Financial Regulation in a Quantitative Model of theModern Banking System∗

Juliane Begenau

Stanford GSB & NBER

Tim Landvoigt

Wharton & NBER

September 2018

Abstract

How does the shadow banking system respond to changes in capital regulation

of commercial banks? We propose a tractable quantitative general equilibrium model

with regulated and unregulated banks to study the unintended consequences of cap-

ital requirements. Tightening the capital requirement from the status quo leads to a

safer banking system despite riskier shadow banking activity. A reduction in aggre-

gate liquidity provision decreases the funding costs of all banks, raising profits and

investment. Calibrating the model to data on financial institutions in the U.S., the op-

timal capital requirement is around 17%.

∗First draft: December 2015. Email addresses: begenau@stanford.edu, timland@wharton.upenn.edu.We would especially like to thank our discussants Dean Corbae, Mark Gertler, Christian Opp, GoncaloPino, David Chapman, and Hendrik Hakenes, as well as Hanno Lustig, Arvind Krishnamurthy, MartinSchneider, Monika Piazzesi, and Joao Gomes for many helpful conversations. We also benefited from com-ments and suggestions from seminar participants at the ASSA 2016 and 2017 meetings, Berkeley-HAAS,Barcelona GSE 2017, Carnegie Mellon, CITE 2015 conference, CREI, Federal Reserve Bank of Boston, FederalReserve Bank of New York, MFM Winter 2016 meeting, MIT Sloan, NBER SI 2016, Northwestern-Kellogg,NYU Junior Macro-Finance 2016 Meeting, SAFE 2016 Conference on Regulating Financial Markets, SEDToulouse 2016, SFI Lausanne, Stanford GSB, Stanford SITE 2017, Stanford 2016 Junior Faculty Workshop, UWisconsin-Madison, Texas Finance Festival, University of Texas at Austin, WFA 2016 and Wharton.

1 Introduction

The goal of the regulatory changes for banks in the aftermath of the Great Recession,notably higher capital requirements, was to reduce the fragility of the financial system.Today’s financial system, however, is much more complex and extends beyond the insti-tutions that are under the regulatory umbrella. Nowadays, the so-called shadow bankingsector fullfills many functions of regulated banks, such as lending and liquidity provision.Do tighter regulations on regulated commercial banks cause an expansion of shadowbanks as they become relatively more profitable? Does a larger shadow banking sec-tor imply an overall more fragile financial system? The example of Regulation Q tells acautionary tale. Introduced after the Great Depression in the 1930s to curb excessive com-petition for deposit funds it had little effect on banks as long as interest rates remainedlow. When interest rates rose in the 1970s, depositors looked for higher yielding alterna-tives and the competition for their savings generated one: money market mutual funds(Adrian and Ashcraft (2016)). This and numerous other examples1 highlight the unin-tended consequences of regulatory policies.

In this paper, we build a tractable general equilibrium model to quantify the costs andbenefits of tighter bank regulation in an economy with regulated commercial banks andunregulated shadow banks.2 The role of banks in this economy is to provide access toassets that require intermediation, such as loans. Banks fund their investment in interme-diated assets through short-term debt, which provides liquidity services to households.Commercial and shadow banks differ in their ability to produce liquidity, but not to in-termediate assets. All banks have the option to default and therefore may not repay theircreditors. However, commercial banks are insured and their depositors are always re-paid, while shadow banks are not insured and thus risky for depositors. This governmentguarantee drives a wedge between the competitive equilibrium and what a social plannerwould choose. Calibrating the model to aggregate data from the Flow of Funds, banks’call reports, and NIPA we find that higher capital requirements indeed increase the sizeof the shadow banking sector. However, instead of becoming more fragile, the aggregatebanking system is safer at the optimal level of bank capital requirements. The optimal

1Asset-backed commercial paper conduits are another example for entities that emerged arguably as aresponse to regulation, more precisely capital regulation (see Acharya, Schnabl, and Suarez (2013)).

2We define shadow banks as financial institutions that share features of depository institutions, eitherby providing liquidity services such as money market mutual funds or by providing credit directly (e.g.finance companies) or indirectly (e.g. security-broker and dealers). At the same time, they are not subject tothe same regulatory supervision as traditional banks. We adopt a consolidated view of the shadow bankingsector, e.g. money market mutual funds invest in commercial paper that fund security brokers-dealers thatprovide credit. We view this intermediation chain as essentially being carried out by one intermediary.

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requirement finds the welfare maximizing balance between a reduction in aggregate liq-uidity provision and an increase in the safety of the financial sector. At the optimum,increased riskiness of shadow banks is more than offset by greater stability of commercialbanks. Welfare is maximized at a capital requirement of roughly 17%.

We derive this result in a production economy with households, commercial banks,shadow banks, and a regulator. To capture the value of liquidity services produced bybank deposits, we assume that households derive utility from bank deposits in additionto consumption. The consumption good is produced with two technologies. One technol-ogy is directly accessed by households through their ownership of a tree that produces anendowment of the good stochastically. The second technology, a Cobb-Douglas, stochas-tic production technology, is operated by banks and uses capital owned by intermediariesand labor rented from households. Banks further possess an investment technology andcompete over capital shares. Their assets are funded by issuing equity and deposits tohouseholds. When either type of bank defaults on its debt, its equity becomes worthlessand a fraction of the remaining bank value is destroyed in bankruptcy. Shadow banks arefragile because some households may randomly lose confidence and run on them akin toAllen and Gale (1994). Run-induced shadow bank defaults cause additional deadweightlosses.

The model’s equilibrium determines the optimal leverage of each type of bank, and therelative sizes of the regulated and unregulated sectors. Households value shadow bankdebt both for its payoff (in consumption goods) and for its liquidity benefits. Greaterleverage increases the default risk of shadow banks, reducing the expected payoff of theirdebt. It also increases the amount of liquidity services shadow banks produce per unit ofassets. Shadow banks internalize this trade-off, choosing leverage to equate the marginalliquidity benefit to households with the marginal bankruptcy cost. Commercial bankdebt is insured, and as a result the regulatory constraint is the only limit on their lever-age. Given these capital structure choices, how does the model pin down the relative sizeof both sectors? A key force is that the liquidity services produced by deposits of commer-cial and shadow banks are imperfect substitutes, with diminishing returns in each.3 Theliquidity premium earned by bank debt is the main source of bank equity value, allowingbanks to issue debt at interest rates that are lower than the expected return of the debt.This links to the second force in the model. Since banks compete with each other, the to-

3Since commercial deposits are safe and shadow deposits are risky, risk averse households in our modelassign an optimal portfolio share to each type of debt. In theory, this force would determine the relative sizeof shadow banks to commercial banks even if commercial and shadow deposits were perfect substitutesin terms of liquidity. However, for realistic risk aversion and shadow bank risk, this mechanism fails toprovide the relatively stable shadow bank share observed in the data.

2

tal quantity of their debt, commercial and shadow, must align such that equity holders –households – are indifferent on the margin between holding each type of equity. Thus, forgiven leverage, the asset share of each bank type expands or shrinks to align their equityvalues.

This core feature of the model also drives the response to higher capital requirements.Increasing capital requirements makes commercial banks less profitable, since it forcesthem to fund each dollar of assets with a smaller share of advantageous deposits. As aresult, shadow bank equity becomes relatively more profitable, inducing households toinvest in shadow banks and raising shadow bank demand for intermediated assets. Asthe shadow sector expands, it issues more debt, lowering its marginal liquidity benefit todepositors and thus its marginal profitability. This process continues until both sectorsare equally profitable again.

We develop this intuition in a simplified static version of our model, which also allowsus to analytically characterize the welfare-maximizing size and leverage of both bankingsectors. We show that a capital requirement alone fails to implement the social optimumin the decentralized economy. Competition forces shadow banks to have too much lever-age in order to overcome the competitive advantage of commercial banks created by de-posit insurance. Our simple model further demonstrates that the welfare consequencesof higher capital requirements crucially depend on the responses of liquidity premia forcommercial and shadow debt, which in turn depend on liquidity preferences. In particu-lar, the questions by how much the shadow sector expands, and whether shadow banksbecome riskier, depend on how elastic liquidity premia of both banks are with respect totheir debt quantities. In the model, these elasticities are functions of the overall returnsto scale in liquidity production, and the degree of substitutability between commercialand shadow liquidity. Thus, in the full quantitative model, we match these elasticitiesto dynamic properties of liquidity premia in the data, building on empirical work by Kr-ishnamurthy and Vissing-Jorgensen (2012), Greenwood, Hanson, and Stein (2016), andNagel (2016). We find that our model matches the data elasticities with moderately de-creasing returns in overall liquidity provision, implying that tightening capital regulationfrom 10% to 17% of assets (the welfare optimum) will raise liquidity premia on both com-mercial and shadow deposits, by 4 basis (11.5%) and 1.2 basis points (5.3%), respectively.We further find a relatively high degree of substitutability, leading to an expansion of theshadow share in debt markets by 2 percentage points (6.2%). We explore the robustnessof these results to different preference specifications.

Overall, our carefully calibrated quantitative model confirms the assertion of some

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commentators that tighter capital regulation will cause an expansion of shadow banking.However, our results also demonstrate that despite this substitution, there is substantialroom for welfare-improving increases in capital requirements, as a greater shadow bank-ing sector does not necessarily imply a riskier financial system.

Finally, we use our calibrated model to evaluate alternative policy measures. Specif-ically, we consider the effects of time-varying capital requirements and higher depositinsurance fees. We also evaluate proposals to impose a tax on shadow bank debt, atthe same time as tightening the capital requirement on traditional banks (the so-calledMinneapolis plan4 outlines such a proposal). We find that this simultaneous taxation ofshadow bank debt achieves a larger welfare gain than the other policies. While such aproposal greatly reduces liquidity provision by both kinds of banks in our model, thisnegative effect is more than offset by the increased stability of the financial system as awhole. This is consistent with the view that the existence of shadow banks limits thescope of regulatory measures that only target traditional banks.

Related Literature. Our paper is part of a growing literature at the intersection of macroe-conomics and banking that tries to understand optimal regulation of banks in a quanti-tative general equilibrium framework.5 Our modeling approach draws on recent workthat analyzes the role of financial intermediaries in the macroeconomy.6 These papersstudy economies with assets that investors can only access through an intermediary, asin our paper. By introducing limited liability and deposit insurance, and by defining therole of banks as liquidity producers, we bridge the gap to a long-standing microeconomicliterature on the function of banks in the economy.7

Our goal is to quantify the unintended consequences of regulating commercial banksfor financial stability and macroeconomic outcomes. Other papers have addressed closelyrelated questions but not in a quantitative setting.8 More broadly, the role of shadowbanks in the recent financial crisis has motivated a number of papers that propose the-ories why the shadow banking system emerged and why it can become unstable (e.g.,

4https://www.minneapolisfed.org/publications/special-studies/endingtbtf5E.g. Begenau (2018), Christiano and Ikeda (2014), Elenev, Landvoigt, and Van Nieuwerburgh (2018),

Gertler, Kiyotaki, and Prestipino (2016), Davydiuk (2017). Nguyen (2014) and Corbae and D’Erasmo (2017)study quantitative models in partial equilibrium.

6E.g. Brunnermeier and Sannikov (2014), He and Krishnamurthy (2013), Garleanu and Pedersen (2011),Moreira and Savov (2017).

7For an overview of standard microeconomic models of banking see Freixas and Rochet (1998). Recenttheoretical contributions with a focus on the role of bank capital include Malherbe (2015) and Harris, Opp,and Opp (2017).

8E.g., Plantin (2015), Huang (2016), Ordoñez (2018), Xiao (2018), Martinez-Miera and Repullo (2017).

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Gennaioli, Shleifer, and Vishny (2013) and Moreira and Savov (2017)). Shadow banks areoften viewed to emerge in response to tighter financial regulation, for example by provid-ing liquid-yet-fragile securities off-balance sheet (e.g., Plantin (2015); Huang (2016); Xiao(2018)). Or they emerge because they produce financial services using a different technol-ogy compared to traditional banks (e.g., Gertler, Kiyotaki, and Prestipino (2016); Ordoñez(2018); Martinez-Miera and Repullo (2017)). Our paper captures both views as shadowbanks can exist independently of how tightly the traditional banking sector is regulated.Yet tighter financial regulation can make shadow banking more attractive and lead to anexpansion of the shadow banking system.9

A key difference to other quantitative work is that we explicitly model moral haz-ard arising from deposit insurance (and more generally government guarantees) akin toBianchi (2016).10 In addition, we account for a key institutional feature of financial in-termediaries by modeling them with limited liability. Hence our setup allows to studywelfare-improving bank regulation in a quantitative framework. Since our focus is onliquidity provision as a fundamental role of banking, we also relate to the literature onthe demand for safe and liquid assets,11 and on the role of financial intermediaries inproviding such assets.12 Pozsar, Adrian, Ashcraft, and Boesky (2012), Chernenko andSunderam (2014), Sunderam (2015), Adrian and Ashcraft (2016), among others, are recentempirical papers documenting the role of shadow banks for liquidity creation.

2 Simple two-period model

Before laying out the dynamic model in Section 3, we build intuition for its main mech-anisms in a simplified static version that we can solve by hand. This version keeps themodel’s essential features, but abstracts from some of its more complicated details suchas shadow bank runs that are necessary to generate a good quantitative fit. The key fea-tures of our model is that two different types of banks provide valuable liquidity servicesto households by issuing debt. Both bank types are risky, and thus can default. But com-mercial bank debt is insured and thus safe for households, while shadow bank debt is

9The paper by Buchak, Matvos, Piskorski, and Seru (2018) empirically estimates how much of the risein shadow banking, most notably Fintech firms, is due to a change in the regulatory system or a change intechnology.

10In contrast to Bianchi (2016), our focus is not on whether the government guarantee itself is optimal.11E.g. Bernanke (2005), Caballero and Krishnamurthy (2009), Caballero, Farhi, and Gourinchas (2016),

Gorton et al. (2012), Krishnamurthy and Vissing-Jorgensen (2012)12There is a large theoretical literature on this subject with seminal papers by Gorton and Pennacchi

(1990) and Diamond and Rajan (2001).

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not. Two equilibrium conditions determine the relative size of the shadow banking sectoras a function of shadow banks’ relative profitability vis-a-vis commercial banks and theleverage of shadow banks as a function of the marginal liquidity value of its debt. In-creasing the capital requirement for commercial banks increases the relative profitabilityof shadow banks, which leads to an expansion of the shadow banking sector. The effecton shadow bank leverage, whether positive or negative, does however depend on house-holds’ preference for liquidity. Ultimately, this leads us to the quantitative part of ourpaper.

2.1 Setup

Production Technology and Preferences. Time is discrete and there are two dates, times0 and 1. The economy is populated by two types of banks, C-banks and S-banks, andhouseholds. Banks can buy capital at time 0 that trades at price p in a competitive market.Each unit of capital produces one unit of the consumption good at time 1. The total supplyof capital is fixed at unity. Banks are financed with equity and uncontingent debt, issuedto households.

Households are endowed with 1 unit of the capital good at time 0. We assume thathouseholds are less efficient at operating the capital stock than banks, in the sense thatwhen owned by households each unit of capital will only produce ζ << 1/2 units ofoutput at time 1. Households have preferences over consumption at time 0, and overconsumption and liquidity services at time 1. We assume that liquidity services are de-rived from holding the liabilities issued by banks

U = C0 + β (C1 + ψH(AS, AC)) , (1)

where H(AS, AC) is the utility13 from liquidity, and Aj, j = S, C is the quantity of depositsof bank type j held by households. The parameter ψ governs the weight of liquidityservices relative to numeraire consumption.

Financial Technology. S-banks and C-banks issue debt Bj and equity shares Sj, j = C, S,in competitive markets to households. Debt trades at prices qj and equity at pj.

13We model liquidity preferences akin to the classic money literature that elicits a demand for money viaa money in the utility function specification (see Poterba and Rotemberg (1986)). Feenstra (1986) showedthat the reduced-form preference specification is functionally equivalent to microfounding a demand formoney with transaction costs.

6

Both types of banks have limited liability and make optimal default decisions at time1. In case of default, bank equity becomes worthless. Debt issued by C-banks is perfectlysafe for households. That is, when the C-bank’s equity is insufficient to pay back itsdepositors in full, the government makes up the shortfall to depositors, and raises thefunds through lump-sum taxes on households.

In contrast, debt issued by S-banks is risky. When a S-bank defaults with insufficientequity to pay out its depositors in full, depositors lose all deposits they held with thedefaulting bank.

2.2 Bank Problem

S-banks. The problem of an individual S-bank at time 0 is to choose how much capital tobuy, KS, and how many deposits to issue, BS. At time 1, the capital produces consumptiongoods that the bank sells to households. Further, the bank needs to pay out its depositorsif it does not default. Individual banks receive idiosyncratic production shocks ρS at time1 that are distributed i.i.d uniform on support [0, 1], such that the total payoff of its capitalat time 1 is ρSKS. Each S-bank maximizes its expected net present value

maxKS≥0,BS≥0

qS(BS, KS)BS − pKS︸ ︷︷ ︸equity raised at t = 0

+β max ρSKS − BS, 0︸ ︷︷ ︸dividend paid at t = 1

.

At time 0, the bank needs to pay pKS to purchase its capital and raises qSBS in deposits.It needs to raise the difference in initial equity from households. At time 1, the capitalproduces ρSKS consumption goods that banks sell to households at a price of one. Theyfurther pay off their deposits at unit face value. The difference is the dividend they payto their equity owners, the households. Due to the default option, the dividend cannot benegative. Banks discount time 1 payments using the discount factor of households (theirshareholders), which, given household preferences in (1), is simply β.

The price at which individual S-banks can issue deposits, qS(BS, KS), depends on theportfolio choice of the bank. Households take into account that the probability of defaultdepends on the bank’s leverage. The bank in turn internalizes this effect of its leveragechoice on its deposit price.

C-banks. The problem of C-banks is analogous to that of S-banks, with two importantdifferences. First, C-banks issue safe deposits due to the government guarantee (depositinsurance). Hence the price at which they raise deposits, qC, is not sensitive to the port-

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folio choice of the individual bank. Secondly, C-banks are subject to a regulatory capitalconstraint that limits the amount of deposits they can issue to a fraction 1− θ of the cap-ital they purchase; put differently, C-banks face an equity capital requirement of θ. Weformulate the capital requirement in terms of the expected payoff E(ρCKC), which is 1

2 KC

since ρC ∼Uniform[0, 1]. C-banks solve the problem

maxKC≥0,BC≥0

qCBC − pKC + βmax ρCKC − BC, 0 ,

subject to the equity capital requirement

BC ≤12(1− θ)KC.

Bank Size and Leverage Choices. Banks make two choices: (1) how much capital tobuy (size), and (2) how much debt to issue (leverage). Defining bank leverage

Lj =Bj

Kj,

for j = S, C, we can write the payoff in period 1 as

max

ρjKj − Bj, 0= Kj

(1− Fj(Lj)

) (E(ρj |ρj > Lj)− Lj

).

In other words, the dividend is the expected dividend per unit of capital conditionalon the bank having survived, scaled by the capital stock. Since ρC ∼Uniform[0, 1], thissimplifies to

12

Kj(1− Lj

)2 ,

and the default rate of banks of type j is given by Lj. The output of defaulting banksis destroyed in bankruptcy, i.e., depositors of S-banks recover zero, and the governmentneeds to “bail out” the full face value of defaulting C-banks’ deposits.

