The Welfare Effects of Bank Liquidity and Capital Requirements

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) The Welfare Effects of Bank Liquidity and Capital Requirements Skander J. Van den Heuvel 2022-072 Please cite this paper as: Van den Heuvel, Skander J. (2022). “The Welfare Effects of Bank Liquidity and Capital Requirements,” Finance and Economics Discussion Series 2022-072. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2022.072. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The Welfare E⁄ects of Bank Liquidity and Capital Requirements Skander J. Van den Heuvel (cid:3) Federal Reserve Board October 30, 2022 Abstract Thestringencyofbankliquidityandcapitalrequirementsshoulddependontheirsocial costsandbene(cid:133)ts. Thispaperinvestigatestheirwelfaree⁄ectsandquanti(cid:133)estheirwelfare costs using su¢ cient statistics. The special role of banks as liquidity providers is embeddedinanotherwisestandardgeneralequilibriumgrowthmodel. Capitalandliquidityrequirementsmitigatemoralhazardfromdepositinsurance, which, ifunchecked, can lead to excessive credit and liquidity risk at banks. However, these regulations are also costly because they reduce the ability of banks to create net liquidity and can distort investment. Equilibrium asset returns reveal the strength of demand for liquidity, yielding two simple su¢ cient statistics that express the welfare cost of each requirement as a function of observable variables only. Based on U.S. data, the welfare cost of a 10 percent liquidity requirement is equivalent to a permanent loss in consumption of about 0.02%, a modest impact. Even using a conservative estimate, the cost of a similarly-sizedincreaseinthecapitalrequirementisroughlytentimesaslarge. Evenso, optimal policy relies on both requirements, as the (cid:133)nancial stability bene(cid:133)ts of capital requirements are found to be broader. (cid:3)Email: skander.j.vandenheuvel@fb.gov. I thank Toni Ahnert, William Bassett, Francesca Carapella, Francisco Covas, Burcu Duygan-Bump, Pedro Gete, Itay Goldstein, Gary Gorton, Gazi Kara, Agnese Leonello, David Martinez-Miera, Mark Mink, Thien Nguyen, Ettore Panetti, Ned Prescott, David Rappoport, Harald Uhlig, Alex Vardoulakis, and seminar participants at CEBRA, Cleveland Fed, Columbia, ECB, Fed Board, FDIC, FIRS, IMF, NBER, SAET, SED, and the Wharton Conference on Liquidity and Financial Crises for valuable comments, and Sorelle Peat, Jacob Fahringer, Olamide Bola, and Tristan D(cid:146)Orsaneo for expert research assistance. A large part of this research was conducted during a secondment at the ECB. The views expressed here do not necessarily represent the views of the Federal Reserve Board, the ECB, or their sta⁄s. 1

1 Introduction Theglobal(cid:133)nancialcrisisspurredkey(cid:133)nancialreforms, includingthestrengtheningofbank capital requirements and the introduction of new liquidity requirements, as part of Basel III. Even so, an important debate continues on the question of whether the strengthening of these requirements has been appropriate, excessive, or insu¢ cient, and their calibration remains at the top of the regulatory agenda (Barr (2022)). Whereas there is widespread agreement that capital requirements can and have helped make banks safer and that liquidity stress exacerbated the crisis through runs and (cid:133)re sales, the ongoing debate in large part re(cid:135)ects di⁄ering views about the existence and magnitude of costs to society from imposing restrictions on banks(cid:146)balance sheets. While some progress has been made in understanding and quantifying the costs of capital requirements,1 a consensus has not yet emerged. Moreover, liquidity regulation, especially its social cost and its interaction with capital regulation,ismuchlesswellunderstood. Somehavearguedfornarrowbanking,wheredeposits are backed exclusively by safe, liquid assets - akin to a 100% liquidity requirement.2 The harm from liquidity stress would presumably be greatly reduced, if not eliminated, if such a policy were adopted. But what would be the cost? Clearly, to determine the optimal levels of liquidity and capital requirements the question of their social cost must be addressed. This paper argues that liquidity and capital regulations can each impose an important cost for a similar reason: they reduce the ability of banks to create net liquidity through the transformation of illiquid loans into liquid deposits (cid:150)a key, traditional function of banks. After all, capital requirements directly limit the fraction of bank loans that can be (cid:133)nanced by issuing liquid, deposit-like liabilities. Liquidity requirements force banks to hold safe, liquid assets against deposits, limiting their liquidity transformation by restricting the asset side of their balance sheet. This can impose a social cost because safe, liquid assets are necessarily in limited supply and have competing uses (see, for example, Krishnamurthy and Vissing-Jorgensen (2012) and Greenwood, Hanson and Stein (2015)). More speci(cid:133)cally, the contribution of this paper is threefold. First, it builds a framework to analyze the social costs and bene(cid:133)ts of liquidity and capital requirements. It provides a rationale for their joint use and characterizes the best division of labor in order 1See, e.g., Begenau (2020), Begenau and Landvoigt (2022), Clerc at al. (2015), Elenev et al. (2021), Martinez-Miera and Suarez(2014),Nguyen (2015),and Van den Heuvel(2008),alldiscussed furtherbelow. 2A classic reference is Friedman (1960). See Cochrane (2014) for a recent proposal. Gorton et al. (forthcoming) note the similarity between the liquidity coverage ratio (LCR), a key Basel III liquidity requirement,andnarrowbanking. TheyarguethatthehistoricalexperiencefromtheU.S.NationalBanking Era suggests that narrow banking is unlikely to be desirable. I examine narrow banking in section 6.4. 2

