigh, a rational expectations equilibrium in which a bank suppliesfirst-best effort obtains if and only if

pb

((γ + k)

[RL −

c

βpb

]− γRC/℘

)− c(γ + k)

βln

(βαpbRL

c

)> 0. (68)

Re-writing this condition in terms of first-best effort, ah,

pb[(γ + k)d(ah)− γRC/℘

]− (γ + k)cah > 0, (69)

we can define the threshold leverage γm for which this condition binds as

γm = kpbd(ah)− cah

pbRC/℘− [pbd(ah)− cah]. (70)

A.2. Proof that High-Effort Profits are Decreasing in Monitoring Cost

πh = (γm + k)[pgRL + pbd(ah)− ciah

]− γmRC (71)

Define Ψ ≡ pbd(ah)− ciah so that γm = k ΨpbRC/℘−Ψ

.

πh =

(k

Ψ

pbRC/℘−Ψ+ k

)[pgRL + Ψ]− k Ψ

pbRC/℘−ΨRC (72)

=kpgRC/℘pbRL −Ψ

pbRC/℘−Ψ

∂πh

∂Ψ=kpgRC

pb[RL −RC/℘]

{pbRC −Ψ/℘}2 > 0 (73)

We then use ∂d(ah)∂ci

= − 1βpb

and ∂ah

∂ci= − 1

ciβto show that

∂Ψ

∂ci=pb

∂d(ah)

∂ci− ah − ci

∂ah

∂ci= −ah < 0. (74)

Hence ∂πh

∂ci= ∂πh

∂Ψ∂Ψ∂ci

< 0.

A.3. Proof of Proposition 1

To prove Proposition 1, we start by showing that there is a unique value of ci thatequates πl to πh. Equating (11) to (12), one obtains equation (14) that can be re-written

39

as

F (ci) ≡ ci

(1

β+ ah(ci)

)=πlpb(RL −RC/℘)

πl − pgkRC/℘(75)

where ah(ci) = 1β

ln(βαpbRL

ci

). Now note that

∂F (ci)

∂ci= ah(ci) > 0 (76)

Consequently, the left-hand-side of equation (75) is an increasing function of ci. Given thatthe right-hand-side of (75) is independent of ci and condition (13) holds, there is a uniquevalue ci for which the left-hand-side of (75) equals the right-hand-side of (75). This isequivalent to there existing a single critical value of ci = c∗ such πl = πh(c∗), and values ofci below c∗ imply πl < πh(ci) while values of ci above c∗ imply πl > πh(ci). Therefore, givenLemma 1, the value c∗ in (14) is the cut-off between choosing high effort, limited leverage,and quasi-safe deposits versus choosing no effort, maximum leverage, and risky deposits.

A.4. Proof of Least-Cost Bank Leverage Limit

Least-cost banks can limit total deposits to a maximum of γmDI,L > γr while still having anincentive to provide high effort. These banks issue insured deposits up to the limit γr ata funding cost of (1 + φh)RF = RF/℘ per deposit. On top of this, they issue uninsureddeposits equal to γmDI,L − γr which promise to return RC/℘. Compared to eq. (69), thecondition for which high effort binds is now

pb[(γ + k)d(ah)− (γ − γr)RC/℘− γrRF/℘

]− (γ + k)cah > 0, (77)

Solving for γ, we obtain

γmDI,L =pbγ

r(RC −RF )/℘+ k[pbd(ah)− cah]pbRC/℘− [pbd(ah)− cah]

. (78)

A.5. Proof of Proposition 3

Recall that cT is the cost of effort such that a bank is indifferent between no effort and higheffort when there is no deposit insurance and no effort banks can tranche their deposits.In other words, πhT

(cT)

= πlT . If with deposit insurance, this bank’s increase in profits isgreater by choosing no effort compared to choosing high effort, then it will unambiguouslychoose no effort. In that case, the cost at which a bank is indifferent between no effort andhigh effort must be less; that is, cDI < cT .

Now note that from equation (31), the increase in profit from no effort after depositinsurance is independent of cost and can be written as

πlDI − πlT = (γr − γs) (RR −RC) + γr (RC −RF ) (79)

40

For the bank with cost ci = cT , equation (39) shows that the corresponding increase inprofit from high effort after deposit insurance is(πhDI,M − πhT

)|ci=cT =

(γmDI,M − γm

) [pgRL + pbd

(ah(cT))− cTah

(cT)−RF

]+γm (RC −RF )

(80)where ah

(cT)

= 1β

ln(βαpbRL/c

T). The difference in this increase in profit from no effort

versus high effort is

πlDI − πlT −(πhDI,L − πhT

)|ci=cT = (γr − γs) (RR −RC) + (γr − γm) (RC −RF ) (81)

−(γmDI,M − γm

) [pgRL + pbd

(ah(cT))− cTah

(cT)−RF

]The first line of equation (81) includes two terms that are both positive and increasingin γr. The second line of equation (81) is negative and independent of γr. Thus, thehigher is γr, the greater is profit difference. The threshold value of γr at which πlDI − πlT −(πhDI,L − πhT

)|ci=cT = 0 is then

γr∗ =

(γmDI,M − γm

) [pgRL + pbd

(ah(cT))− cTah

(cT)−RF

]+ γs (RR −RC) + γm (RC −RF )

RR −RF(82)

B. Deposit Insurance Solution

The cost threshold which equates high-effort and no-effort bank profits, cDI , and the tax-limit on insured deposits, γr, solve

πhM,DI(cDI) = πlDI(γ

r), (83)

where the definition of the profit equations may depend on the solutions to γr detailedbelow, and

L(γr, cDI) = tω, (84)

where the tax base is reduced by γnH under the hybrid regime of both narrow banking anddeposit insurance.