We exploit the fact that the problem of banks is homogeneous in capital to separatethe problem of each bank into a leverage and a size choice. For C-banks, the leverageproblem is

vC = maxLC∈[0,1]

qCLC − p + β12(1− LC)

2 (2)

subject to

LC ≤12(1− θ) ,

8

and for S-banks it is

vS = maxLS∈[0,1]

qS(LS)LS − p + β12(1− LS)

2 . (3)

The capital purchase decision for each bank is then given by

maxKj

Kjvj, (4)

subject to Kj ≥ 0.

2.3 Households

Households are endowed with one unit of capital. They optimally sell this capital tobanks at price p. They buy deposits Aj and equity shares Sj of bank type j at time 0, suchthat their time 0 budget constraint is

C0 = p− qS AS − qC AC − pSSS − pCSC. (5)

The time-1 consumption is therefore

C1 = (1− LS)AS + AC +12

SSKS (1− LS)2 +

12

SCKC (1− LC)2 − T, (6)

where T denotes government lump-sum taxation to bail out deposits, i.e., T = LCBC, (1−LS)AS and AC are deposit redemptions for S- and C-banks, respectively, and 1

2 SjKj(1− Lj

)2

denotes the expected cash-flow from owning bank type j equity.

Households choose C0, C1, Sj, and Aj, j = C, S to maximize utility (1) subject to con-straints (5) and (6).

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2.4 Equilibrium Definition

Definition. Market clearing requires

SS = 1

SC = 1

AC = BC

AS = BS

1 = KS + KC.

By Walras law, consumption at time 0 is14

C0 = 0, (7)

and consumption at time 1 is

C1 = KC (E(ρC)− F(LC)E(ρC |ρC < LC)) + KS (E(ρS)− F(LS)E(ρS |ρS < LS))

=12

(1− KCL2

C − KSL2S

). (8)

The resource constraint for period-1 consumption (8) clarifies the fundamental trade-offof the model. If banks did not issue any debt, then LC = LS = 0 and household consump-tion of the numeraire good would be maximized at the full payoff of capital, E(ρj) = 1/2.However, in that case banks would produce no liquidity services from which householdsalso derive utility. To produce liquidity services, banks need to issue debt and take onleverage, which causes a fraction Lj of them to default. In the process, some payoffs ofthe numeraire good, KjLj, are destroyed.

2.5 Equilibrium Characterization

Now, we are ready to describe how the equilibrium in this model works. We are par-ticularly interested in the leverage choice of banks as this determines the equilibriumresponse to higher capital requirements.

14The funds households spend on their portfolio of bank securities, qC AC + pCSC + qS AS + pSSS, areequal to the market value of the capital they sell to banks in equilibrium, p.

10

Household Optimality. Before taking the first-order conditions for households’ bondpurchases, we denote the partial derivatives of the liquidity utility with respect to thetwo types of liquidity as

HC(AS, AC) =∂H(AS, AC)

∂AC,

HS(AS, AC) =∂H(AS, AC)

∂AS.

Using this notation, the household’s first-order condition for C-bank debt is

qC = β(1 + ψHC(AS, AC)). (9)

The FOC for S-bank debt is

qS = β(1− LS + ψHS(AS, AC)). (10)

Equation (9) states that the bond price qC must equal the expected discounted bond pay-off, β, plus the discounted marginal liquidity benefit βψHS(AS, AC). Since C-bank debtis insured, the expected payoff is unaffected by C-bank default and hence risk-free. Incontrast, equation (10) shows that the expected discounted payoff to uninsured S-bankdebt is β(1− LS), reflecting that in expectation a fraction LS of S-banks defaults (with arecovery value of zero).

S-bank Optimality. Each individual S-bank recognizes that the price of its debt is afunction of its leverage according to households’ valuation in (10). However, S-banks donot internalize the effect of their leverage choice on the aggregate marginal benefit of S-bank liquidity ψHS(AS, AC). The following proposition characterizes the leverage choiceand debt price of S-banks.

Proposition 1. The price of S-bank debt is

qS = β, (11)

and S-bank leverage is equal to the aggregate marginal benefit of S-bank liquidity

LS = ψHS(AS, AC). (12)

Proof. See appendix A.

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Proposition 1 states that S-bank debt is priced in equilibrium like riskfree debt withno liquidity benefit. This is because S-banks optimally choose leverage such that themarginal benefit of S-bank liquidity to households, on the RHS of (12), is equal to themarginal loss due to defaulting S-banks (LS on LHS).

S-bank’s capital valuation. The constant returns assumption in the scale decisions ofbanks in (4) implies that in equilibrium, S-banks must have zero expected value, i.e.KSvS = 0. The condition for an equilibrium with an S-bank sector, in the sense of KS > 0,therefore requires that the per-capital profits in (3) are zero:

vS = 0,

which leads to the zero-profit condition

p− qSLS = β12(1− LS)

2 . (13)

Condition (13) states that the initial equity required to start an S-bank on the left, per unitof capital acquired by S-banks, must be equal to the expected payoff to equity holders onthe right. Since qS = β by proposition 1, we get the following equilibrium restriction onthe capital price

p =12

β(1 + L2S). (14)

Equation (14) states that S-bank demand for capital is perfectly elastic at a price p, thatincorporates both the direct expected payoff of a unit of capital, 1

2 β, and the benefit eq-uity owners derive from debt financing, 1

2 βL2S. This benefit stems from the fact that the

marginal benefit of S-bank liquidity for households (see equation (12)) lowers the debtfinancing costs of S-banks. The liquidity demand of households leads to an increase inthe value of capital in equilibrium.

C-bank Optimality. The following proposition characterizes C-bank leverage and debtpricing:

Proposition 2. If there is a positive marginal benefit of C-bank liquidity, ψHC(AS, AC) > 0, theC-bank leverage constraint is always binding

LC =12(1− θ), (15)

12

and C-banks debt sells at a higher price than S-bank debt

qC − qS > 0. (16)

Proof. See appendix A.

Since C-banks can issue insured debt that also generates utility for households, thereis no interior optimum to their capital structure choice. They issue debt at a price that isstrictly greater than its expected payoff β due to the liquidity premium it earns.

Analogous to S-banks, the scale invariance of the C-bank problem requires that KCvC =

0. Any equilibrium with a C-bank sector (KC > 0) thus requires vC = 0, or

p− qCLC = β12(1− LC)

2 . (17)

Combining this condition with the price of C-bank debt required by HH optimization in(9) gives

p =12

β(1 + L2C) + βψLCHC(AS, AC). (18)

C-bank’s capital valuation. Comparing the “zero-profit” condition for C-banks in (18)to that of S-banks in (14) demonstrates how the different debt financing costs for bothbanks affect their demand for capital. Like S-banks, C-banks value capital at the expectedpayoff 1

2 β plus an additional term that represents the advantage of debt funding due tothe liquidity benefit households derive from holding debt, 1

2 βL2C. Note that this advantage

of debt arises even in absence of deposit insurance. C-banks assign additional value to debtfunding, since their debt is insured by the government and thus its price is insensitive toC-bank default risk, βψLCHC(AS, AC). This additional debt advantage increases C-bankdemand for capital that serves as collateral for debt.

Equilibrium determination. To characterize the leverage of both banks and their rela-tive market shares in equilibrium, we assume that the liquidity function has the constantelasticity of substitution (CES) form

H(AS, AC) = (αAεS + (1− α)Aε

C)1/ε , (19)

where α ∈ [0, 1] is the weight on S-bank liquidity and 1/(1− ε) is the elasticity of sub-stitution between both types of liquidity. Since H is homogeneous of degree one, the

13

marginal liquidity benefit functions are homogeneous of degree zero and only depend onthe ratio of S-bank to C-bank debt RS = AS

AC, i.e. Hj(AS, AC) = Hj(RS) for j = S, C. Fur-

ther, the function H has diminishing returns in each type of debt, such that H′S(RS) < 0and H′C(RS) > 0 for RS ∈ (0, ∞). We can characterize the equilibrium as a system of twoequations in the two variables LS and RS, which are S-bank leverage and debt ratio. Weobtain the first equation by applying the definition of the debt ratio to the condition foroptimal S-bank leverage, (12):

LS = ψHS (RS) . (20)

We obtain the second equation by equating the capital demand conditions for S-banks(14), and C-banks (18). After simplifying, we get the zero-profit condition

LS =

(14(1− θ)2 + ψ(1− θ)HC (RS)

)1/2

. (21)

Equations 20 and 21 define two functions mapping RS ∈ [0, ∞) into S-bank leverageLS ∈ [0, 1]. The range of values for both functions is bounded below by zero since themarginal liquidity benefit is weakly positive and θ ≤ 1. The upper bound is one, sincethe idiosyncratic shocks ρS are uniformly distributed over [0, 1], and the probability ofdefault of the S-bank is equal to its leverage LS. Denote by R0

S the debt ratio such thatthe leverage condition implies 1 = ψHS

(R0

S). Further, denote by L0

S the leverage ratioimplied by the zero-profit condition at this point, i.e. L0

S = 14(1− θ)2 + ψ(1− θ)HC

(R0

S).

Since equation (20) defines a decreasing function and (21) an increasing function, a uniqueequilibrium with S-bank leverage LS ∈ [0, 1] exists if L0

S < 1. This condition is satisfiedfor a large range of parameter values.

Graphically, the equilibrium is at the intersection of both curves in (RS, LS) space, il-lustrated in Figure 1.

The shape of both curves is a consequence of decreasing returns to scale in each typeof liquidity. The upward-sloping (red) curve is the zero-profit condition in (21). Thecondition says that in equilibrium, equity owners must be indifferent between owningS-banks and C-banks. Thus, S-banks need to generate the same (zero) profit for per unitof capital as C-banks. Because of diminishing returns, the marginal benefit of C-bankdebt, HC (RS), is decreasing in the amount of C-bank liquidity provided. This means thatC-bank profitability is increasing in S-bank ratio RS. Hence, to “keep up” with C-banks,the zero-profit condition for equity owners requires that S-banks increase their leverageas their capital share rises along the curve. Basically, S-banks need to lever up to boosttheir return to equity.

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Figure 1: Equilibrium Existence and Comparative Static in θ

1

𝐿𝑆

𝑅𝑆

𝐿𝑆0

𝑅𝑆0

1

𝐿𝑆

𝑅𝑆𝑅𝑆0

𝐿𝑆0

Left panel: The unique equilibrium S-bank leverage and debt ratio satisfy optimal leverage (blue) and zero-profit (red) conditions. Right panel: Higher capital requirement causes a downward shift in the zero-profitcondition, implying lower S-bank leverage and greater share.

The downward-sloping (blue) curve is the optimal leverage condition (20). The con-dition says that S-bank leverage in equilibrium is equal to the marginal benefit of S-bankdebt, HS (RS). For given leverage, this marginal benefit is decreasing in S-bank ratio RS,again because of diminishing returns.

2.6 The effect of C-bank capital requirements

We can now ask how higher capital requirements affect the economy. To this end, westudy the comparative static of the equilibrium characterized by equations (20) and (21)with respect to θ, as shown by the right-hand panel of Figure 1 and formalized in proposi-tion 5 in Appendix A. Raising θ leads to a drop in shadow bank leverage and an expansionin the shadow banking share.

The zero-profit condition (the red curve) shifts down because a rise in θ decreases theprofitability of C-bank equity (for a given level of KS), as C-banks are now further re-stricted in their ability to take advantage of cheap deposit funding.15 In equilibrium, this

15An increase in θ reduces the amount of C-bank debt for given KS. With decreasing returns, this causes arise in the marginal benefit of C-bank debt, which by itself makes issuing debt more attractive for C-banks.However, a higher θ also directly lowers the fraction of debt funding C-banks can use per dollar of capital.The second effect dominates. This can be seen analytically by inspecting the RHS of equation (21), whichrepresents the benefit of debt funding to C-bank equity owners.

15

means that S-bank equity also needs to be less profitable. However, after the increasein θ and at the old S-bank share, S-bank equity is now more profitable than C-bank eq-uity. Hence households (equity holders) invest in new S-banks, which results in a greaterquantity of S-bank deposits produced at constant leverage. As the quantity of S-bank debtrises, the marginal liquidity benefit of S-bank debt declines and S-banks reduce leverage.The declining benefit requires S-banks to pay higher interest on their deposits, which inturn lowers the profitability of S-bank equity until equilibrium is restored.

To summarize, higher capital requirement shrink the C-bank sector and expand theS-bank sector. The policy makes the S-bank sector initially relatively more profitable byreducing C-banks’ access to subsidized deposit funding. In that sense, our model’s pre-diction lines up with the intuitive reason why regulators fear that tighter capital require-ments will cause substitution towards shadow banking. However, our simple model alsoclarifies that shadow bank incentives for risk taking in the form of higher leverage do notnecessarily rise with tighter regulation.

Welfare. To understand under which conditions higher capital requirements improvewelfare, we first solve for the optimal allocation of capital and leverage of each type ofbank from the perspective of a social planner that maximizes household welfare. Theplanner is restricted to the same resource constraint as the decentralized economy, and hasto use the same risky intermediation technology to produce liquidity services. Therefore,numeraire consumption in periods 0 and 1 is restricted by the resource constraints ofthe decentralized economy in equations (7) and (8). The liquidity production technologyimplies

AS = LSKS and (22)

AC = LCKC = LC(1− KS). (23)

Therefore the planner’s optimization problem is

maxKS,LS,LC

12

(1− (1− KS)L2

C − KSL2S

)+ ψH (LSKS, LC(1− KS)) . (24)

Proposition 3 characterizes the solution to this problem.

16

Proposition 3. The optimal ratio of S-bank and C-bank capital is given by

KS

KC=

(α

1− α

) 11−ε

.

Optimal leverage is equalized across bank types and given by

LS = LC = L∗,

where L∗ is a function of parameters and given in the appendix.

Proof. See appendix A.

Since both banks have equally good technologies for producing liquidity of their owntype for a given unit of capital, the planner chooses equal leverage for both. Further, thisoptimal leverage is equal to the marginal utility of liquidity for each type. The alloca-tion of capital reflects the weight each type of liquidity receives in the utility function. Ahigher elasticity of substitution 1/(1− ε) “tilts” the optimal allocation towards the banktype that receives a greater weight in the utility function. Two key properties of the de-centralized equilibrium highlight the difference to the planner allocation.

Proposition 4. Index competitive equilibria by the factor m > −1, such that

LC = (1 + m)ψHC(RS),

and the function θ = f (m) that determines the value of θ implementing equilibrium m.

(i) There is no θ ∈ [0, 1] that implements the planner allocation from Proposition 3.

(ii) In any equilibrium with m ≥ 0, an increase in the capital requirement θ is welfare-improving.

Proof. See appendix A.

The factor m is the wedge between the social marginal benefit of C-bank liquidityψHC(RS), and the cost to society of producing this liquidity LC. The social planner solu-tion requires that m = 0, such that cost and benefit are equal. In the competitive equilib-rium, the C-bank constraint is always binding and consequently LC = (1− θ)/2. A highvalue of m, resulting from a low capital requirement, implies that C-banks overproduce

17

liquidity in the sense that LC > ψHC(RS). This is caused by the combination of limitedliability and deposit guarantees, which lead to excessive C-bank leverage.16

What if the regulator chooses θ = f (0), such that LC = ψHC(RS) and C-banks pro-duce liquidity services efficiently? Part (i) of Proposition 4 states that even in such a case,the competitive equilibrium does not achieve overall efficiency. The reason is deposit in-surance for C-banks and competition between S- and C-banks, as formally expressed bycondition (21). Since C-banks can issue insured deposits while S-banks cannot, C-bankshave a competitive advantage. To compensate for this disadvantage, S-banks always op-erate at higher leverage than C-banks, which is incompatible with the planner solution.Thus, the capital requirement θ is sufficient to align C-bank leverage with the social op-timum. However, absent additional policy tools to “regulate” S-banks, it cannot achieveoverall efficiency.

Nonetheless, part (ii) of Proposition 4 provides a sufficient condition under which anincrease in θ improves welfare. In any equilibrium with m ≥ 0, C-banks (weakly) over-produce liquidity, and raising θ will shrink the wedge m towards zero. Even at m = 0, amarginal increase in θ is still unambiguously welfare-improving. The reason is once morecompetition between both types of banks. As a result of C-banks’ competitive advantage,S-banks’ market share is too small relative to the planner allocation in the decentralizedequilibrium. The analysis in Figure 1 shows that raising θ will cause an expansion in theS-bank share, moving the allocation of capital closer to the planner solution.

Decreasing returns in liquidity. In anticipation of our quantitative results, we general-ize the liquidity function to allow for decreasing returns in overall liquidity

H(AS, AC) =

(αAε

S + (1− α)AεC) 1−γH

ε

1− γH, (25)

parameterized by γH ≥ 0. The marginal liquidity benefit of S-bank and C-bank liquidityare given by

ψHj(AS, AC) = ψHj(RS)((αAε

S + (1− α)AεC)

1ε

)−γH, (26)

16We take these basic institutional features as given, with the reasons for their existence outside themodel. In the quantitative version of the model, we take into account that S-banks face the risk of largewithdrawals (banks runs) due to the lack of deposit insurance.

18

for j=S, C, respectively. For γH = 0, we get the CES case with constant returns in (19),

whereas γH > 0 yields decreasing marginal returns in total liquidity(αAε

S + (1− α)AεC) 1

ε ,as can be seen from the rightmost term in conditions (26).

Analytical characterization of equilibrium in leverage LS and debt ratio RS is no longertractable for this case. However, we can still express equilibrium as two nonlinear func-tions in two unknowns, leverage LS and S-bank assets KS. Market clearing and a bindingC-bank leverage constraint imply that AS = LSKS and AC = 1

2(1− θ)(1− KS), yieldingthe two equations

LS = ψHS

(LSKS,

12(1− θ)(1− KS)

), (27)

LS =

(14(1− θ)2 + ψ(1− θ)HC

(LSKS,

12(1− θ)(1− KS)

))1/2

. (28)

The interpretation of (27) and (28) is analogous to (20) and (21), with the asset share KS

replacing the debt ratio RS. We solve the system (27) – (28) numerically. For a wide rangeof parameter values, allowing decreasing returns leaves the basic properties of equilib-rium unchanged.17 However, the effect of an increase in the capital requirement θ is nowambiguous: as with γH = 0, a higher capital charge reduces C-bank liquidity, whichin turn lowers the marginal benefit of S-bank liquidity. But with γH > 0, the reduc-tion in C-bank liquidity also causes a rise in the marginal benefit of aggregate liquidity(αAε

S + (1− α)AεC) 1

ε , effectively raising demand for both types of liquidity. If the secondeffect is strong enough, it can dominate the first effect and both the leverage and zero-profit schedule in Figure 1 shift upwards. As a result, both equilibrium S-bank leverageand share increase. More generally, the model with γH > 0 can generate any combinationof S-bank leverage and asset share changes in response to a higher capital requirement,depending on parameter values. The response of the financial sector to changes in reg-ulation is therefore a quantitative question in the model with more general preferencesover liquidity.

3 The quantitative model

Building on the static model, this section presents the quantitative model that we take tothe data.