to foster (cid:133)nancial stability in the least costly way. Second, and this is the most important contribution, it derives two simple formulas for the magnitude of the welfare costs of capital and liquidity requirements. These formulas are su¢ cient statistics for the marginal welfare costs and are functions of observable variables only, sidestepping the di¢ culties inherent to full-model calibration or estimation (Chetty, 2009). The third contribution is quantitative: the paper deploys the su¢ cient statistics, using U.S. data, in order to measure the welfare cost of each requirement. TheframeworkisbasedonVandenHeuvel(2008)andembedsliquidity-creatingbanks in an otherwise standard general equilibrium growth model. Due to their role in the provision of liquidity services, bank liabilities are special and, as a result, the Modigliani-Miller theorem fails to hold for banks: their capital structure is not irrelevant. The welfare costs of the capital and liquidity requirements depend crucially on the value of the liquidity provided by bank deposits and by government bonds (cid:150)a safe and liquid asset that can be used to satisfy the liquidity requirement. For this reason, investors(cid:146)preferences for liquidity are modeled in a (cid:135)exible way. A key insight is that equilibrium (cid:133)nancial spreads reveal the strength of these preferences for liquidity and this allows us to quantify the welfare costs without imposing restrictive assumptions on preferences. Furthermore, the analysis shows how capital and liquidity requirements can a⁄ect capital accumulation and the size of the banking sector. The formulas for the welfare costs take these general equilibrium feedbacks into account. The model also incorporates a rationale for the use of both regulations, based on a moral hazard problem created by deposit insurance (or similar types of government guarantees), which if unchecked, can lead banks to take on excessive credit and liquidity risk. Themain(cid:133)ndingsareasfollows. Thepreferenceforliquidityimpliesthatthepecuniary returns on liquid assets (cid:150)bank deposits and Treasuries (cid:150)are lower than the returns on nonliquid assets (cid:150)equity in the model. For banks, this departure from Modigliani-Miller can result in a binding capital requirement. The liquidity requirement (cid:150)a minimum ratio of banks(cid:146)holdings of Treasuries to their deposit liabilities (cid:150)binds if the convenience yield on Treasuries exceeds the convenience yield on bank deposits, net of the non-interest cost of servicing those deposits. Because of competition, banks pass on the cheap deposit funding to borrowers in the form of a lower lending rate. However, if binding, both the capital requirement and the liquidity requirement limit the extent of this pass-through. Possible non-interest costs of (cid:133)nancial intermediation can also increase the lending rate. If the net impact of these factors is such that bank loans are still relatively inexpensive, (cid:133)rms will borrow exclusively from banks. Otherwise, the equilibrium will be one of both bank and 3

non-bank (cid:133)nance, and the size of the banking sector will be determined endogenously. As a consequence, in the model, liquidity and capital regulation can each lead to migration of (cid:133)nancial activity to non-bank intermediaries, such as shadow banks, or to disintermediation. Forliquidityregulation, thisoutcomeismorelikelyifthesupplyofhighquality liquid assets is low relative to the demand for such assets, so that their convenience yield is high. Moreover, these regulations can alter not only the composition of the (cid:133)nancial sector, but also the size of the economy, through their e⁄ect on (cid:133)rm investment. Turning to normative results, both capital and liquidity regulations are helpful in mitigating the moral hazard from deposit insurance and thereby preventing (cid:133)nancial crises. First, moral hazard can lead banks to take on excessive credit risk. A capital requirement is helpful in limiting this problem by ensuring that shareholders internalize potential losses. Second, moral hazard can lead banks to take on excessive liquidity risk. A liquidity requirement and a capital requirement are each helpful in mitigating this problem. However, a liquidity requirement addresses this problem more directly and e¢ ciently and is therefore sociallydesirable. Insum,themodelsuggestsasimpledivisionoflabor: liquidityregulation should address liquidity risk, and capital regulation should address credit risk. Thesebene(cid:133)tsarenotafreelunch,however,astheseregulationsalsoentailsocialcosts. If binding, each requirement reduces banks(cid:146)ability to perform liquidity transformation, a socially valuable activity. The model can be used as a lens to see how the magnitude of these costs can be measured with real-world data. As equilibrium asset returns reveal the strength of investors(cid:146)preferences for liquidity, two su¢ cient statistics can be derived for the marginal welfare costs of the two regulations (cid:150)two simple formulas that are functions of observable variables only, shown in section 5 (propositions 5 and 6). First, the cost of the capital requirement scales with the convenience yield on bank deposits. Second, the cost of the liquidity requirement scales with the di⁄erence in the convenience yields on Treasuries and on bank deposits. (In each case, there is an adjustment for banks(cid:146)net non-interest costs.) The intuition for the second result is that the liquidity requirement essentially removes Treasuries from non-bank investors and puts them in banks (cid:150)but banks can (cid:133)nance these new assets with deposits which, like Treasuries, also provide liquidity services. This entails a net social cost only to the extent that the liquidity services of bank deposits (net of their non-interest costs) are, at the margin, valued less than the liquidity services of Treasuries to non-bank investors. I then use U.S. data to measure these cost-revealing (cid:133)nancial spreads and the other variables in the su¢ cient statistics. The welfare cost of a 10 percent liquidity requirement 4