Before solving these equations, we must first consider that all three types of insuredbank may not exist.32 If the tax base is sufficiently high that the lowest-cost bank cannotcredibly exert effort and lever above the limit on insurance, γr > γmDI,M(c) then least-costbanks do not exist and cm = c. This is the case that corresponds to our calibration whichassumes a tax-base large enough to support high levels of γr seen in the data. As stated inthe text, this limit prevents no-effort banks from tranching, γr > γs.

32High cost banks do not exert effort; moderate-cost high-effort banks restrict leverage to below the limiton insurance; and least-cost high-effort banks issue insured deposits up to the limit as well as quasi-safedeposits.

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By contrast, if the tax base is limited such that such that high-effort banks are ableto offer leverage above the insurance limit even at the highest cost, γr < γmDI,L(cDI), thenmoderate-cost banks do not exist and cm = cDI . In this case, the cost threshold betweenhigh and no effort banks equates no-effort bank profits to least-cost bank profits, suchthat πhL,DI(c

DI) replaces the LHS of eq. (83). Both least and moderate cost banks existonly if the tax base is such that the insurance limit is between these extremes such thatγr ∈

(γmDI,M(cDI), γmDI,L(c)

). In either of the two previous cases, if the tax limit is such

that γr < γs then no-effort banks will be able to issue quasi-safe senior deposits, increasingtheir profits to πlDI,T given by eq. (55), which replaces the RHS of eq. (83). These casesare applicable when considering the restricted tax-base available to the insurer under thehybrid regime.

Secondly, high-effort banks do not exist if a bank with lowest costs gains more profitfrom not monitoring. πhDI(c) < πlDI . If from the conditions above we verify that least-costbanks exist, this condition is πhDI,L(c) < πlDI otherwise the condition is πhDI,M(c) < πlDI .If this condition is satisfied, then all banks do not monitor and cDI = c and γr = tω/RF .Similarly, no-effort banks do not exist if a bank with highest costs finds it profitable to exerteffort, πhDI(c) > πlDI , in which case cDI = c and γr = tω/RF .

Under the hybrid regime with the assumption of government investment, banks face aleverage limit of 1− γnH instead of 1 which will reduce no-effort bank profits and also least-cost bank profits if γmL,DI(ci) > 1 − γnH . In both cases, these banks lever up to the limit of1− γnH

Denoting least,moderate and high cost bank with ‘L’,‘M’ and ‘H’ respectively, there areseven combinations of bank types: LMH,LM,MH,LH,L,M and H. For the first four combina-tions, where both high- and no-effort banks exist, we solve eqs. (83) and (84) simultaneouslyfor {γr, cDI}. If the solution does not match the constraints on the tax-limit and profits forthe case in question set out above, we discard this solution.

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C. Welfare Measures

Government intervention produces safe assets equal to γn under both deposit insurance andnarrow banking. Aggregate production of quasi-safe assets under each regime is

ScT =1

c− c

[∫ cT

cγm(ci)dci + γs(c− cT )

], (85)

ScDI =1

c− c

[∫ cm

c

γmDI,L(ci)dci − γr(cm − c)

], (86)

ScNB,I =1

c− c

[∫ cn

c

min [γm(ci), γd]dci + γsNB(c− cn)

], (87)

ScNB,R =ScT , (88)

(89)

where the limits on tranching are given by γs = (γ+k)RL(1−α)RC/℘

and γsNB = (γd+k)RL(1−α)RC/℘

.

Under narrow banking without the rebate, maximum leverage is γd = 1− γn. The surplusfrom issuing quasi-safe assets for regime r is

SurplusSr = (RR −RC)Scr . (90)

No-effort banks issue default-risky deposits equal to

RT =c− cT

c− c(1− γs), (91)

RDI =c− cDI

c− c(1− γr), (92)

RNB,I =c− cNB

c− c(γd − γsNB), (93)

RNB,R =RT . (94)

The quantity of loans provided by the banking industry is equal to total leverage extendedby private banks

LT =1

c− c

[k +

∫ cT

cγm(ci)f(ci)dci + (c− cT )

], (95)

LDI =1

c− c

[k +

∫ cm

cγmDI,L(ci)f(ci)dci +

∫ cDI

cm

γmDI,M(ci)f(ci)dci + (c− cDI)], (96)

LNB,I =1

c− c

[k +

∫ cn

cmin[γm(ci), γ

d]f(ci)dci + γd(c− cn)

], (97)

LNB,R =LT . (98)

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As only first-best banks exert effort, total monitoring effort is

A(c∗) =1

c− c

∫ c∗r

cahdci, (99)

where c∗r is the threshold cost above which banks do not exert effort in each of the regimes{cT , cDI , cn, cT}. Based on previous logic, we have effort A(cn) > A(cT ) > A(cDI). Totalexpected bank profit is given by

ΠT =1

c− c

[∫ cT

cπh(ci)d(ci) + (c− cT )πlT

], (100)

ΠDI =1

c− c

[∫ cm

cπhDI,L(ci)d(ci) +

∫ cDI

cmπhDI,M(ci)d(ci) + (c− cDI)πlDI

], (101)

ΠNB,I =1

c− c

[∫ cn

cπh(ci)d(ci) + (c− cn)πlNB,I

], (102)

ΠNB,R =ΠT . (103)

Loan surplus is

SurplusLT =ΠT − SurplusST , (104)

SurplusLDI =ΠDI − SurplusSDI − (RR −RF )SfDI , (105)

SurplusLNB,I =ΠNB,I − SurplusSNB,I , (106)

SurplusLNB,R =ΠNB,R − SurplusSNB,R = SurplusLT . (107)

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