17For very large values of γ, the zero-profit condition becomes non-monotonic and an equilibrium withboth sectors holding a positive amount of capital may not exist.

19

Time is discrete and infinite. The household sector owns a Lucas tree (non-bank de-pendent sector) and all claims on banks. Households value consumption and liquidityservices according to the utility function

U(

Ct, H(

ASt , AC

t

))=

C1−γt

1− γ+ ψ

([α(AS

t )ε + (1− α)(AC

t )ε] 1

ε

)1−γH

1− γH, (29)

where γ is the inverse of the intertemporal elasticity of substitution for consumption. Thepreferences for liquidity are the same as in the simple model (equation (25)), with themeaning of liquidity parameters ψ, α, γH, and ε explained in section 2.

Two types of banks, C-banks and S-banks, provide liquidity services to householdsby issuing deposits under limited liability. Banks operate a Cobb-Douglas productiontechnology combining capital and labor. As in the simple model, C-banks have capitalrequirements and deposit insurance, while S-banks have neither.

To move closer to the data, we introduce capital- and investment adjustment costs, aswell as stochastic deposit redemption shocks (bank runs) for S-banks. The main trade-offin the quantitative model is between liquidity provision and financial fragility as dis-cussed in section 2.

3.1 Production Technology

There is a continuum of mass one of each type of bank, j = C, S. Each bank owns produc-tive capital K j

t at the beginning of the period. Even though banks receive idiosyncraticproductivity shocks, we can solve the problem of a representative bank of each type. Weprovide details on banks’ intertemporal optimization problem and aggregation in sec-tions 3.2 and 3.3 below.

Banks hire labor N jt from households at competitive wage wt and combine it with their

capital to produceY j

t = Zt(Kjt)

1−η(N jt )

η,

where η is the labor share and Zt is an aggregate productivity shock common to all banks.After production, capital depreciates at rate δK. Banks can also invest using a standardconvex technology. Creating I j

t units of the capital good requires

I jt +

φI

2

(I jt

K jt

− δK

)2

K jt

20

units of consumption. Capital trades in a competitive market among all banks at price pt.Defining the investment rate ij

t = I jt /K j

t and the labor-capital ratio njt = N j

t /K jt, the gross

payoff per unit of capital is

Πjt = Zt(n

jt)

η + (1− δK)pt − wtnSt + ij

t(pt − 1)− φI

2(ij

t − δK)2

= (1− η)Zt(njt)

η + pt − δK +(pt − 1)2

2φI, (30)

which already embeds banks’ optimal labor demand and investment decisions. House-holds can also hold capital and produce directly. However, they have an inferior abilityto operate the asset, leading to lower productivity Zt < Zt and a higher depreciation rateδK > δK. They further do not have access to an investment technology. When householdshold capital and produce, the gross payoff per unit is hence

ΠHt = (1− η)Zt(n

Ht )

η + pt(1− δK). (31)

3.2 S-banks

In addition to labor input and investment, each period S-banks choose the amount ofcapital to purchase for next period KS

t+1 and the amount of deposits to issue to householdsBS

t+1 at price qSt .

Bank Runs and Timing To capture the fragility of S-banks, we introduce bank runsin the S-bank sector similar to Allen and Gale (1994). A fraction of shadow bank deposits$t is withdrawn early within a given period (affecting all shadow banks equally), where$t ∈ 0, $∗ > 0, following a two-state Markov chain. When deposits are withdrawn,shadow banks need to liquidate a fraction of their assets to pay out depositors. Assetsthat are liquidated early are sold to households at price ΠH

t defined in equation (31) anddo not yield any output to the bank. Households sell the assets again in the regular capitalmarket later in the same period.18

The timing of decisions within each period is now as follows:

1. Aggregate productivity shocks Zt, Zt and the early withdrawal shock $t are realized.

18Since both transactions take place within the same period and households are unconstrained, it imme-diately follows that ΠH

t is the marginal value households attach to capital. The marginal product of capitalto households is always lower than that of banks, so households never optimally own any capital at theend of the period.

21

2. If $t = $∗, S-banks sell capital worth $∗BSt to households at price ΠH

t .

3. Production of all banks and households and investment decisions of banks ensue.

4. Idiosyncratic payoff shocks of banks are realized. Default decisions.

5. Trade occurs in asset markets. Surviving banks pay dividends and new banks areset up to replace liquidated bankrupt banks.

6. Households consume.

To pay out its depositors in case of a withdrawal shock ($St = $∗) at stage 2, the fraction

of assets that needs to be liquidated is

`St =

$St BS

t

KSt ΠH

t.

Thus, the capital available for production at stage 3 is KSt = (1− `S

t )KSt .

S-banks In appendix B.1, we show that at the time banks choose their new portfo-lio (at step 5 of the intraperiod sequence of events), all banks have the same value andface the same optimization problem. The two properties of the bank problem sufficientto obtain this aggregation result are that (i) idiosyncratic profit shocks ρS

t ∼ FS are uncor-related over time, and (ii) the value function of S-banks is homogeneous in capital. Weuse these properties to write the value of a bank with capital KS

t , VS(KSt , Zt), in terms of

the value per unit of capital vS(Zt) = VS(KSt , Zt)/KS

t . The two intertemporal choices are

the deposit-capital ratio bSt+1 =

BSt+1

KSt+1

and capital growth kSt+1 =

KSt+1

KSt

, which is subject to

quadratic adjustment costs φK(kS

t+1 − 1)2 /2. Defining leverage

LSt =

bSt

ΠSt

,

we write the bank problem as

vS(Zt) = maxbS

t+1≥0,kSt+1≥0

kSt+1

(qS(bS

t+1)bSt+1 − pt

)− φK

2

(kS

t+1 − 1)2

+ kSt+1Et

[Mt,t+1 ΠS

t+1ΩS(LSt+1)

], (32)

where Mt,t+1 is the stochastic discount factor of households. The function ΩS(LSt ), de-

fined in equation (50) in appendix B.1, reflects the bank’s continuation value including

22

the option to default and possible losses from bank runs and depends on the capital struc-ture choice bS

t+1 through leverage LSt+1. Intuitively, banks choose asset growth kS

t+1 andcapital structure bS

t+1 to maximize shareholder value under limited liability. As in thesimple model of section 2, S-banks internalize that the price of their deposits, qS(bS

t+1), isa function of their default risk and thus their capital structure. S-banks optimally defaultat stage 4 in the intraperiod time line when ρS

t < ρSt , with

ρSt =

LSt − (1− `S

t )vS(Zt)

ΠSt− δS

1− `St(1−ΠH

t /ΠSt) , (33)

where δS ≥ 0 is a default penalty parameter. The probability of default is thus FSρ,t ≡

FS (ρSt). We derive Euler equations for S-banks in appendix B.3.

3.3 C-banks and Government

C-banks. The problem of C-banks is analogous to S-banks, but they differ from S-banksin four ways: (i) they issue short-term debt that is insured and risk free from the perspec-tive of creditors, (ii) they do not experience runs (as result of (i)), (iii) they are subject toregulatory capital requirements, and (iv) they pay an insurance fee of κ for each bondthey issue. Using the same notation as for S-banks, C-banks solve

vC(Zt) = maxbC

t+1≥0,kCt+1≥0

kCt+1

((qC

t − κ)

bCt+1 − pt

)− φK

2

(kC

t+1 − 1)2

Et

[Mt,t+1 kC

t+1 ΠCt+1max

ρC

t+1 − LCt+1 +

vC(Zt+1)

ΠCt+1

,−δC

], (34)

subject to the capital requirement

(1− θ)pt ≥ bCt+1. (35)

C-banks optimally default at stage 4 in the intraperiod time line when ρCt < ρC

t , with

ρCt = LC

t −vC(Zt)

ΠCt− δC, (36)

where δC ≥ 0 is a default penalty parameter. The probability of default is thus FCρ,t ≡

FC (ρCt). We state the full optimization problem of C-banks including Euler equations in

appendix B.4.

23

Bankruptcy, Bailout and Government Budget Constraint. If a bank declares bankruptcy,its equity (and continuation value) becomes worthless, and creditors seize all of the banksassets, which are liquidated. The recovery amount per bond issued is

rjt = (1− ξ j)

ρj,−t

(1− `

jt

(1− ΠH

t

Πjt

))Lj

t

,

for j = S, C. A fraction ξ j of assets is lost in the bankruptcy proceedings, with ρj,−t ≡

E(

ρjt | ρ

jt < ρ

jt

)being the average idiosyncratic shock of defaulting banks. Since C-banks

do not experience runs, `Ct = 0 ∀t. Bankruptcy losses ξ jρ

j,−t

(`

jtΠ

Ht +

(1− `

jt

)Πj

t

)K j

t arereal losses to the economy. They reflect both greater capital depreciation of foreclosedbanks, and real resources destroyed in the bankruptcy process that reduce bank profits.

After the bankruptcy proceedings are completed, a new bank is set up to replace thefailed one. This bank sells its equity to new owners, and is otherwise identical to a sur-viving bank after asset payoffs.

If a S-bank defaults, the recovery value per bond is used to pay the claims of bondhold-ers to the extent possible. We further consider the possibility that the government bailsout the bond holders of the defaulting S-bank with a probability πB, known to all agentsex-ante. If a C-bank declares bankruptcy, the bank is taken over by the government thatuses lump-sum taxes and revenues from deposit insurance, κBC

t+1, to pay out the bank’screditors in full. Summing over defaulting C-banks and S-banks that are bailed out, wedefine lump sum taxes as

Tt = FCρ,t

(1− rC

t

)BC

t − κBCt+1 + πBFS

ρ,t

(1− rS

t

)BS

t .

3.4 Households and Equilibrium

Households. Each period, households receive an endowment from a Lucas tree Yt andthe payoffs from owning all equity and debt claims on intermediaries, yielding financialwealth Wt. They further inelastically supply their unit labor endowment at wage wt andpay lump-sum taxes Tt. Households choose consumption Ct, deposits of both banks forredemption next period, AS

t+1 and ACt+1, and bank equity purchases SS

t and SCt , to maxi-

mize utility (29) subject to their intertemporal budget constraint

Wt + Yt + wt − Tt ≥ Ct + ∑j=S,C

pjtS

jt + ∑

j=S,Cqj

t Ajt+1, (37)

24

where pjt, j = S, C, denote the market price of bank equity of type j. As for banks, we state

the full optimization problem of household including Euler equations in appendix B.2.

Equilibrium. The full definition of competitive equilibrium is provided in appendix B.5.Market clearing requires that households purchase all securities issued by banks, whichimplies Bj

t+1 = Ajt+1, for j = S, C, in deposit markets, and Sj

t = 1 in equity markets. Laborsupply by households has to equal labor demand by banks, and by producing householdsin case of fire sales, implying NS

t + NCt + NH

t = 1. The market clearing conditions for cap-ital and consumption are provided in appendix B.5. In the capital market, bank failureslead to endogenous depreciation in addition to production-induced depreciation δK. Sim-ilarly, bank failures also cause a loss of resources in the goods market. Appendix B.6 liststhe full set of nonlinear equations characterizing the equilibrium.

3.5 Stochastic Environment and Solution Method

Stochastic Processes. The stochastic process for the Y-tree (not intermediated by banks)is an AR(1) in logs

log(Yt+1) = (1− ρY)log(µY) + ρYlog(Yt) + εYt+1,

where εYt is i.i.d. N with mean zero and volatility σY. To capture the correlation of asset

payoffs with fundamental income shocks, we model the payoff of the intermediated assetas

Zt = νZYt exp(εZt ),

where εZt is i.i.d. N with mean zero and volatility σZ, independent of εY

t , and νZ > 0 isa parameter. This structure of the shocks implies that Zt inherits all stochastic propertiesof aggregate income Yt and is subject to a temporary shock reflecting risks specific tointermediated assets, such as credit risk.

Solution Method. We solve the dynamic model using nonlinear methods. To this end,we write the equilibrium of the economy as a system of nonlinear functional equations ofthe state variables, with the unknown functions being the agents’ choices, the asset prices,and the Lagrange multiplier on the C-bank’s leverage constraint. We parametrize thesefunctions using splines and iterate on the system until convergence. We check the relativeEuler equation errors at the solution we obtain to make sure the unknown functions are

25

well approximated. We then simulate the model for many periods and compute momentsof the simulated series.

The model features three exogenous state variables, the stochastic endowment Yt, pro-ductivity Zt, and the run shock $t. These shocks are jointly discretized as a first-orderMarkov chain with three nodes for Yt and three nodes for Zt. We assume that runs onlyoccur in low productivity states, yielding a total of 12 different discrete states.

The endogenous state variables are (1) the aggregate capital stock Kt = KCt + KS

t , theoutstanding amount of bank debt of each type (2) BC

t and (3) BSt , and the share of the

capital stock held by S-banks (4) KSt /Kt. Appendix B.7 contains a description of the com-

putational solution method.

4 Mapping the model to the data

In this section, we discuss the parametrization of the model and its fit with the data.

Our calibration relies on various data sources, including the data from bank call reportscollected by the Federal Reserve Board and the Federal Deposit Insurance Corporation,the Flow of Funds, Compustat and NIPA. We match our model to quarterly data from1999 Q1 to 2017 Q4. We choose 1999 as the start date because it marked the passageof the Gramm-Leach-Bliley Act that deregulated the banking sector. For example, thislegislation removed the mandated separation between commercial and investment banks.For some calibration targets, we use longer time series to reduce measurement noise (forexample, if only annual data are available).

We organize the description of our parametrization into four parts: (1) parametersgoverning bank leverage and default, (2) liquidity preference parameters, (3) bank-runrelated parameters, and (4) all remaining parameters. This section discusses how weselect the first three sets of parameters (see Table 1), while Appendix C.1 contains thedescription of the remaining parameter choices.

Bank leverage and defaults. Banks have the option to default. The default penaltiesδj with j ∈ C, S determine the default threshold. Typically the default threshold isassumed to be zero with the reasoning that default occurs whenever equity holders arewiped out. Mapping this concept precisely to the data is difficult because the distressedfirm’s franchise value is often difficult to measure. We therefore choose to set δC and δS tomatch the default rates of the assets held by commercial and shadow banks, respectively.

26

Table 1: Parametrization

Values Target Data Model

Bank leverage and defaultδS 0.300 Quarterly corp. bond default rate 0.36% 0.31%δC 0.175 Quarterly net loan charge-offs 0.25% 0.26%ξC 0.515 Recovery rate Moody’s 63% 62%ξS 0.415 Recovery rate Moody’s 63% 63%πB 0.905 Shadow bank leverage 93% 93%

Liquidity preferencesβ 0.993 C-bank debt rate 0.39% 0.39%α 0.330 Shadow banking share; Gallin (2015) 35% 34%ψ 0.0103 Liquidity premium C-banks; KV2012 0.18% 0.17%γH 1.700 Corr(GDP, C-bank liquid. premium) -0.28 -0.39ε 0.420 S-bank liquidity elasticity 0.17% 0.16%

RunsδK 0.10 Max. haircut non-subprime; Gorton and Metrick (2009) 20% 19%Z 26%× Z Real asset foreclosure discount; Campbell et al. (2011)$ [0, 0.3] Fraction of households run; Covitz et al. (2013)

Prob$

[0.97 0.030.33 0.67

]Uncond. run prob. 3% 3%

In case of δC, we match the charge-off rate on loans held by commercial banks, and forthe case of δS, we match the default rate on corporate bonds.19

The bankruptcy costs parameters ξ j with j ∈ C, S determine how much of banks’asset value can be recovered to pay out their creditors in case of default. We choose theseparameters to match Moody’s financial sector corporate bond recovery rate of 63%.20

The shadow bank bailout probability πB affects the optimal leverage of S-banks. Ahigh value of πB means that a large fraction of S-bank debt is insured. For this reason,creditors do not fully price the default risk of S-banks, lowering S-banks’ incentives tointernalize default costs. S-banks can then increase their equity valuation by increasingtheir leverage. We choose πB to match the book value leverage of shadow banks in Com-

19Loan charge-offs are from Federal Reserve Board data, and corporate bond defaults are from the S&P2017 Annual Global Corporate Default Study. We use 1985-2017 data for both targets, the longest availableseries. Since our model consolidates bank-dependent producers and banks, it does not explicitly includecorporate loans or bonds. To capture the fundamental source of risk from the intermediation activity of C-and S-banks, we calibrate the default rates of both types to the risk of the key assets they own.

20We use Moody’s 1984-2004 reports. Exhibit 9 in the report presents the recovery rates of defaultedbonds for financial institutions. We use the average for financial institutions over all bonds and preferredstocks.

27

pustat. To this end, we compute the value weighted leverage of publicly traded shadowbanks as debt over assets weighted by the relative market value of each institution andaverage across time and banks.21 This procedure leads to a value-weighted book leverageratio of 93%.

Liquidity Preferences. The parametrization of households’ liquidity preferences is atthe core of our quantitative model as they admit a range of relationships between com-mercial bank- and shadow bank liquidity services.

Our model determines two interest rates, one for C-banks and one for S-banks, whichcan be understood from the household Euler equations (9) and (10) in the simple model,with their quantitative counterparts (56) and (57) in appendix B.2. Both rates are affectedby the representative consumer’s stochastic discount factor, and both contain a liquiditypremium. In addition, the S-bank rate reflects the default risk of S-banks.

To compute the liquidity premium in our model, it is useful to define the price of ahypothetical asset

qt = Et Mt,t+1 ,

which is a short-term riskfree bond without any liquidity benefits.

The most direct measure of the liquidity premium in the model is the marginal benefitof C-bank liquidity qC

t − qt = Et Mt,t+1MRSC,t+1, with MRSC,t+1 as defined in equation(55). The weight on liquidity services ψ in the utility function directly scales this liquid-ity premium. Unfortunately, a measure for qC − q is difficult to obtain in the data, sincealmost all short-term safe interest rates convey some form of liquidity benefit. Krishna-murthy and Vissing-Jorgensen (2012) estimate a liquidity premium of 73 bps per annum(18 bps quaterly) based on the spread between the yield on commercial paper (CP) andTbills for a long time series. We choose ψ such that the marginal value of commercialbank liquidity matches this premium (net of the deposit insurance fee).22 This calibra-tion strategy relies on the assumption that Tbills and commercial bank deposits are closesubstitutes in terms of their liquidity benefits, as has been argued by Nagel (2016) amongothers. It further requires that the CP yield is close to a “pure” riskfree rate without liq-

21We define shadow banks as all institutions with SIC codes 6111-6299, 6798, 6799, 6722, 6726, excludingSIC codes 6200, 6282, 6022, 6199.

22C-banks pay a deposit insurance fee κC for each bond they issue, with more details on the calibrationof κC below. In equilibrium, they pass on this fee to consumers in the form of lower deposit rates. The fee iswhat makes bank deposits free of credit-risk and comparable to Tbills. Further, Tbill investors do not paysuch a fee (directly or indirectly). This is why we match the premium measured by Krishnamurthy andVissing-Jorgensen (2015) for Tbills to MRSC,t − κC in the model.

28

uidity benefits, which is less likely to be true. To the extent that commercial paper alsoprovides liquidity benefits and is closer in spirit to the S-bank interest rate in our model,the calibration of ψ is a lower bound on the level of the liquidity premium of deposits.

The parameter β governs households’ time preferences and the level of the C- and S-bank interest rates for a given liquidity premium. We choose β to match the C-bank ratein the model to the deposit rate in the data, calculated as the total interest expense ondeposits at the end of period t, divided by total amount of deposits at the beginning ofperiod t. We use call report data for this calculation.

The parameter α is the weight on shadow bank liquidity services in households’ pref-erences and therefore governs how much shadow bank debt contributes to aggregateliquidity. This in turn determines the relative size of the shadow banking sector. We cal-ibrate α to the share of shadow bank funding of real production activity, as estimated byGallin (2015).23

The curvature parameter γH determines how the marginal value of liquidity moveswith the total amount of liquidity. When the marginal value of liquidity decreases in theamount of liquidity provision (γH > 0), households how a downward-sloping demandcurve for liquidity. Since good economic times are typically characterized by an abun-dance of liquidity, we can use the correlation between GDP and the liquidity premiumas a target for γH. Consistent with the calibration of ψ discussed above, we define theliquidity premium as the difference between the rate on 3-month AA rated financial com-mercial paper and the rate on 3-month Tbills.24 Over our sample period, the correlation ofreal per capita GDP growth with this liquidity premium is -0.28. We use this value as ourcalibration target. The value of γH = 1.7 minimizes the distance between this data targetand its model analogue (the correlation between GDP growth and the liquidity premiumon C-bank debt).

The parameter ε determines the elasticity of substitution between S-bank and C-bankdebt. To calibrate ε, we target the effect of changes in the supply of S-bank debt on thespread between both interest rates. We derive an equation for the S−C debt spread by log-

23Alternatively, we could have used the share of liquid shadow bank debt (i.e., money market mutualfund shares, REPO funding, and short term commercial paper) relative to the sum of liquid shadow bankdebt and commercial bank deposits. The average share is 38% over our sample period using Flow of Fundsdata and therefore close to the 35% estimate by Gallin (2015).

24Instead of the Tbill rate, we could have directly used the bank deposit rate that we computed as targetfor the C-bank rate. However, the combination of accounting rules and lower reporting frequency meansthat bank accounting data lags behind market data. Any sensitivity measure based on deposits is thereforedownward biased. To avoid this problem, we follow Nagel (2016), who finds deposits and Tbills to be nearperfect substitutes in terms of their liquidity premia, and use the Tbill rate instead of the deposit rate in ourdefinition of the liquidity premium.

29

linearizing households’ combined first-order conditions for liquidity holdings for bothbank types, with details given in Appendix C.2. Based on the equation, imperfect sub-stitutability between both types (ε < 1) implies that an increase in S-bank (C-bank) debtsupply should widen (compress) the spread, everything else equal. The intuition is sim-ple: if both types of liquidity are imperfect substitutes, then the liquidity benefit providedby each type does not only depend on the total amount of debt, but also on the individualsupply of each type. Decreasing returns in each type of liquidity then imply that a greatersupply of S-bank debt will lower the liquidity premium earned by this debt, such that thespread will widen, and vice versa for C-banks. If both types were perfect substitutes, thenthe spread would not depend on the supply of each type of bank, since both banks earn apremium that only depends on the total supply.

Hence, controlling for S-bank default risk, in a regression of the spread on the supplyof each type, the coefficient on each type is informative about ε. We run such a regressionin both real and model-generated data, and set ε to match the coefficient on S-bank debtsupply in the model regression to the corresponding coefficient in the data regression.In the data, we regress the CP−Tbill spread described above on the logarithms of Tbillsupply divided by GDP and a measure of short-term money market debt divided byGDP, while including the VIX to control for default risk. In the model, we run analogousregressions, directly controlling for the S-bank default rate.25

Consistent with our theory, the CP−Tbill spread in the data reacts positively to anincrease in money market debt supply, with a coefficient of 20bp. The model matches thiscoefficient with a value of ε = 0.42, implying a relatively high degree of substitutability.The coefficient means that a one percent rise in the supply of S-bank debt leads to anincrease of the spread by 20bp. Regression results are available in appendix C.2.

Bank runs. The introduction of bank runs to the dynamic model is a significant depar-ture from the simple two-period model outlined in Section 2. Bank runs make shadowbanks more risky compared to commercial banks.

During the run-state the capital stock of affected shadow banks is transfered to house-holds. We assume that the depreciation rate of the capital stock held by households is fourtimes higher than when capital is held by banks, implying a value of run-state depreci-

25Neither the Tbill nor the CP yield are direct measures of C-bank and S-bank interest rates in the senseof our model. However, we already argued above that Tbills are likely close enough substitutes to deposits,and commercial paper is an important intermediate asset in shadow bank liquidity production. Thus webelieve that the sensitivity of the data spread to the respective supply of each security is a good approxima-tion of the sensitivities in our model (even though the level of the CP−Tbill spread is not a good measure ofthe level of the S−C model spread).

30

Figure 2: The effect of Bank Runs on the Economy

0 4 8 12 16 20-5

-4

-3

-2

-1

0

1Z

0 4 8 12 16 20-50

-40

-30

-20

-10

0

10Investment

0 4 8 12 16 20-1

-0.8

-0.6

-0.4

-0.2

0Consumption

0 4 8 12 16 20-4

-3

-2

-1

0Output

0 4 8 12 16 20-30

-25

-20

-15

-10

-5

0Liquidity

0 4 8 12 16 20-60

-40

-20

0

20

40

60DWL

This figure presents the impulse response functions to a productivity shock (in black) and a productivityshock together with a run shock (red). The x-axis denotes quarters. The shocks occur in the first quarter.The y-axis denotes percentage deviations from the stationary equilibrium.

ation of δK = 0.10. When households hold the financial assets of run-impaired shadowbanks the productivity of financial assets falls to 26%, consistent with the foreclosurediscounts documented by Campbell, Giglio, and Pathak (2011). Together this implies amaximum haircut of 20% for non-subprime assets as documented by Gorton and Metrick(2009).26 We set the fraction of households that run in the run-state to 30% (Covitz, Liang,and Suarez (2013)). The probability of entering a bank run state from a non-bank run stateis set to 3% and the probability of staying in a bank run state is 67%, the latter implies thatthe average crisis lasts for about three quarters.

How bad are bank runs for the economy? Figure 2 compares the impulse responsefunctions of key model variables to a typical productivity crisis (in black) with the im-pulse response functions to a productivity crisis coupled with a bank run (in red). Theyshow that a shadow bank run significantly worsens recessions, leading to higher lossesin output, consumption, and investment. This is summarized by 10 percentage points

26See the haircut for non-subprime bonds during the crisis in Figure 2 in Gorton and Metrick (2009).

31

higher deadweight losses. A bank run forces shadow banks to delever, resulting in a liq-uidity crunch. The lower productivity of physical capital during a run reduces the valueof the intermediated assets, making investments less attractive.

5 Effect of Bank Capital Requirements

In this section, we answer two questions. (1) How do static bank capital requirementsaffect the economy, and (2) what is the value of the optimal capital requirement? We laterdiscuss a range of alternative regulatory policies in Section 6.

The optimal static capital requirement. How do higher capital requirements affect liq-uidity provision and do they improve overall financial stability? A safer financial systemis naturally the desired outcome of tighter bank regulation since the 2008 financial cri-sis. But what if tighter bank regulation shifts activity to the shadow banking sector? Weanswer this question by solving our model numerically and simulating the economy for5,000 periods under different levels of commercial bank capital requirements. All otherparameters stay at their benchmark level. The results are in Table 2 and Figure 3.

Figure 3: Welfare in consumption equivalent units

10 15 20 25 30

%

0

0.05

0.1

0.15

Wel

fare

: con

sum

ptio

n eq

uiv.

uni

ts %

This figure plots the welfare gain relative to the benchmark (θ = 10%) in consumption equivalent unitsover a range of different values for θ capital requirements.

We calculate welfare based on households’ value function. Figure 3 shows that theoptimal capital requirement level is around 17%. This level trades off an increase in con-

32

sumption against a reduction in liquidity services (see Table 2). For capital requirementlevels exceeding 17%, the loss in liquidity services exceeds 5%, which outweights theincrease in consumption. This is why welfare declines after 17%. The increase in con-sumption is driven by (i) a reduction in commercial bank defaults that lowers deadweightlosses and (ii) an increase in GDP that is caused by an increase in the capital stock. As inthe simple model, the capital requirement is binding (see equation 15 in Section 2), be-cause deposits are insured and enjoy a liquidity premium. A higher capital requirementtightens this constraint and consequently forces commercial banks to lower their lever-age. While lower leverage makes commercial banks safer, it also restricts their abilityto produce liquidity. The presence of shadow banks means that, in principle, aggregateliquidity would not need to fall when commercial banks lower their supply of liquidityservices. Indeed, shadow banks expand their liquidity production by 6.2% when the cap-ital requirement is increased to 17%. This is driven by an increase in leverage (+.40%) andin the size of S-banks’ balance sheet (+1.2%). However, this increase in S-bank liquidityproduction does not compensate for the loss in C-bank liquidity services. Thus, aggregateliquidity services fall.

Do higher capital requirements make the system overall safer? This question is ofgreat concern to policy makers who worry about the unintended consequences of tighterregulation. Undoubtedly, regulated C-banks get safer with tighter regulation as high-lighted by the reduction in deadweight losses and leverage. The question is rather howshadow banks are going to react. If S-banks are more fragile, an expansion of the shadowbanking sector could undo the gains in financial stability caused by higher restrictionson C-banks’ leverage. Table 2 shows that shadow banks indeed partially fill the void byproviding more liquidity. They do this by both expanding their balance sheet (both ag-gregate capital and the S-bank capital share increase) and increasing leverage. Higherleverage makes S-banks riskier. The deadweight losses caused by S-banks increase by11.7%, which however does not offset the 94.6% reduction in C-bank deadweight losses.Table 2 shows that even though S-banks take up a higher share of financial intermediationactivity and become riskier themselves, the net effect of higher capital requirements stillimproves overall financial stability.

To get a better sense for what happens on the transition to the new capital requirementregime, we plot the transition paths in Figures 4 and 5 of key model variables from thebenchmark capital requirement of 10% to the optimal capital requirement of 17%. Liq-uidity drops sharply the moment the new capital requirement is introduced and slowlyincreases in the following quarters until it reaches its new stationary equilibrium levelthat is 5.2% below the benchmark level. The S-bank debt share transition mirrors aggre-

33

Table 2: Effect of higher capital requirements

Ben

chm

ark

13%

15%

17%

20%

25%

30%

Cap

ital

and

Deb

t

Cap

ital

4.00

5+0

.2%

+0.4

%+0

.6%

+0.9

%+1

.4%

+2.2

%D

ebts

hare

S0.

349

+3.4

%+4

.9%

+6.2

%+8

.1%

+11.

7%+1

6.0%

Cap

ital

shar

eS

0.34

2+1

.1%

+1.0

%+0

.6%

-0.1

%-1

.0%

-1.7

%Le

vera

geS

0.93

3+0

.1%

+0.3

%+0

.4%

+0.6

%+0

.9%

+1.0

%Le

vera

geC

0.90

0-3

.3%

-5.6

%-7

.8%

-11.

1%-1

6.7%

-22.

4%Ea

rly

Liqu

idat

ion

(run

s)0.

004

+0.3

%+0

.5%

+0.8

%+1

.2%

+1.8

%+2

.3%

Pric

es

Dep

osit

rate

S0.

49%

-0.7

%-1

.3%

-2.0

%-3

.0%

-4.8

%-6

.5%

Dep

osit

rate

C0.

39%

-4.1

%-6

.6%

-9.1

%-1

3.0%

-20.

3%-2

8.6%

Con

veni

ence

Yiel

dS

0.23

%+2

.0%

+3.6

%+5

.3%

+8.0

%+1

2.7%

+17.

8%C

onve

nien

ceYi

eld

C0.

31%

+5.2

%+8

.4%

+11.

5%+1

6.4%

+25.

4%+3

6.2%

Wel

fare

DW

LS

0.00

1+5

.2%

+8.5

%+1

1.7%

+16.

7%+2

5.1%

+31.

8%D

WL

C0.

003

-68.

6%-8

6.5%

-94.

6%-9

8.8%

-99.

9%-1

00.0

%G

DP

1.36

5+0

.0%

+0.0

%+0

.1%

+0.1

%+0

.1%

+0.2

%Li

quid

ity

Serv

ices

1.96

9-2

.3%

-3.8

%-5

.2%

-7.3

%-1

0.8%

-14.

4%C

onsu

mpt

ion

1.26

1+0

.13%

+0.1

7%+0

.19%

+0.2

0%+0

.21%

+0.2

4%Vo

l(Li

quid

ity

Serv

ices

)0.

068

-2.5

%-4

.3%

-6.1

%-8

.9%

-13.

6%-1

8.7%

Vol(

Con

sum

ptio

n)0.

005

+0.4

%+0

.5%

+0.6

%+0

.7%

+2.1

%+1

0.5%

Wel

fare

+0.1

06%

+0.1

27%

+0.1

29%

+0.1

16%

+0.0

74%

+0.0

48%

34

Figure 4: Transition path from the benchmark economy to the economy with the optimalcapital requirement: quantities

0 4 8 12 16 200

0.05

0.1

0.15Consumption

0 4 8 12 16 20-6

-5

-4

-3

-2

-1

0Liquidity

0 4 8 12 16 20-0.1

0

0.1

0.2

0.3

0.4Capital

0 4 8 12 16 200

0.1

0.2

0.3

0.4

0.5S Cap Share

0 4 8 12 16 200

0.1

0.2

0.3

0.4

0.5

0.6S Leverage

0 4 8 12 16 200

2

4

6

8S Debt Share

This figure plots the transition paths from the benchmark capital requirement to the optimal capital re-quirement of 17% over 20 quarters (x-axis). The y-axis marks the percentage change relative to benchmarkeconomy. We plot consumption, aggregate liquidity, the aggregate capital stock, the S-bank capital share (SCap Share), the leverage of S-banks, and the share of S-bank debt in total debt.

gate liquidity. It increases on impact by almost 6%. This is at first driven by an increase inS-bank leverage and then subsequently by an expansion in the balance sheet of S-banks.What moves the leverage of S-banks? As in the simple model, S-bank leverage is a func-tion (see equation 12 in Section 2) of the marginal value of S-bank liquidity services. WithγH > 0, the marginal value of liquidity is decreasing in total liquidity services. Thus,when total liquidity falls enough in response to higher capital requirements and whenhouseholds’ preference for liquidity provision is sufficiently inelastic, the marginal valueof shadow bank liquidity services also increases. We can see this in the top-right panelof Figure 5 that shows S-banks’ deposit rate. The higher marginal value of S-bank debtlowers S-banks’ debt financing costs. We also see from the bottom-middle panel thatthe reduction in the debt financing costs reduces S-banks’ overall funding costs: theirweighted-average-cost-of-capital (WACC) falls by 0.4% the instant the new capital re-quirement is introduced and keeps on falling subsequently. This funding cost reduction

35

makes S-banks more profitable and leads to more entry of S-banks, hence increasing theircapital share and the aggregate balance sheet size of S-banks. It also increases the valueof the collateral for liquidity production: the physical capital stock. Since the value of liq-uidity services increases, the collateral value of capital also increases, leading to a highercapital stock in equilibrium.

Figure 5: Transition path from the benchmark economy to the economy with the optimalcapital requirement: default rates and prices

0 4 8 12 16 200

5

10

15S Default Rate

0 4 8 12 16 20-100

-80

-60

-40

-20

0C Default Rate

0 4 8 12 16 20-1.5

-1

-0.5

0S Deposit Rate

0 4 8 12 16 20-10

-8

-6

-4

-2

0C Deposit Rate

0 4 8 12 16 20-1.5

-1

-0.5

0wacc

S

0 4 8 12 16 20-2.5

-2

-1.5

-1

-0.5

0wacc

C

This figure plots the transition paths from the benchmark capital requirement to the optimal capital re-quirement of 17% over 20 quarters (x-axis). The y-axis marks the percentage change relative to benchmarkeconomy. We plot the default rate of shadow banks and commercial banks, the deposit rate of shadowbanks and commercial banks, and the weighted average cost of capital (funding costs of assets) for bothshadow- and commercial banks.

Our general equilibrium model shows that changes in the capital requirement alsoaffect financial institutions that were not targeted by this policy. These effects were causedby a reduction in C-banks’ liquidity services. But why do commercial bank liquidityservices fall when the capital requirement constraint is tightened? To answer this questionit is useful to visualize the balance sheet. A tighter constraint means that the ratio of bankequity to bank assets has to increase. It does not necessarily follow that bank debt hasto fall. Instead, C-banks could increase their equity-asset ratio while keeping bank debtconstant by funding an asset expansion with equity. Banks would only find it optimal to

36

do this if (1) the marginal benefit of assets exceeded their marginal funding costs, and (2)at the margin equity was cheaper than debt. The marginal benefit of assets is not affectedby changes in the bank capital requirement. Thus, to incentivize banks to expand theirbalance sheet their funding costs have to fall substantially. The bottom-right panel ofFigure 5 shows that indeed C-banks’ overall funding costs (WACCC) fall in response tohigher capital requirements. This is a general equilibrium effect that is also present inBegenau (2018).27 This effect indeed motivates commercial banks to increase their assetsand fund new investments with equity. However, commercial banks merely increase theirassets by .30%, which is not large enough to offset the higher equity ratio.

How liquidity preferences shape our welfare results. Households’ liquidity prefer-ences are at the heart of our model and critical for the response of shadow banks to highercapital requirements on commercial banks. Higher shadow bank leverage can counteractthe benefits of tighter requirements for commercial banks. In section 2, we show that withconstant returns in overall liquidity (γH = 0), shadow banks’ leverage decreases unam-biguously with higher C-bank capital requirements. However, our benchmark calibrationhas moderately diminishing returns (γH = 1.7), and S-bank leverage rises as we raise θ.Another take-away from the simple model is that the expansion of the shadow bank sharein response to higher θ depends on the elasticity of substitution 1/(1− ε). The larger ε,the better are S-banks in replacing reduced C-bank liquidity provision by expanding theirasset share.

To explore the robustness of our main conclusions, we analyze the effect of increasing θ

under two alternative specifications: a calibration in which ε = 0 (i.e., low substitutabilitybetween S-bank and C-bank debt), and one in which γH = 0 (i.e., no decreasing returns toscale in aggregate liquidity). We present detailed results in Table 7 of Appendix C.3. Themain take-away is that these variations in preferences affect the results consistent with theintuition developed in the simple model of Section 2.6. A lower degree of substitutability(ε = 0) leads to a smaller expansion in shadow banking, and thus to a greater reductionin overall bankruptcy losses. The net effect is a slightly higher welfare gain. Constantreturns in total liquidity (γH = 0) cause a larger drop both in liquidity provision anddeadweight losses. However, they also lead to a smaller rise in the aggregate liquiditypremium, which in general equilibrium causes an increase in both banks’ cost of capital,

27The reason for the reduction in overall funding costs is simple. Households like commercial bankdeposits. Their preferences (see 29) imply that the marginal value of commercial bank liquidity servicesis positive and decreasing in the amount of commercial bank debt. To put it simply, households have adownward-sloping demand curve for commercial bank liquidity. The increase in the convenience yield inresponse to a higher capital requirement in Table 2 reflects this demand curve.

37

resulting in less investment and output. Thus, the net effect is a smaller welfare gain de-spite a more stable financial system. In both cases, we see qualitatively clear differencesin the S-bank response. However, the magnitudes of the differences in welfare are rela-tively small, leading us to conclude that our quantitative results are robust to reasonablevariations of our liquidity preference calibration.

6 Other Regulatory Policies

So far we analyzed the effects of a simple static capital requirement. Since the financialcrisis, policy-makers and academics have proposed several alternatives, including timevarying capital requirements, fairly priced deposit insurance premia, and more recentlya proposal that has become known as the Minneapolis plan.28

Table 3 shows how the economies under these alternative policies differ from thebenchmark economy. The first column presents the results for the optimal static capitalrequirement that we discussed in the previous section. In the second column, we presentthe effects of a fair insurance premium. This means that we set the deposit insurancepremium κ on C-bank debt to the expected loss of the deposit insurance fund. In otherwords, κ is set such that the deposit insurance fund breaks even in expectations. This doeslittle to reduce C-banks’ leverage and deadweight losses, as they pass on most the fee todepositors in the form of lower rates. Therefore, this policy does not lift consumption likehigher capital requirements. The reduction in leverage, albeit tiny, is sufficient to cause areduction in aggregate liquidity provision. Without gains in consumption, the net effecton welfare of this policy is negative.

In the third column, we experiment with a countercyclical capital requirement. Moreconcretly, we set the average of this time varying capital requirement to the optimal staticcapital requirement value of 17%, and increase it to 22% during bad aggregate statesand decrease it to 12% in good aggregate states. The fourth column experiments witha procyclical capital requirement that is on average 17%, and 12% in bad states and 22%in good states. Interestingly, both types of capital requirements have similar effects on theeconomy as a static requirement and do not significantly differ from each other. Relativeto the static capital requirement they increase the volatility of liquidity provision and leadto a smaller expansion of the shadow banking system.

The last column presents the results from the Minneapolis Plan. In contrast to all other

28https://www.minneapolisfed.org/publications/special-studies/endingtbtf/

final-proposal.

38

Table 3: Effects of Alternative Bank Regulation Policies

θ=17% fair κ Corr(θt, Yt) Corr(θt, Yt) Minn. plan< 0 > 0

Capital +0.56% -0.30% +0.59% +0.58% -0.40%Debt share S +6.18% +9.04% +5.95% +6.12% -63.88%Capital share S +0.60% +9.34% +0.26% +0.43% -68.86%Leverage S +0.39% -0.22% +0.42% +0.37% +0.02%Leverage C -7.78% -0.00% -7.77% -7.79% -14.45%

Deposit rate S -1.97% +1.15% -2.04% -2.03% -59.99%Deposit rate C -9.11% -3.92% -9.33% -9.23% -7.40%Convenience Yield S +5.34% -3.06% +5.49% +5.53% +130.87%Convenience Yield C +11.51% +4.78% +11.74% +11.67% +9.42%

DWL S +11.70% +2.57% +11.83% +11.99% -70.70%DWL C -94.63% -5.49% -80.47% -83.73% -99.72%GDP +0.05% -0.03% +0.05% +0.05% -0.03%Liquidity Services -5.16% -1.43% -5.08% -5.16% -14.02%Consumption +0.19% -0.00% +0.16% +0.17% +0.28%Vol(Liquidity Services) -6.15% -0.10% +45.19% +20.89% -47.54%Vol(Consumption) +0.59% +1.37% +0.23% +0.87% -1.29%Welfare +0.129% -0.011% +0.115% +0.118% +0.144%

This table shows how alternative policies change the behavior of the economy relative to the benchmarkcalibration. The first column shows the results for the optimal static capital requirement, the second fora fair deposit insurance premium, the third for a countercyclical capital requirement, the fourth for a pro-cyclical capital requirement, the last column for the Minneapolis Plan. Each value represents the percentagechange relative to the benchmark economy in the average of the variable of interest.

39

plans, this proposal also targets the shadow banking sector leading to pronounced differ-ences in the results. Specifically, this plan combines higher capital requirements for com-mercial banks (23%) with a 30 bps quarterly tax on shadow bank borrowings. Despitetougher capital requirements on commercial banks, fewer shadow banks find financialintermediation and liquidity provision profitable due to the tax. Instead of filling the gap,S-banks’ debt and capital share falls by 64% and 69%, respectively. Thus, the result ofthis experiment provides an insight into an economy with a much smaller role of shadowbanks. Given the reduction in C-bank leverage and lower S-bank liquidity provision itis not surprising that liquidity production takes a large hit relative to the other propos-als, a fall by 14% in contrast to only 5% with the static requirement. A lower supply ofshadow bank debt drives up its value and thus lowers S-banks’ debt funding costs (grossof the tax). S-bank leverage remains roughly unchanged. Because S-banks’ become muchsmaller, the aggregate amount of deadweight losses from shadow banks is drastically re-duced. In this world with fewer shadow banks, liquidity services and consumption aremuch less volatile. Together with an overall larger reduction in deadweight losses it leadsto the highest welfare gain of all policies considered.

7 Conclusion

As the relative market share of traditional depositary institutions has been declining, bothin terms of loan originations and safe asset provision, the question of optimal bank reg-ulation requires modeling both these traditional banks and less regulated shadow banks.We propose a quantitative framework that views shadow banks not just as regulatoryavoidance schemes of traditional banks, but as alternative providers of intermediationservices and safe and liquid assets. Our approach draws the main distinction betweenboth types of intermediaries on the funding side: while traditional banks issue insureddeposits, shadow banks fund their balance sheets through unsecured short-term debt.Our theory is compatible with many underlying reasons why investors want to hold bothtraditional deposit accounts and other forms of (uninsured) money-like assets.

The model allows us to analyze how higher capital requirements on commercial bankschange the relative profitability of shadow banking and the relative attractiveness ofshadow bank debt, affecting the size and riskiness of the shadow banking sector. Theeffect on financial stability depends on investor demand for liquidity provision, whichwe capture through household preferences for safe and liquid assets. If household de-mand is sufficiently inelastic, the withdrawal of liquidity produced by traditional banks

40

following tighter capital regulation causes shadow bank leverage and risk to increase.We let the data guide our specification of preferences for liquidity to quantify the effectof higher capital requirements. At the optimal static capital requirement of 17% of (risky)assets, aggregate liquidity provision is lower compared to the benchmark capital require-ment of 10%. As liquidity is more valuable, shadow banking becomes more profitable,leading to an expansion in the asset share and leverage of shadow banks. However, eventhough shadow banks become riskier, traditional banks become sufficiently safer, suchthat the net effect for financial stability is positive.

Our analysis shows that the substitution towards shadow banking in response to tighterregulation is indeed a valid concern. However, it also clarifies that a financial system withmore shadow banking is not necessarily riskier. It further highlights a positive effect ofthe shift to shadow banks: the increased liquidity provision by shadow banks can offsetsome of the reduction in traditional bank liquidity production, causing a smaller overalldecline in liquid assets. Furthermore, since production of safe assets in our framework iscostly to society, both types of banks may overproduce liquidity relative to the social opti-mum. In particular, if shadow banks also enjoy implicit government guarantees, policiesaimed at reducing both traditional and shadow banks’ leverage and risk, such as a tax onunsecured short-term debt funding, lead to higher welfare gains than policies targeted attraditional banks alone.

While the focus on differences in funding of traditional and shadow banks sharpensour conclusions, our framework abstracts away from differences in technology or exper-tise on the asset side. Further, we only indirectly account for the government’s role in safeasset provision. We leave these important questions as extensions for future work.

41

References

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45

A Simple Model

Proof of proposition 1.

Proof. To obtain the S-bank FOC for leverage, we differentiate the S-bank objective in (3) to get

qS + q′S(LS)LS = β(1− LS).

Differentiating the HH FOC (10) with respect to LS, and under the restriction that individual S-banks do not internalize their effect on aggregate S-bank liquidity AS, gives

q′S(LS) = −β.

Combining the two yields qS = β. Substituting this result back into the HH FOC (10) results inequation (12).

Proof of proposition 2.

Proof. Differentiating the C-bank objective in (2) with respect to LC gives

qC = µC + β(1− LC), (38)

where µC is the Lagrange multiplier on the leverage constraint. Combining equations (9) and (38)yields

µC = β(ψHC + LC) > 0. (39)

The LC term on the RHS of equation (39) denotes the probability of default FC(LC) and is therefore≥ 0. Under the assumption that ψHC > 0, this implies that the multiplier is positive and theconstraint is binding, with leverage given by equation (15). Combining equations (11) and (9)under the same assumption implies the result in (16).

Proposition 5. An increase in the capital requirement θ causes lower S-bank leverage and greater S-bankdebt and capital shares, i.e., dLS/dθ < 0, dRS/dθ > 0 and dKS/dθ > 0.

Proof. Differentiation of the leverage condition (20) with respect to θ gives

dLS

dθ= ψH′S(RS)

dRS

dθ, (40)

and differentiation of the scale condition (21) yields

2LSdLS

dθ= −1

2(1− θ)− ψHC(RS) + ψ(1− θ)H′C(RS)

dRS

dθ. (41)

Combining (40) and (41) to eliminate dRS/dθ and solving for dLS/dθ gives

dLS

dθ=

ψH′S(RS)( 1

2 (1− θ) + ψHC(RS))

(1− θ)H′C(RS)− 2LSH′S(RS)

,

46

which is negative sinceH′S(RS) < 0 andH′C(RS) > 0. By (40),

dRS

dθ=

1ψH′S(RS)

dLS

dθ,

which is positive given dLS/dθ < 0. Finally, the equilibrium S-bank debt ratio can be expressed interms of LS and KS

RS =LSKS

12 (1− θ)(1− KS)

.

Again differentiation with respect to θ and solving for dKS/dθ yields

dKS

dθ=

dRSdθ −

dLSdθ

KS12 (1−θ)(1−KS)

KS12 (1−θ)(1−KS)

+ 112 (1−θ)(1−KS)2

,

which given dRS/dθ > 0 and dLS/dθ < 0 is positive.

Proof of proposition 3. The first-order conditions for S- and C-bank leverage are given by

Lj = ψHj(LSKS, LC(1− KS)) = ψHj(RS), (42)

for j = S, C respectively, i.e. the leverage ratio of each bank is equal to the marginal benefit ofliquidity to households. The first-order condition for the S-bank capital share is

12

L2S − LSψHS(LSKS, LC(1− KS)) =

12

L2C − LCψHC(LSKS, LC(1− KS)),

which combined with the FOCs in (42) implies

LS = LC.

Setting equal the conditions in (42) and using the functional form for utility in (25) yields

α

[α + (1− α)

1Rε

S

] 1−εε

= (1− α) [αRεS + 1− α]

1−εε ,

which can be rearranged to

α(1− α)ε

1−ε R2εS +

((1− α)

11−ε − α

11−ε

)Rε

S − (1− α)αε

1−ε = 0.

This is an exponential polynomial of the form

Aexp(2εx) + Bexp(εx) + C = 0,

47

with x = log(RS) and

A = α(1− α)ε

1−ε

B =((1− α)

11−ε − α

11−ε

)C = −(1− α)α

ε1−ε .

Given α ∈ [0, 1], the unique real root is

x =1ε

log

α1

1−ε (1− α)−ε

1−ε − (1− α) +

√(1− α)2

(1 + α

11−ε (1− α)−

11−ε

)2

2α

=

1ε

log

(1− α)(

α1−α

) 11−ε − (1− α) + (1− α) + (1− α)

(α

1−α

) 11−ε

2α

=

1ε

log

(1− α)(

α1−α

) 11−ε

α

,

which simplifies to

RS =

(α

1− α

) 11−ε

.

SinceRS =

LSKS

LCKC=

KS

KC,

we obtain the solution for the capital shares in the proposition. Plugging this solution back intoeither condition (42) and setting LS = LC gives

L∗ ≡ ψ(1− α)

[α

(α

1− α

) ε1−ε

+ 1− α

] 1−εε

. (43)

Proof of proposition 4. Applying the definition of the wedge m to the zero-profit condition(21) gives

L2S = L2

C + 2(1 + m)(ψHC(RS))2,

which when combined with the condition for S-bank leverage (20) implies

LS =

√m + 3m + 1

LC,

or equivalently

HS(RS) =√(m + 1)(m + 3)HC(RS).

48

Defining the wedge factorM =√(m + 1)(m + 3) and using the functional form for utility in (25)

yields

α

[α + (1− α)

1Rε

S

] 1−εε

=M(1− α) [αRεS + 1− α]

1−εε ,

which can be rearranged to

α(1− α)ε

1−εM ε1−ε R2ε

S +((1− α)

11−εM ε

1−ε − α1

1−ε

)Rε

S − (1− α)αε

1−ε = 0.

This is an exponential polynomial of the form

Aexp(2εx) + Bexp(εx) + C = 0,

with x = log(RS) and

A = α(1− α)ε

1−εM ε1−ε

B =((1− α)

11−εM ε

1−ε − α1

1−ε

)C = −(1− α)α

ε1−ε .

Given α ∈ [0, 1], the unique real root is

x =1ε

log

(1− α)(

α1−α

) 11−ε

αM ε1−ε

,

which simplifies to

RS =

(1M

) 11−ε(

α

1− α

) 11−ε

.

To prove part (i), note that at m = 0, which leads to efficient C-bank leverage, we getM =√

3,

which implies a smaller S-bank ratio RS by factor 1/√

31

1−ε compared to the planner solution inproposition 3.

To prove part (ii), we differentiate the household objective given by (24) in the decentralizedequilibrium with respect to θ. After collecting terms, the derivate is

dU(θ)

dθ=

dKC

dθ

(ψHC(RS)LC −

12

L2C

)+

dKS

dθ

(ψHS(RS)LS −

12

L2S

)+

dLC

dθ(ψHC(RS)KC − KCLC) +

dLS

dθ(ψHS(RS)KS − KSLS) .

Since in equilibrium LS = ψHS(RS) and LC = (1 + m)ψHC(RS), this expression becomes

dU(θ)

dθ=

(1

1 + m− 1

2

)L2

CdKC

dθ+

12

L2S

dKS

dθ+

(1

1 + m− 1)

KCLCdLC

dθ.

49

Further noting that LS =√

m+3m+1 LC and dKC/dθ = −dKS/dθ (since KC = 1− KS), this becomes

dU(θ)

dθ= L2

CdKS

dθ−(

1− 11 + m

)KCLC

dLC

dθ.

Since dKS/dθ > 0 by Proposition 5, and dLC/dθ < 0, this expression is positive for any m ≥ 0.

B Quantitative Model

B.1 Bank Optimization and Aggregation

This section describes details of the bank optimization problem resulting in equations (32) and (34)in the main text. Anticipating the result that all banks are equal due to i.i.d. shocks and a valuefunction that is homogeneous in capital, we suppress individual bank subscripts throughout.

Production. After aggregate productivity Zt and the run shock $t are realized, all banks pro-duce and invest. Denote by K j

t = (1− `jt)K

jt the capital banks retain after a possible fire sale due

to runs. The profits generated by the two lines of bank business (production and investment) are

Djt = Zt

(K j

t

)1−η (N j

t

)η+ (1− δK)ptK

jt − wtN

jt + I j

t (pt − 1)− φI

2

(I jt

K jt

− δK

)2

K jt, (44)

for j = S, C. Banks choose labor input N jt and investment I j

t to maximize (44). Note that the profitalso includes the proceeds from selling depreciated capital after production, (1− δK)ptK

jt.

The first-order condition for labor demand is the usual intratemporal condition equating thewage to the marginal product of labor

wt = Ztη

(N j

t

K jt

)η−1

= Ztη(

njt

)η−1. (45)

Similarly, the first-order condition for investment yields the usual relationship between the capitalprice and the marginal value of a unit of capital

pt = 1 + φI

(I jt

K jt

− δK

)= 1 + φI

(ijt − δK

). (46)

We can substitute both conditions back into the definition of profit in (44) to eliminate the wageand investment and define the gross payoff per unit of capital in equation (30) to get

Djt = Πj

tKjt.

50

The total dividend banks pay to shareholders is given by

Djt = ρ

jtD

jt − Bj

t + (qjt − κj)Bj

t+1 − ptKjt+1 −

φK

2

(K j

t+1

K jt

− 1

)2

K jt.

It scales the profit banks receive from their real business, Djt, by the idiosyncratic shock, ρ

jt, and

also includes redemptions of last period’s deposits, Bjt, and the equity cost of the portfolio for

next period, (qjt− κj)Bj

t+1− ptKjt+1−

φK2

(K j

t+1

K jt− 1)2

K jt, where the deposit insurance fee κj = 0 for

S-banks in the benchmark model.

Bank value function. We define the value function of a bank that did not default, at the timeit chooses its portfolio for next period as

V j(K jt, Zt, ρ

jt) = max

K jt+1,Bj

t+1

Djt

+ Et

[Mt,t+1max

V j((1− `

jt+1)K

jt+1, Zt+1, ρ

jt+1) + `

jt+1ρ

jt+1ΠH

t+1K jt+1 ,−δjΠ

jt+1K j

t+1

]. (47)

We assume that the default penalty −δjΠjt+1K j

t+1 in (47) is proportional to the asset value of thebank with parameter δj. This is reasonable and also retains the homogeneity of the problem in

capital K jt. Further note that capital sold to households in a run-induced fire sale yields ΠH

t perunit, as defined in equation (31).

To simplify the problem, we recognize that profits from real business and deposits redemptionsρ

jtD

jt − Bj

t are irrelevant for the bank’s choice after the default decision. After the bankruptcydecision, non-bankrupt banks choose their portfolio for next period, and households set up newbanks to replace those banks who defaulted. With respect to the portfolio choice for period t + 1,the optimization problem of all banks is identical conditional on having the same capital K j

t. Thuswe define the value function

V j(K jt, Zt) = V j(K j

t, Zt, ρjt)− ρ

jtD

jt + Bj

t,

such that we can write the problem in (47) equivalently as

V j(K jt, Zt) = max

K jt+1,Bj

t+1

(qjt − κj)Bj

t+1 − ptKjt+1 −

φK

2

(K j

t+1

K jt

− 1

)2

K jt

+ Et

[Mt,t+1max

ρ

jt+1

(Πj

t+1K jt+1 + `

jt+1ΠH

t+1K jt+1

)− Bj

t+1 + V j(K jt+1, Zt+1) ,−δSΠj

t+1K jt+1

].

(48)

Aggregation. Next, we conjecture that V j(K jt, Zt) is homogeneous in capital K j

t of degree one.

This allows us to define the scaled value function vj(Zt) =V j(K j

t ,Zt)

K jt

. Further defining the fire sale

51

discount

xjt =

ΠHt

Πjt

,

we can write vj(Z) as

vj(Zt) = maxkj

t+1,bjt+1

((qj

t − κj)bjt+1 − pt

)kj

t+1 −φK

2

(kj

t+1 − 1)2

+ Et

[Mt,t+1Πj

t+1kjt+1max

ρ

jt+1

(1− `

jt+1

(1− xj

t+1

))− Lj

t+1 + (1− `jt+1)

vj(Zt+1)

Πjt+1

,−δj

].

(49)

where the choice variables have been redefined as asset growth kjt+1 =

K jt+1

K jt

and capital structure

bjt+1 =

Bjt+1

K jt+1

.

We use the definition of the default threshold ρjt in the main text in equation (33) and take the

expectation with respect to ρjt+1 to rewrite the max operator in equation (49). This term represents

the leverage-adjusted payoff of banks’ portfolio including the default option and is

Ωj(Ljt) = (1− Fj

ρ,t)

(ρ

j,+t

(1− `

jt

(1− xj

t

))− Lj

t + (1− `jt)

vj(Zt)

Πjt

)− Fj

ρ,tδj, (50)

where Fjρ,t = Fj(ρ

jt) is the probability of default and ρ

j,+t = E

(ρ

jt | ρ

jt > ρ

jt

)is the expected value of

the idiosyncratic shock conditional on not defaulting.

We can thus rewrite (49) more compactly as

vj(Zt) = maxkj

t+1,bjt+1

((qj

t − κj)bjt+1 − pt

)kj

t+1 −φK

2

(kj

t+1 − 1)2

+ kjt+1Et

[Mt,t+1Πj

t+1Ωj(Ljt+1)

]. (51)

Equation (51) corresponds to equations (32) and (34) in the main text. It shows that the expectationonly depends on the capital structure choice through leverage Lj

t+1 = bjt+1/Πj

t+1. The first-order

condition for asset growth kjt+1 is

pt − (qjt − κj)b

jt+1 + φK

(kj

t+1 − 1)= Et

[Mt,t+1Πj

t+1Ωj(Ljt+1)

]. (52)

Substituting (52) into (51) yields

vj(Zt) = kjt+1φK(k

jt+1 − 1)− φK

2

(kj

t+1 − 1)2

=φK

2

((kj

t+1)2 − 1

).

The solution for vj(Zt) confirms the conjecture that

V j(K jt, Zt) = K j

tvS(Zt),

52

and we can thus solve the problem of a representative bank of each type. Note that the scaledvalue function vj(Zt) only depends on asset growth kj

t+1, which is a choice variable. Intuitively, ifthe bank expects high future capital growth, then the continuation value is large.

B.2 Household Problem

Denoting household wealth at the beginning of the period by Wt, the complete intertemporalproblem of households is

VH(ASt , AC

t , Wt, Yt) = maxCt,AS

t+1,ACt+1,SS

t ,SCt

U(Ct, H(

ASt , AC

t

)) + β Et

[V(AS

t+1, ACt+1, Wt+1, Yt+1)

]subject to the budget constraint in (37).

The transition law for household financial wealth Wt is

Wt+1 = ∑j=S,C

(1− Fjρ,t+1)

(Dj,+

t+1 + pjt+1

)Sj

t

+ ASt+1

[1− FS

ρ,t+1 + FSρ,t+1

(πB + (1− πB) rS

t+1

)]+ AC

t+1.

The beginning-of-period dividend paid by banks to households, conditional on survival, forS-banks is

DS,+t = ρS,+

t KSt

(`S

t ΠHt +

(1− `S

t

)ΠS

t

)− BS

t + KSt

(kS

t+1

(qS

t bSt+1 − pt

)− φK

2

(kS

t+1 − 1)2)

,

and for C-banks

DC,+t = ρC,+

t KCt ΠC

t − BCt + KC

t

(kC

t+1

((qC

t − κ)bCt+1 − pt

)− φK

2

(kC

t+1 − 1)2)

.

Households’ first-order conditions for purchases of bank equity are, for j = S, C,

pjt = Et

[Mt,t+1

(1− Fj

ρ,t+1

) (Dj,+

t+1 + pjt+1

)], (53)

where we have defined the stochastic discount factor

Mt,t+1 = βUC(Ct+1, Ht+1)

UC(Ct, Ht)= β

(Ct+1

Ct

)−γ

.

The marginal rate of substitution between consumption and liquidity services of bank type j isdefined as

MRSj,t =UH(Ct, Ht)

UC(Ct, Ht)

∂H(ASt , AC

t )

∂Ajt

,

53

and given by

MRSS,t = αψCγt H−γH

t

(Ht

ASt

)1−ε

, (54)

MRSC,t = (1− α)ψCγt H−γH

t

(Ht

ACt

)1−ε

. (55)

for S- and C-bank debt, respectively.

Then the first-order conditions for purchases of bonds of either type of bank are

qSt = Et

Mt,t+1

[1− FS

ρ,t+1 + FSρ,t+1(πB + (1− πB)rS

t+1) + MRSS,t+1

]. (56)

qCt = Et Mt,t+1 [1 + MRSC,t+1] , (57)

The payoff of commercial bank bonds is 1, whereas the payoff of shadow bank bonds dependson their default probability, recovery value, and th probability of a government bailout πB. Thelast term in each expectation represents the marginal benefit of liquidity services to households,as defined in (54) and (55).

B.3 S-bank Optimality Conditions

Each period, S-banks choose investment ISt , labor input NS

t , capital growth kSt+1 and capital struc-

ture bSt+1. The first-order conditions for investment and labor are given by (46) and (45), respec-

tively. They are incorporated into the gross payoff of capital ΠSt in (30). The first-order condition

for capital growth is given by (52).

It remains to derive the first-order condition for capital structure. Before doing so, we firstrecognize that individual S-banks take into account the effect of their capital structure choice theprice of their debt. Thus, they optimally respond to households’ valuation of idiosyncratic S-bankrisk, such that we replace qS

t in (51) by the function

qS(bSt+1) = Et

Mt,t+1

[1− FS

ρ,t+1 + FSρ,t+1(πB + (1− πB)rS

t+1) + MRSS,t+1

],

which is households’ first-order condition for S-bank debt purchases in (56). Differentiating equa-tion (51) with respect to bS

t+1 after this substitution yields

qSt + bS

t+1 q′S(bSt+1) = −Et

[Mt+1ΠS

t+1∂LS

t+1

∂bSt+1

Ω′(LSt+1)

].

Using ∂LSt+1

∂bSt+1

= 1ΠS

t+1, this reduces to

qS(bSt+1) + bS

t+1 q′S(bSt+1) = −Et

[Mt+1Ω′(LS

t+1)]

.

54

In section B.3.1 we calculate Ω′(LS+1), such that the first-order condition becomes

qSt + bS

t+1 q′S(bSt+1) = Et

[Mt+1(1− FS

t+1)

(1 +

`t+1

LSt+1

((1− xt+1)ρ

S,+t+1 +

vSt+1

ΠSt+1

))], (58)

To obtain the partial derivative q′S(bSt+1), we differentiate equation (56) to get

q′S(bSt+1) = −(1− πB)Et

[Mt+1

∂LSt+1

∂bSt+1

(∂FS

t+1rSt+1

∂LSt+1

−∂FS

t+1

∂LSt+1

)].

In section B.3.2 we calculate ∂FSt+1rS

t+1∂LS

t+1− ∂FS

t+1∂LS

t+1, such that the derivative becomes

q′S(bSt+1) = −

1− πB

bSt+1

Et

Mt,t+1

[FS

t+1rSt+1

1− `t+1(1− xt+1)

+ f St+1LS(LS

t+1)

((1− ξS)

(δS + (1− `t+1)

vSt+1

ΠSt+1

)+ ξSLS

t+1

)], (59)

where

LS(LSt ) =

∂ρSt

∂LSt

is the derivative of the default threshold with respect to leverage (equation (60)).

The full first-order condition for the S-bank capital structure choice is obtained by substituting(59) into (58).

It is useful to examine the FOC for the case of no run `t+1 = 0, zero default penalty δS = 0, nocapital adjustment cost φK = 0 (implying vS

t+1 = 0), and zero bailout probability πB = 0. In thatcase, the derivative in (59) reduces to

q′S(bSt+1) = −

1bS

t+1Et

Mt,t+1

[FS

t+1rSt+1 + ξS f S

t+1LS(LSt+1)LS

t+1

]and the first-order condition (58) becomes

qSt + bS

t+1 q′S(bSt+1) = Et

[Mt+1(1− FS

t+1)]

.

Combining these two equations, we get

qSt = Et

[Mt+1

(1− FS

t+1 + FSt+1rS

t+1 + ξS f St+1LS(LS

t+1)LSt+1

)].

We can equate this with the household first-order condition (56) and collect terms to get

Et

[Mt+1ξS f S

t+1LS(LSt+1)LS

t+1

]= Et [Mt+1MRSS,t+1] .

This equation is the analogue to equation (12) in the simple model. The S-bank chooses leverageto equalize the expected marginal liquidity benefit to households on the RHS with the expectedmarginal losses due to bankruptcy on the LHS.

55

B.3.1 Computing Ω′(LSt+1)

Rewrite the function ΩS(LSt ) as

ΩS(LSt ) = (1− FS

t )

(ρS,+

t (1− `t(1− xt))− LSt + (1− `t)

vSt

ΠSt

)− FS

t δS

= (1− `t(1− xt))

ˆ ∞

ρSt

ρ f St (ρ) dρ− (1− FS

t )

(LS

t − (1− `t)vS

t

ΠSt

)− FS

t δS.

Recall the default threshold for S-banks is given by

ρSt =

LSt − (1− `S

t )vS(Zt)

ΠSt− δS

1− `St(1−ΠH

t /ΠSt) .

We denote the derivative with respect to LSt as

LSt (LS

t ) =∂ρS

t

∂LSt=

1− `tLS

t

(δS(1− xt)− xt

vSt

ΠSt

)(1− `t(1− xt))2 . (60)

First computing∂(1− `t(1− xt))

∂LSt

= −$St

1− xt

xt= −`t

1− xt

LSt

,

we can further calculate (using Leibniz’s rule)

∂ΩS(LSt )

∂LSt

= − `t1− xt

LSt

(1− FSt )ρ

S,+t − f S

t LSt (LS

t )

(LS

t − δS − (1− `t)vS

t

ΠSt

)− (1− FS

t )

(1 +

`t

LSt

vSt

ΠSt

)+ f S

t LSt (LS

t )

(LS

t − (1− `t)vS

t

ΠSt

)− f S

t LSt (LS

t )δS

= −(1− FSt )

(1 +

`t

LSt

((1− xt)ρ

S,+t +

vSt

ΠSt

)).

B.3.2 Computing q′S(bSt+1)

Computing∂FS

t+1rSt+1

∂LSt+1

. Recall the definition of the recovery value for S-banks as

rS(LSt ) = (1− ξS)

ρS,−t (1− `t(1− xt))

LSt

,

with the conditional expectation ρS,−t = E

[ρ | ρ < ρS

t].

First compute

∂

∂LSt

1− `t(1− xt)

LSt

=−$S

t1−xt

xtLS

t − (1− `t(1− xt))

(LSt )

2= − 1

(LSt )

2.

56

We can rewrite the recovery value times the probability of default as

FSt rS

t =(1− ξS)(1− `t(1− xt))

LSt

ˆ ρSt

−∞ρ dFS(ρ).

Differentiating this expression with respect to LSt gives

∂FSt rS

t

∂LSt

=− (1− ξS)

(LSt )

2FS

t ρS,−t +

f St

LSt(1− ξS)LS(LS

t )(LSt − δS − (1− `t)

vSt

ΠSt)

= − FSt rS

t

LSt (1− `t(1− xt))

− f St

LStLS(LS

t )(1− ξS)(δS + (1− `t)vS

t

ΠSt− LS

t ),

where we use the function LS(LSt ) defined in (60).

Combining. Using that∂FS

t+1

∂LSt+1

= f St LS(LS

t ),

we get

∂FSt+1rS

t+1

∂LSt+1

−∂FS

t+1

∂LSt+1

= − FSt rS

t

LSt (1− `t(1− xt))

− f St

LStLS(LS

t )

((1− ξS)

(δS + (1− `t)

vSt

ΠSt

)+ ξSLS

t

).

B.4 C-bank Optimality Conditions

Like S-banks, C-banks choose investment ICt , labor input NC

t , capital growth kCt+1 and capital struc-

ture bCt+1. The first-order conditions for investment and labor are given by (46) and (45), respec-

tively. They are incorporated into the gross payoff of capital ΠCt in (30). The first-order condition

for capital growth is given by (52).

To derive the C-bank first-order condition for capital structure, first note that C-banks are sub-ject to the regulatory constraint in (35). Denote the Lagrange multiplier associated with the con-straint by λC

t . Differentiating (51) subject to the constraint with respect to bCt+1 gives

qCt − κC = λC

t − Et

[Mt+1ΠC

t+1∂LC

t+1

∂bCt+1

Ω′(LCt+1)

].

Using ∂LCt+1

∂bCt+1

= 1ΠC

t+1, this reduces to

qCt − κC = λC

t − Et

[Mt+1Ω′(LC

t+1)]

.

57

In section B.4.1 we calculate Ω′(LC+1), such that the first-order condition becomes

qCt − κC = λC

t + Et

[Mt+1(1− FC

t+1)]

. (61)

B.4.1 Computing Ω′(LCt+1)

Rewrite the function ΩC(LCt ) as

ΩC(LCt ) = (1− FC

t )

(ρC,+

t − LCt +

vCt

ΠCt

)− FC

t δC

=

ˆ ∞

ρCt

ρ f Ct (ρ) dρ− (1− FC

t )

(LC

t −vC

t

ΠCt

)− FC

t δC.

Recall the default threshold for C-banks is given by

ρCt = LC

t −vC(Zt)

ΠCt− δC,

implying that ∂ρCt

∂LCt= 1.

We calculate (using Leibniz’s rule)

∂ΩC(LCt )

∂LCt

= − f Ct

(LC

t − δC −vC

t

ΠCt

)− (1− FC

t ) + f Ct

(LC

t −vC

t

ΠCt

)− f C

t δC

= − (1− FCt ).

B.5 Equilibrium Definition

Given a sequence of aggregate Yt, Zt, $t and idiosyncratic shocks ρSt,i, ρC

t,i, a competitive equi-librium consists of a sequence of prices wt, pt, qS

t , qCt , pS

t , pCt , household choices Ct, AS

t+1, ACt+1, SS

t , SCt , NH

t ,S-bank choices IS

t , NSt , BS

t+1, KSt+1, and C-bank choices IC

t , NCt , BC

t+1, KCt+1 such that households

and banks optimize given prices, and markets clear.

There is market clearing for capital

KSt+1 + KC

t+1 = ISt + IC

t + (1− δK) ∑j=S,C

(1− ξ jFj

ρ,tρj,−t

)K j

t

(1− `

jt

)+ (1− δK)KS

t `St , (62)

securities issued by banks

BSt+1 = AS

t+1,

BCt+1 = AC

t+1,

SSt = 1,

SCt = 1,

58

the goods market

Ct + ∑j=S,C

(I jt + Φ(I j

t , K jt))+ ∑

j=S,CDWLj

t

= Yt + Zt ∑j=S,C

(N jt )

η((1− `

jt)K

jt

)1−η+ Zt(NH

t )η(`St KS

t )1−η , (63)

and the labor market

NSt + NC

t + NHt = 1. (64)

The deadweight losses for each bank type are

DWLjt = ξ jFj

ρ,tρj,−t

(`

jtΠ

Ht +

(1− `

jt

) (Πj

t − (1− δK)pt

))K j

t,

and the adjustment costs for capital and investment amount to

Φ(I jt , K j

t) = K jt

(1− `

jt

)(φI

2

(ijt − δK

)2+

φK

2

(kj

t − 1)2)

.

Note that commercial banks are isolated from bank runs, and so `Ct = 0 ∀t. The market clearing

condition for capital in Equation (62) is also the transition law for the aggregate capital stock.

B.6 Equilibrium Conditions

The solution of the model can be written as a system of 17 nonlinear functional equations inequally many unknown functions of the state variables. The model’s state variables are St =(Yt, Zt, $t, KC

t , KSt , AC

t , ASt ).

The functions are aggregate consumption C(St), prices of C-bank and S-bank equity (pC(St), pS(St)),prices of C-bank and S-bank deposits (qC(St), qS(St)), C-bank and S-bank deposit issuance perunit of capital (bC(St+1), bS(St+1)), the Lagrange multiplier on C-bank leverage λC(St), C-bankand S-bank capital purchases (KC(St+1), KS(St+1)), the capital price p(St), C-bank and S-bank in-vestment (IC(St), IS(St)), labor demand of C-bank, S-bank and households (NC(St), NS(St), NH(St)),and the wage w(St). For the equations, we will use time subscripts und suppress the dependenceon state variables. All variables can be expressed as functions of current (St) or one-period ahead(St+1) state variables.

The equations are

59

pCt = Et

[Mt,t+1FC

ρ,t+1

(DC,+

t+1 + pCt+1

)](E1)

pSt = Et

[Mt,t+1FS

ρ,t+1

(DS,+

t+1 + pSt+1

)](E2)

qCt = Et

[Mt,t+1

(1 + MRSC

t+1

)](E3)

qSt = Et

[Mt,t+1

(1− FS

ρ,t+1

(1− (πB + (1− πB)rS

t+1))+ MRSS

t+1

)](E4)

Ct + ICt + IS

t + Φ(ICt , KC

t ) + Φ(ISt , (1− `S

t )KSt ) = Yt + YC

t + YSt + YH

t

− ξCFCt ρC,−

t (ΠCt − (1− δK)pt)KC

t

− ξSFSt ρS,−

t

(`S

t ΠHt + (1− `S

t )(ΠSt − (1− δK)pt)

)KS

t (E5)

qSt + bS

t+1 q′S(bSt+1) = Et

[Mt+1(1− FS

ρ,t+1)

(1 +

`t+1

LSt+1

((1− xt+1)ρ

S,+t+1 +

vSt+1

ΠSt+1

))](E6)

qCt − κ = λC

t + Et

[Mt,t+1(1− FC

ρ,t+1)]

(E7)

λCt

(pt − (1− θ)bC

t+1

)= 0 (E8)

pt − qSt bS

t+1 + φK

(kS

t+1 − 1)= Et

[Mt,t+1ΠS

t+1 ΩS(

LSt+1

)], (E9)

pt − (qCt − κ)bC

t+1 + φK

(kC

t+1 − 1)= Et

[Mt,t+1ΠC

t+1 ΩC(

LCt+1

)](E10)

KCt+1 + KS

t+1 = ICt + IS

t + (1− δK)(

1− ξCFCρ,tρ

C,−t

)KC

t

+ (1− δK)(

1− ξSFSρ,tρ

S,−t

)(1− `S

t )KSt + (1− δK)`

St KS

t (E11)

ICt =

(pt − 1

φI+ δK

)KC

t (E12)

ISt =

(pt − 1

φI+ δK

)(1− `S

t )KSt (E13)

wt = ηZt(nCt )

η−1 (E14)

NCt =

KCt

KCt + (1− `S

t )KSt + `S

t KSt (Zt/Zt)1/(1−η)

(E15)

NSt =

(1− `St )K

St

KCt

NCt (E16)

NHt = 1− NC

t − NSt (E17)

(E1) – (E4) are the household Euler equations for bank equity and debt from equations (53)applied to j = C, S, (56), and (57). (E5) is the resource constraint from (63). (E6) is the S-bankcondition for leverage from (58). (E7) is the C-bank condition for leverage (61), with (E8) beingthe complementary slackness condition for the leverage constraint (35). (E9) and (E10) are theS-bank and C-bank conditions for capital growth from (52), applied to either bank type. (E11) isthe market clearing condition for capital (62), and (E12) – (E13) are the first-order conditions for

60

investment by banks from (46), applied to j = C, S. (E14) – (E16) are the first-order conditions forlabor demand by banks and households, from (45) applied to j = C, S, H, and (E17) is the marketclearing condition for labor.

B.7 Computational Solution Method

The equilibrium of dynamic stochastic general equilibrium models is usually characterized recur-sively. If a stationary Markov equilibrium exists, there is a minimal set of state variables that sum-marizes the economy at any given point in time. Equilibrium can then be characterized using twotypes of functions: transition functions map today’s state into probability distributions of tomor-row’s state, and policy functions determine agents’ decisions and prices given the current state.Brumm et al. (2018) analyze theoretical existence properties in this class of models and discussthe literature. Perturbation-based solution methods find local approximations to these functionsaround the “deterministic steady-state”. For applications in finance, there are often several prob-lems with local solution methods. First, portfolio restrictions such as leverage constraints may beoccasionally binding in the true stochastic equilibrium. Generally, a local approximation aroundthe steady state (with a binding or slack constraint) will therefore inaccurately capture nonlin-ear dynamics when constraints go from slack to binding. Further, local methods have difficultiesin dealing with highly nonlinear functions within the model such as probability distributions oroption-like payoffs, as is the case for the quantitative model in this paper. Finally, in models withrarely occurring bad shocks (such as the runs in our model), the steady state used by local methodsmay not properly capture the ergodic distribution of the true dynamic equilibrium.

Global projection methods (Judd (1998)) avoid these problems by not relying on the determin-istic steady state. Rather, they directly approximate the transition and policy functions in therelevant area of the state space.

B.8 Solution Procedure

The projection-based solution approach used in this paper has three main steps.

Step 1. Define approximating basis for the policy and transition functions. To approximate theseunknown functions, we discretize the state space and use multivariate linear interpolation.Our general solution framework provides an object-oriented MATLAB library that allowsapproximation of arbitrary multivariate functions using linear interpolation, splines, orpolynomials. For the model in this paper, splines or polynomials of various orders achievedinferior results due to their lack of global shape preservation.

Step 2. Iteratively solve for the unknown functions. Given an initial guess for policy and transi-tion functions, at each point in the discretized state space compute the current-period opti-mal policies. Using the solutions, compute the next iterate of the transition functions. Re-peat until convergence. The system of nonlinear equations at each point in the state space issolved using a standard nonlinear equation solver. Kuhn-Tucker conditions can be rewrittenas equality constraints for this purpose. This step is completely parallelized across points inthe state space within each iterate.

Step 3. Simulate the model for many periods using approximated functions. Verify that the sim-ulated time path stays within the bounds of the state space for which policy and transition

61

functions were computed. Calculate relative Euler equation errors to assess the computa-tional accuracy of the solution. If the simulated time path leaves the state space boundariesor errors are too large, the solution procedure may have to be repeated with optimized gridbounds or positioning of grid points.

We will now provide a more detailed description for each step.

Step 1 The state space consists of

- two exogenous state variables [Yt, $t], and

- four endogenous state variables [Kt, KSt , BS

t , BCt ].

The banking sector specific shock Zt does not contain any persistent shocks in addition to Yt andis therefore not an additional state variable. We first discretize Yt into a NY-state Markov chainusing the Rouwenhorst (1995) method, where NY is an odd number. The procedure chooses theproductivity grid points YjNY

j=1 and the NY × NY Markov transition matrix ΠY between themto match the volatility and persistence of GDP growth of the bank independent sector. The runshock $t can take on two realizations 0, $∗ as described in the calibration section. The 2 x 2Markov transition matrix between these states is given by Π$. We assume that run shocks onlyoccur in states with negative GDP growth. Denote the set of the Nx = NY + (NY − 1)/2 valuesthe exogenous state variables can take on as Sx, and the associated Markov transition matrix Πx.

Our solution algorithm requires approximation of continuous functions of the endogenousstate variables. Define the “true” endogenous state space of the model as follows: if each endoge-nous state variable St ∈ Kt, KS

t , BSt , BC

t can take on values in a continuous and convex subsetof the reals, characterized by constant state boundaries, [Sl , Su], then the endogenous state spaceSn = [Kl , Ku]× [KS

l , KSu ]× [BS

l , BSu ]× [BC

l , BCu ]. The total state space is the set S = Sx × Sn.

To approximate any function f : S → R, we form an univariate grid of (not necessarilyequidistant) strictly increasing points for each endogenous state variables, i.e., we choose KjNK

j=1,

KSk

NKSk=1 , BS

mNBSm=1, and BC

n NBCn=1. These grid points are chosen to ensure that each grid covers

the ergodic distribution of the economy in its dimension, and to minimize computational errors,with more details on the choice provided below. Denote the set of all endogenous-state gridpoints as Sn = KjNK

j=1 × KSk

NKSk=1 × BS

mNBSm=1 × BC

n NBCn=1, and the total discretized state space

as S = Sx × Sn. This discretized state space has NS = Nx · NK · NKS · NBS · NBC total points, whereeach point is a 5 x 1 vector as there are 5 distinct state variables (counting the exogenous state asone). We can now approximate the smooth function f if we know its values f jNS

j=1 at each points ∈ S, i.e. f j = f (sj) by multivariate linear interpolation.

Our solution method requires approximation of of three sets of functions defined on the do-main of the state variables. The first set of unknown functions CP : S → P ⊆ RNC

, with NC

being the number of policy variables, determines the values of endogenous objects specified inthe equilibrium definition at every point in the state space. These are the prices, agents’ choicevariables, and the Lagrange multipliers on the portfolio constraints. Specifically, the 8 policy func-tions are bond prices qS(S), qC(S), capital price p(S), debt issued by banks in the current periodBS(S), BC(S), the capital purchased by S-banks KS(S), labor demand of S-banks nS(S), and theLagrange multiplier for the C-bank leverage constraint λC(S). There is an equal number of theseunknown functions and nonlinear functional equations, to be listed under step 2 below.

62

The second set of functions CT : S × Sx → Sn determine the next-period endogenous statevariable realizations as a function of the state in the current period and the next-period realiza-tion of exogenous shocks. There is one transition function for each endogenous state variable,corresponding to the transition law for each state variable, also to be listed below in step 2.

The third set are forecasting functions CF : S → F ⊆ RNF, where NF is the number of forecast-

ing variables. They map the state into the set of variables sufficient to compute expectations termsin the nonlinear functional equations that characterize equilibrium. They partially coincide withthe policy functions. In particular, the forecasting functions for our model are the capital pricep(S), S-bank labor input nS(S), capital growth of both types of banks kS(S), kC(S), and the valuefunction of households VH(S) (to compute welfare).

Step 2 Given an initial guess C0 = C0P, C0

T, C0F, the algorithm to compute the equilibrium takes

the following steps.

A. Initialize the algorithm by setting the current iterate Cm = CmP , Cm

T , CmF = C0

P, C0T, C0

F.

B. Compute forecasting values. For each point in the discretized state space, sj ∈ S , j =

1, . . . , NS, perform the steps:

i. Evaluate the transition functions at sj combined with each possible realization of theexogenous shocks xi ∈ Sx to get s

′j(xi) = Cm

T (sj, xi) for i = 1, . . . , Nx, which are thevalues of the endogenous state variables given the current state sj and for each possiblefuture realization of the exogenous state.

ii. Evaluate the forecasting functions at these future state variable realizations to get f 0i,j =

CmF

(s′j(xi), xi

).

The end result is a Nx × NS matrix F m, with each entry being a vector

f mi,j = [pi,j, nS

i,j, kSi,j, kC

i,j, VHi,j ] (F)

of the next-period realization of the forecasting functions for current state sj and futureexogenous state xi.

C. Solve system of nonlinear equations. At each point in the discretized state space, sj ∈ S ,j = 1, . . . , NS, solve the system of nonlinear equations that characterize equilibrium in theequally many “policy” variables, given the forecasting matrix F m from step B. This amountsto solving a system of 12 equations in 12 unknowns

Pj = [qSj , qC

j , pj, BSj , BC

j , KSj , nS

j , λCj ] (P)

63

at each sj. The equations are

qCj = Es′i,j|sj

[Mi,j

(1 + MRSC

i,j

)](C1)

qSj = Es′i,j|sj

[Mi,j

(1− FS

i,j

(1− (πB + (1− πB)rS

i,j))+ MRSS

i,j

)](C2)

qSj + bS

j q′S(bSj ) = Es′i,j|sj

[Mi,j(1− FS

i,j)

(1 +

`i,j

LSi,j

((1− xi,j)ρ

S,+i,j +

vSi,j

ΠSi,j

))](C3)

pj − qSj bS

j + φK

(kS

j − 1)= Es′i,j|sj

[Mi,jΠS

i,j ΩS(

LSi,j

)](C4)

qCj − κ = λC

j + Es′i,j|sj

[Mi,j(1− FC

i,j)]

(C5)

pj − (qCj − κ)bC

j + φK

(kC

j − 1)= Es′i,j|sj

[Mi,jΠC

i,j ΩC(

LCi,j

)](C6)

λCj

(pj − (1− θ)bC

j

)= 0 (C7)

1 = NHj + NC

j + NSj . (C8)

(C1) and (C2) are the household Euler equations for purchases of deposits. (C3) and (C4)are the intertemporal optimality conditions for S-banks, and (C5) and (C6) are those forC-banks. (C7) is the leverage constraint for C-banks. Finally, (C8) is the market clearingcondition labor.

Expectations are computed as weighted sums, with the weights being the probabilities oftransitioning to exogenous state xi, conditional on state sj. Hats (·) in (C1) – E(C8) indicatevariables that are direct functions of the vector of unknowns (P). These are effectively thechoice variables for the nonlinear equation solver that finds the solution to the system (C1)– (C8) at each point sj. All variables in the expectation terms with subscript i,j are directfunctions of the forecasting variables (F).

The latter values are fixed numbers when the system is solved, as they we pre-computed instep B. For example, the stochastic discount factor Mi,j depends on both the solution and theforecasting vector, i.e.

Mi,j = β

(Ci,j

Cj

)−γ

,

since it depends on future and current consumption. To compute the expectation of theright-hand side of equation (C1) at point sj, we first look up the corresponding column j inthe matrix containing the forecasting values that we computed in step B, F m. This columncontains the Nx vectors, one for each possible realization of the exogenous state, of theforecasting values defined in (F). From these vectors, we need consumption Ci,j. Further, weneed current consumption Cj, which is a policy variable chosen by the nonlinear equationsolver. MRSC

i,j is a function of future consumption Ci,j, and the future state variables BSi,j and

BCi,j (since market clearing implies Aj

t = Bjt for j = S, C). Denoting the probability of moving

from current exogenous state xj to state xi as πi,j, we compute the expectation of the RHS of(C1)

Es′i,j|sj

[Mi,j

(1 + MRSC

i,j

)]= ∑

xi | xj

πi,j Mi,j

(1 + MRSC

i,j

).

64

The mapping of solution and forecasting vectors (P) and (F) into the other expressions inequations (C1) – (C8) follows the same principles and is based on the equations in modelappendix B. In particular, the system (C1) – (C8) implicitly uses the budget constraints of allagents, and the market clearing conditions for capital and debt of both banks.

Note that we could exploit the linearity of the market clearing condition in (C8) to eliminateone more policy variable, nS, from the system analytically. However, in our experience thealgorithm is more robust when we explicitly include labor demand of all agents as policyvariables, and ensure that these variables stay strictly positive (as required with CD produc-tion functions) when solving the system. To solve the system in practice, we use a nonlinearequation solver that relies on a variant of Newton’s method, using policy functions Cm

P asinitial guess. More on these issues in subsection B.9 below.

The final output of this step is a NS× 12 matrix Pm+1, where each row is the solution vectorPj that solves the system (C1) – (C8) at point sj.

D. Update forecasting, transition and policy functions. Given the policy matrix Pm+1 fromstep B, update the policy function directly to get Cm+1

P . All forecasting functions with theexception of the value functions are also equivalent to policy functions. The householdvalue function is updated based on the recursive definition

VHj = U(Cj, Hi,j) + βEs′i,j|sj

VHi,j (V)

using the same notation as defined above under step C. Note that the value function com-bines current solutions from Pm+1 (step C) for consumption with forecasting values fromF m (step B). Using these updated value functions, we get Cm+1

F .

Finally, update transition functions for the endogenous state variables using the followinglaws of motion, for current state sj and future exogenous state xi as defined above:

Km+1i,j = IC

j + ISj + (1− δK)

(1− ξCFC

i,jρC,−i,j

)KC

i,j

+ (1− δK)(

1− ξSFSi,jρ

S,−i,j

)(1− `S

i,j)KSi,j + (1− δK)`

Si,jK

Si,j (T1)

(KSi,j)

m+1 = kSj KS

i,j (T2)

(BCi,j)

m+1 = BCj (T3)

(BSi,j)

m+1 = BSj . (T4)

(T1) is simply the law of motion for aggregate capital, and (T2) is the definition of capitalgrowth kS

t . (T3) and (T4) follow directly from the direct mapping of policy into state variablefor bank debt. Updating according to (T1) – (T4) gives the next set of functions Cm+1

T .

E. Check convergence. Compute distance measures ∆F = ||Cm+1F − Cm

F || and ∆T = ||Cm+1T −

CT Fm||. If ∆F < TolF and ∆T < TolT, stop and use Cm+1 as approximate solution. Oth-erwise reset policy functions to the next iterate i.e. Pm → Pm+1 and reset forecastingand transition functions to a convex combination of their previous and updated values i.e.Cm → Cm+1 = D× Cm + (1− D)× Cm+1, where D is a dampening parameter set to a valuebetween 0 and 1 to reduce oscillation in function values in successive iterations. Next, go tostep B.

65

Step 3 Using the numerical solution C∗ = Cm+1 from step 2, we simulate the economy forT = Tini + T period. Since the exogenous shocks follow a discrete-time Markov chain with transi-tion matrix Πx, we can simulate the chain given any initial state x0 using T − 1 uniform randomnumbers based on standard techniques (we fix the seed of the random number generator to pre-serve comparability across experiments). Using the simulated path xtT

t=1, we can simulate theassociated path of the endogenous state variables given initial state s0 = [x0, K0, KS

0 , BS0 , BC

0 ] byevaluating the transition functions

[Kt+1, KSt+1, BC

t+1, BCt+1] = C∗T(st, xt+1),

to obtain a complete simulated path of model state variables stTt=1. To remove any effect of the

initial conditions, we discard the first Tini points. We then also evaluate the policy and forecastingfunctions along the simulated sample path to obtain a complete sample path st, Pt, ftT

t=1.

To assess the quality and accuracy of the solution, we perform two types of checks. First, weverify that all state variable realizations along the simulated path are within the bounds of thestate variable grids defined in step 1. If the simulation exceeds the grid boundaries, we expandthe grid bounds in the violated dimensions, and restart the procedure at step 1. Secondly, wecompute relative errors for all equations of the system (C1) – (C8) and the transition functions(T1) – (T4) along the simulated path. For equations involving expectations (such as (C1)), thisrequires evaluating the transition and forecasting function as in step 2B at the current state st. Foreach equation, we divide both sides by a sensibly chosen endogenous quantity to yield “relative”errors; e.g., for (C1) we compute

1 =1

qCj

Es′i,j|sj

[Mi,j

(1 + MRSC

i,j

)],

using the same notation as in step 2B. These errors are small by construction when calculated at thepoints of the discretized state grid S , since the algorithm under step 2 solved the system exactlyat those points. However, the simulated path will likely visit many points that are between gridpoints, at which the functions C∗ are approximated by interpolation. Therefore, the relative errorsindicate the quality of the approximation in the relevant area of the state space. We report average,median, and tail errors for all equations. If errors are too large during simulation, we investigatein which part of the state space these high errors occur. We then add additional points to the statevariable grids in those areas and repeat the procedure.

B.9 Implementation

Solving the system of equations. We solve system of nonlinear equations at each point inthe state space using a standard nonlinear equation solver (MATLAB’s fsolve). This nonlinearequation solver uses a variant of Newton’s method to find a “zero” of the system. We employseveral simple modifications of the system (C1) – (C8) to avoid common pitfalls at this step ofthe solution procedure. Nonlinear equation solver are notoriously bad at dealing with comple-mentary slackness conditions associated with a constraint. Judd, Kubler, and Schmedders (2002)discuss the reasons for this and also show how Kuhn-Tucker conditions can be rewritten as addi-tive equations for this purpose. Consider the C-bank’s Euler Equation for risk-free bonds and the

66

Kuhn-Tucker condition for its leverage constraint:

qCj − κ = λC

j + Es′i,j|sj

[Mi,j(1− FC

i,j)]

0 = λCj

(pj − (1− θ)bC

j

)Now define an auxiliary variable hj ∈ R and two functions of this variable, such that λC,+

j =

max0, hj3 and λC,−j = max0,−hj3. Clearly, if hj < 0, then λI,+

j = 0 and λC,−j > 0, and vice

versa for hj > 0. Using these definitions, the two equations above can be transformed to:

qCj − κ = λC,+

j + Es′i,j|sj

[Mi,j(1− FC

i,j)]

0 = pj − (1− θ)bCj − λC,−

j .

The solution variable for the nonlinear equation solver corresponding to the multiplier is hj. Thesolver can choose positive hj to make the constraint binding (λC,−

j = 0), in which case λC,+j takes on

the value of the Lagrange multiplier. Or the solver can choose negative hj to make the constraintnon-binding (λC,+

j = 0), in which case λC,−j can take on any value that makes (K2) hold.

Similarly, certain solution variables are restricted to positive values due to the economic struc-ture of the problem. For example, given the Cobb-Douglas production function, optimal S-bankcapital for next period KS

j is always strictly positive. To avoid that the solver tries out negativecapital values (and thus output becomes ill-defined), we use log(KS

j ) as solution variable for thesolver. This means the solver can make capital arbitrarily small, but not negative.

Grid configuration. We choose to include the relative capital share of S-banks KSt = KS

t /Kt asstate variable instead of borrower debt KS

t such that the total set of endogenous state variables is[Kt, KS

t , BCt , BS

t ]. The reason is that the capital share is more stable in the dynamics of the modelthan the level, since total capital and S-bank capital are strongly correlated. For similar reasons,we choose to include S-bank and C-bank book leverage bS

t = BSt /KS

t and bCt = BC

t /KCt instead of

the levels of debt. For the benchmark case, the grid points in each state dimension are as follows

• Y: We discretize Y and Z jointly into a 9-state Markov chain (with three possible realizationsfor each) using the Rouwenhorst (1995) method. The procedure chooses the productivitygrid points Y3

j=1 and Z3j=1 and the 9× 9 Markov transition matrix ΠY,Z between them

to match the volatility and persistence of GDP growth. This yields the possible realizationsfor Y: [0.9869, 1.0000, 1.0132], and for Z: [0.9698, 1.0000, 1.0312].

• $: [0.0, 0.3] (see calibration)

• K: [3.8, 3.87, 3.94, 4.01, 4.08, 4.15]

• KS: [0.26, 0.28, 0.30, 0.32, 0.34, 0.36, 0.38]

• bS: [0.10, 0.22, 0.34, 0.46, 0.58, 0.71, 0.83, 0.95]

• bC: [0.87, 0.882, 0.894, 0.906, 0.918, 0.93]

67

The total state space grid has 24,192 points. The grid boundaries and the placement of pointshave to be readjusted for each experiment, since the ergodic distribution of the state variables de-pends on parameters. Finding the right values for the boundaries is a matter of experimentation.

Generating an initial guess and iteration scheme. To find a good initial guess for the policy,forecasting, and transition functions, we solve the deterministic “steady-state” of the model underthe assumption that the bank leverage constraint is binding and no runs are occurring. We theninitialize all functions to their steady-state values, for all points in the state space. Note that theonly role of the steady-state calculation is to generate an initial guess that enables the nonlinearequation solver to find solutions at (almost) all points during the first iteration of the solutionalgorithm. In our experience, this steady state delivers a good enough initial guess.

In case the solver cannot find solutions for some points during the initial iterations, we revisitsuch points at the end of each iteration. We try to solve the system at these “failed” points usingas initial guess the solution of the closest neighboring point at which the solver was successful.This method works well to speed up convergence and eventually finds solutions at all points.

To determine convergence, we check absolute errors in the value function of households, (V).Out of all functions we approximate during the solution procedure, it exhibits the slowest con-vergence. We stop the solution algorithm when the maximum absolute difference between twoiterations, and for all points in the state space, falls below 1e-3 and the mean distance falls below1e-4. For appropriately chosen grid boundaries, the algorithm converges within 120 iterations.

We implement the algorithm in MATLAB and run the code on a high-performance computing(HPC) cluster. As mentioned above, the nonlinear system of equations can be solved in parallelat each point. We parallelize across 28 CPU cores of a single HPC node. From computing theinitial guess and analytic Jacobian to simulating the solved model, the total running time for thebenchmark calibration is about 2 hours and 40 minutes. Calibrations that exhibit more financialfragility and/or macro volatility converge up to 15% slower.

Simulation. To obtain the quantitative results, we simulate the model for 5,000 periods after a“burn-in” phase of 500 periods. The starting point of the simulation is the ergodic mean of thestate variables. As described in detail above, we verify that the simulated time path stays withinthe bounds of the state space for which the policy functions were computed. We fix the seed ofthe random number generator so that we use the same sequence of exogenous shock realizationsfor each parameter combination.

To produce impulse response function (IRF) graphs in Figure 2, we simulate 10,000 differentpaths of 25 periods each. In the initial period, we set the endogenous state variables to severaldifferent values that reflect the ergodic distribution of the states. We use a clustering algorithm torepresent the ergodic distribution non-parametrically. We fix the initial exogenous shock realiza-tion to mean productivity (Y = Z = 1) and no run ($ = 0). The “impulse” in the second periodis either only a bad productivity shock, or both low productivity and a run shock ($ = 0.3). Forthe remaining 23 periods, the simulation evolves according to the stochastic law of motion of theshocks. In the IRF graphs, we plot the median path across the 10,000 paths given the initial condi-tion. The transition dynamics in Figures 4 and 5 are constructed similarly, with the difference thatthe economy does not experience a productivity or run shock, but rather an unanticipated changein θ.

68

Table 4: Computational Errors

Equation Percentile50th 75th 95th 99th Max

C1 1.60797E-05 1.98285E-05 2.33846E-05 0.00011435 0.000609577C2 1.64198E-05 2.02402E-05 2.38312E-05 0.000114068 0.000600429C3 1.81838E-05 2.22974E-05 2.62468E-05 0.000115962 0.000590462C4 3.51337E-06 6.07882E-06 1.18653E-05 0.000108445 0.000234119C5 3.89366E-06 6.20534E-06 1.15129E-05 0.000122232 0.000844679C6 1.03779E-05 1.3081E-05 1.92823E-05 6.36015E-05 0.000318513C7 0.000161026 0.000196303 0.000229379 0.000250181 0.003953529C8 2.9076E-05 3.57655E-05 3.86194E-05 4.05661E-05 0.004224141

The table reports median, 75th percentile, 95th percentile, 99th percentile, and maximum abso-lute value errors, evaluated at state space points from a 5,000 period simulation of the bench-mark model. Each row contains errors for the respective equation of the nonlinear system (C1)– (C8) listed in step 2 of the solution procedure.

Evaluating the solution. Our main measure to assess the accuracy of the solution are relativeequation errors calculated as described in step 3 of the solution procedure. Table 4 reports themedian error, the 95th percentile of the error distribution, the 99th, and 100th percentiles during the5,000 period simulation of the model. Median errors are very small for all equations, with evenmaximum errors only causing small approximation mistakes. Errors are comparably small for allexperiments we report.

C Calibration Appendix

C.1 Bank Technology and Other Parameters

This section describes the remaining parameter choices not yet covered in the main text, listed intable 5. First, we can normalize the average output of the bank-independent sector µY to one. Wealso set the average idiosyncratic shock received by banks µ

jρ, for j = S, C, to one, which means

that absent default these shocks would aggregate away.

Bank technology and adjustment costs. We need to take a stand on the share of firms thatdepend on banks. In the U.S. a large part of the economy is in fact not directly dependent on banksas a large fraction of firms (in a value-weighted sense) can issue debt and equity in public markets.Yet, a sizeable share of the economy does depend on bank funding. We follow the definition inKashyap, Lamont, and Stein (1994), where bank-dependent firms are those without a S&P long-term credit rating. Because mortgages make up the largest share of the bank loan portfolio, wealso add construction and real estate firms as identified by SIC codes 6500-6599 (real estate), 1500-1599 (construction), and 1700-1799 (construction contractors, special trades) to the set of bankdependent firms. We operationalize this definition on Compustat data with the aggregate sales

69

data of bank dependent firms.29 We use this definition to parametrize the relative size of bankdependent productivity νZ and the investment technology parameters. We set νZ such that wematch the ratio of bank-dependent firms’ value-added to total GDP. The GDP of bank dependentfirms is measured as the aggregate revenue of bank-dependent firms. Accordingly, we set σZ tomatch the volatility of this series (for a given σY).

The labor share η in bank-dependent production is set to standard value of 67%. Similarly, weset the depreciation rate δK to 2.5% quarterly.

Our model features two types of adjustment costs that are governed by the parameters φI andφK, respectively. The value of φI determines the marginal cost of investment. We calibrate it tomatch the investment-asset ratio volatility of the bank dependent sector. The investment-assetratio is mapped to the capital expenditure-asset ratio of bank-dependent firms in Compustat.Capital growth is also subject to adjustment costs that are governed by the parameter φK. Thisintroduces frictions into the capital flow between shadow– and commercial banks. We do notassume bank-type specific capital growth adjustment costs and thus could target either the assetgrowth volality of shadow banks or commercial banks. We have much better data on the latterand use asset growth of commercial banks as a target that is measured as total financial assets ofdepository institutions from the Flow of Funds.

Table 5: Parametrization

Parameter Value Target Data Model

NormalizationµY = µC

ρ = µSρ 1

Bank technologyνZ 0.23 Value-added of bank-dependent sector 26% 27%σZ 0.0217 Bank dependent output volη 0.67 NIPA labor shareδK 0.025 10% annual depreciationφI 1.3 Vol(Investment) 0.30% 0.31%φK 0.0185 Vol(C-bank asset growth) 0.50% 0.54%σρC 0.11 SD(Dividend per share) 10.6% 11%σρS 0.15 SD(Dividend per share) 14.9% 15%

Bank regulationθ 10% Basel capital requirementκ 0.142% FDIC insurance fee

Bank-independent sectorσY 0.0093 Bank-independent sector volρY 0.366 Bank-independent sector AC

Preferencesγ 2 Risk aversion

To calibrate the standard deviation of idiosyncratic shocks for C- and S-banks, we estimates the

29We use data from Compustat quarterly fundamentals (compm/fundq/) as well as Compustat’s creditrating database (compm/rating/).

70

cross-sectional dispersion in dividends per share for publicly traded commercial and “shadowbanks” in the data. As for the calculation of shadow bank leverage, we define shadow banks as allinstitutions with SIC codes 6111-6299, 6798, 6799, 6722, 6726, excluding SIC codes 6200, 6282, 6022,6199. The definition of commercial banks in the data is simpler, since here can use publicly tradeddepository institutions and bank holding companies. We compute the cross-sectional standarddeviation of the dividend/share ratio for each quarter from 1999 to 2017 and take the time-seriesaverage of these standard deviations, which gives 10.6% for banks and 15% for shadow banks.These ratios directly correspond to the parameters σρC and σρS in the model.

In the model, we parameterize the idiosyncratic ρ shocks as gamma distributions. Let thegamma cumulative distribution function be given by Γ(ρ; χ0, χ1) with parameters (χ0, χ1). Theseparameters map into means µ

jρ and variances σ2

ρj as follows:

χ1 = σ2ρ /µρ,

χ0 = µρ/χ1.

A standard result in statistics states that the conditional expectations are

E(ρ | ρ < x) = µρΓ(x; χ0 + 1, χ1)

Γ(x; χ0, χ1),

E(ρ | ρ > x) = µρ1− Γ(x; χ0 + 1, χ1)

1− Γ(x; χ0, χ1),

which we use to compute the conditional expectations ρj,− and ρj,+ used in bank payoffs to share-holders and recovery values for creditors.

Bank regulation. We set the regulatory capital ratio θ in the baseline of the model to 10%. Tocalibrate the deposit insurance fee κC, we use data from the FDIC on banks’ insurance contribu-tions. According to the 2016 FDIC report, banks paid $10 billion in FDIC insurance fees on aninsurance fund balance of $83.162 billion, which represents 1.18% of insured deposits. This meansthat banks paid 14.2 bps per dollar of insured deposits (see also: https://www.fdic.gov/about/strategic/report/2016annualreport/ar16section3.pdf), which is the value we assign to κC.

Bank-independent sector and preferences. We calibrate the stochastic process for the en-dowment income Yt that households receive directly to the volatility and autocorrelation of realGDP per capita, net of the contribution of the bank-dependent sector. We further set risk aversionγ to the standard value of 2.

C.2 Details on Calibration of ε

To motivate our regression design for calibrating ε, we derive an equation for the model spreadbetween the rate on S-bank and C-bank debt. The starting point are the household first-orderconditions for holdings of the two types of debt, (56) and (57), under the simplifying assumptionsthat πB = 0 and ξS = 1 (these assumptions do not affect the fundamental conclusions from thederivation). Since we are looking for a simple empirical relationship, we further suppress the

71

expectations operators. Under these assumptions, the equations are

qCt = Mt,t+1

(1 + MRSC

t+1

)qS

t = Mt,t+1

(1− FS

ρ,t+1 + MRSSt+1

),

where

MRSS,t = αψCγt H−γH

t

(Ht

ASt

)1−ε

, (65)

MRSC,t = (1− α)ψCγt H−γH

t

(Ht

ACt

)1−ε

, (66)

and

Mt,t+1 = β

(Ct+1

Ct

)−γ

.

We perform a first-order log-linear expansion of both conditions around the deterministic steadystate of the model. Variables without time subscript denote steady state values, and hatted (·)variables denote log-deviations from steady state. The usual log-linearization techniques give

qCt = −γ∆Ct+1 +

βMRSC

qCˆMRSC

t+1,

and

qSt = −γ∆Ct+1 +

βMRSS

qSˆMRSS

t+1 −βFS

qS FSt+1.

Further expandingˆMRSj

t = γCt + (1− ε− γH) Ht − (1− ε) Ajt

and

Ht = α(AS)ε

HεAS

t + (1− α)(AC)ε

HεAC

t ,

we can compute the spread qCt − qS

t and collect terms to get

qCt − qS

t = β

(MRSC

qC − MRSS

qS

)γCt+1 +

βFS

qS FSt+1

+ β

((1− ε− γH)

(MRSC

qC − MRSS

qS

)α

(AS

H

)ε

+ (1− ε)MRSS

qS

)AS

t+1

+ β

((1− ε− γH)

(MRSC

qC − MRSS

qS

)(1− α)

(AC

H

)ε

− (1− ε)MRSC

qC

)AC

t+1. (67)

The coefficients before ASt+1 and AC

t+1 in equation (67) reveal the role of γH and ε for the effect ofdebt quantities on the spread. Clearly, if ε = 1 (perfect substitutes) and γH = 0 (constant returns

72

in total liquidity), the quantity terms vanish. If ε = 1 and γH > 0, the equation becomes

qCt − qS

t = β

(MRSC

qC − MRSS

qS

)γCt+1 +

βFS

qS FSt+1

− γH β

(MRSC

qC − MRSS

qS

)(α

AS

HAS

t+1 + (1− α)AC

HAC

t+1

), (68)

or equivalently

qCt − qS

t = β

(MRSC

qC − MRSS

qS

)γCt+1 +

βFS

qS FSt+1 − γH β

(MRSC

qC − MRSS

qS

)Ht+1,

i.e., in that case only total liquidity Ht matters. Further, as we can see from equation (68), if bothquantities are included separately in the equation, they should enter with a coefficient of the samesign.

Table 6: Elasticity of Substitution Regression

Data Model

Coef t-stat Coef t-stat

Constant -0.0006 -0.34 0.0049 8.60log(S-bank liquidity/GDP) 0.0019 1.44 0.0020 11.04log(Tbills/GDP) -0.0013 -2.31log(C-bank liquidity/GDP) -0.0019 -1.80GDP growth -0.0002 -0.22 -0.0085 -14.37VIX -0.0080 -0.47S-bank defaults 0.0124 10.71

R2 0.29 0.85

However, the empirically relevant case is a combination of imperfect substitutes (ε < 1) anddecreasing returns (γH > 0). If further the difference between liquidity premia, MRSC

qC − MRSS

qS ,

is small relative to the level MRSj/qj, then we would expect the second term in each quantitycoefficient in (67), (1− ε)MRSj

qj , to dominate. In that case, we would expect a positive coefficienton the S-bank quantity, and a negative coefficient on the C-bank quantity. This is exactly what wefind in the data, and we choose ε in the model (for a given γH) to match this pattern.

In particular, for the sample from 1999:Q1 to 2017:Q4, we regress the CP/Tbill spread on logof the TBill/GDP ratio, the log of “shadow bank” debt relative to GDP, the VIX, and log GDPgrowth. The quantity of shadow bank liabilities is the sum of REPO claims held by shadow banks(Flow of Funds table L.207), total money market mutual fund assets (table L.206), and commercialpaper held by the domestic financial sector less depository institutions (table L.209). The Tbillsupply series is directly computed from Treasury department reports, as in Greenwood, Hanson,and Stein (2016) and Nagel (2016).

Table 6 shows regression results for data and model in the baseline calibration (with γH = 1.7).By setting ε = 0.42, the model matches the coefficient on S-bank debt to GDP in the data. Note

73

that both in data and model, the coefficient on the other source of liquidity (Tbills in the data andC-bank debt in the model) has the expected sign.

C.3 Results Appendix

Table 7: Effect of Liquidity Preference Parametrization on Capital Requirement Results

Benchmark ε = 0 γH = 0θ = 15% θ = 20% θ = 15% θ = 20% θ = 15% θ = 20%

% change in mean relative to θ = 10%

1. Capital +0.4% +0.9% +0.3% +0.8% -0.4% -0.7%2. Debt share S +4.9% +8.1% +2.8% +4.6% +7.4% +12.6%3. Capital share S +1.0% -0.1% -0.9% -3.1% +3.8% +5.1%4. Leverage S +0.3% +0.6% +0.2% +0.5% -0.3% -0.5%5. Leverage C -5.6% -11.1% -5.6% -11.1% -5.6% -11.1%

6. Deposit rate S -1.3% -3.0% -1.2% -2.7% +1.6% +2.7%7. Deposit rate C -6.6% -13.0% -6.4% -12.6% -1.8% -2.9%8. Convenience Yield S +3.6% +8.0% +3.1% +7.2% -4.0% -6.9%9. Convenience Yield C +8.4% +16.4% +8.0% +15.8% +2.2% +3.6%

10. Default Rate S +6.7% +15.1% +5.8% +13.5% -7.3% -12.3%11. Default Rate C -85.6% -98.7% -85.5% -98.7% -85.7% -98.7%12. GDP +0.0% +0.1% +0.0% +0.1% -0.0% -0.1%13. Liquidity Services -3.8% -7.3% -3.5% -6.8% -5.0% -9.5%14. Consumption +0.17% +0.20% +0.17% +0.19% +0.17% +0.19%15. Welfare +0.127% +0.116% +0.128% +0.120% +0.123% +0.109%

This table presents the results from increases in the capital requirement using different liquidity preferencecalibrations. With each table entry we show for each variable of interest the percentage change in the meanrelative to the economy with an capital requirement of 10%. The benchmark calibration reprints the resultsin Table 2 in columns 1 and 2. The columns headed with “epsilon = 0” show the results for the capitalrequirement increase experiment for an economy with a low elasticity of substitution between C-bank andS-bank debt. The columns headed with “gammaH = 0” show the results for the capital requirement increaseexperiment for an economy without decreasing returns to scale in total liquidity.

Table 7 shows how the results for the capital requirement experiment depend on the liquidityparameters ε and γH. We present the results for our main capital requirement experiment usingour benchmark calibration, and for the two variations described in the main text (ε = 0 andγH = 0). For each variable in the table, we report the percentage change of its mean relative to anincrease in the capital requirement from 10% to 15% and from 10% to 20%.

First of all, the table shows that the welfare results are not affected by changes in these param-eters. The optimal capital requirement remains around 15% as the increase in consumption andthe decrease in liquidity services are roughly identical across parametrizations. However, for the

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question of financial fragility looking at the results using different liquidity function parametriza-tions is quite instructive.

When the capital requirement is increased, commercial banks reduce their leverage and there-with their supply of liquidity services. All else equal, aggregate liquidity falls in the economy.This increases the convenience yield of C-banks across all parametrizations and lowers C-banks’deposit rate. What is the effect on shadow banks? We know from Section 2 that S-bank leverageis a function of the marginal benefit of leverage. With γH = 0, we turned off the feature thatthe marginal value of aggregate liquidity services is decreasing in the amount of liquidity ser-vices. Thus a reduction in aggregate liquidity services does not increase the marginal benefit ofS-bank liquidity services. Without this feature, the convenience yield on S-banks’ falls, causingthe deposit rate of S-banks to rise. Faced with higher debt-financing costs, S-banks reduce theirleverage, ,which makes them safer. In this scenario, the effect of capital requirements on financialfragility are unambigious: the economy is less risky even though more financial intermediation isperformed by now less risky shadow banks. However, the overall amount of financial interme-diation decreases, as can be seen from the reduction in the aggregate capital stock. This could beinterpreted as a reduction in lending, which also translates into a small reduction in GDP. Togetherwith a larger crunch in liquidity production, higher capital requirements lead therefore to lowerwelfare gains.

When we set ε to zero, households do not like to substitute between the two types of liquidityservices. However, because γH > 0, a reduction in aggregate liquidity raises the marginal valueand thus the convenience yield of any type of liquidity service. Relative to the benchmark, lowersubstitutability means that the increase (decrease) in S-banks’ convenience yield (deposit rate) isslightly smaller. Lower debt financing costs provide S-banks with an incentive to ramp up lever-age and risk. The effect on overall financial fragility is thus ambiguous. In this calibration (asin the benchmark calibration) it turns out that aggregate deadweight losses decrease with tighterregulation. Another effect of γH > 0 is that the capital stock increases. This is due to the collateralvalue of capital in liquidity production. When the marginal value of liquidity increases, the col-lateral value of capital increases as well. This is also the reason for the small positive response inGDP to higher capital requirements.

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