is found to be equivalent to a permanent loss in consumption of about 0.02%, a modest cost.3 Even using a conservative method, the cost of a similarly-sized increase in the capital requirement is found to be roughly ten times as large. The cost of a complete move to narrow banking would be another order of magnitude higher, about 2.4% of consumption. A comparison of these costs to existing estimates of the bene(cid:133)ts of capital and liquidity requirements suggests that the post-crisis reforms to capital and liquidity requirements have resulted in net increases in welfare, especially with regard to capital requirements. As a caveat, the model does not feature a lender of last resort that could save solvent banks with liquidity problems, which could lessen the need for ex-ante liquidity regulation. Becauseofthat,theanalysismayoverstatethebene(cid:133)cialroleofliquidityregulation,though thisdoesnotmatterfortheresultsonthecost. Thatsaid, inreality, thelenderoflastresort function of central banks is not completely free of challenges. Deciding whether a bank only experiences liquidity problems or liquidity and solvency problems can be di¢ cult in crisis times. And it has been argued that interventions by a lender of last resort can themselves lead to moral hazard problems (see, e.g., Farhi and Tirole (2012)). To the extent that the lender of last resort function entails economic costs, these could be compared to the costs of liquidity regulation, which this paper attempts to quantify.4 Several recent papers present quantitative, macroeconomic models of optimal bank capital regulation, including Begenau (2020), Begenau and Landvoigt (2022), Clerc at al. (2015), Elenev, Landvoigt, and Van Nieuwerburgh (2021), Martinez-Miera and Suarez (2014), and Nguyen (2015).5 In their calibrated versions, these models each yield an interior level of the capital requirement that maximizes a welfare criterion, with the optimal levels ranging from 6 percent in Elenev et al., whose model features (cid:133)nancially constrained producers, to 16 percent in Begenau and Landvoigt, whose model features unregulated as well as regulated banks. There are three main di⁄erences with the model developed here. First and most obviously, the above-mentioned papers do not aim to examine liquidity requirements, which is a focus in this paper. Second, the reason the Modigliani-Miller theorem fails for banks is di⁄erent, except for Begenau (2020) and Begenau and Landvoigt 3This is for a liquidity requirement that is modelled after the liquidity coverage ratio (LCR), one of two liquidityrulesintroducedbyBaselIII.Theotherruleisthenetstablefundingratio(NSFR),whichisoutside the scope of this paper. 4See Carlson et al. (2015) and Hoerova et al. (2018) for discussions of the relation between liquidity regulation and the lender of last resort. 5In addition, recent studies also provide quantitative examinations of optimal time-varying capital requirements; see, e.g., Canzoneri et al. (2020) and Davydiuk (2017). 5

(2022), in which, as in this paper, it fails chie(cid:135)y because banks provide liquidity services.6 Third,thesestudiesallrelyonafull-modelcalibrationtodrawoutquantitativeimplications, whereas the main results in this paper are obtained without calibration, using a (cid:147)su¢ cient statistics(cid:148)approach instead (based on a revealed preference logic).7 Chetty (2009) argues that such a su¢ cient statistics approach (cid:147)combines the advantages of reduced-form empirics (cid:150)transparent and credible identi(cid:133)cation(cid:150)with an important advantage of structural models (cid:150)the ability to make precise statements about welfare.(cid:148) This could be viewed as especially attractive in the context of macroeconomic models with (cid:133)nancial intermediation, because such models tend to have many parameters that are notoriously di¢ cult to calibrate or estimate. That said, a limitation of the approach used here is that it only quanti(cid:133)es the welfare costs of regulation, as sizing the bene(cid:133)ts does not lend itself readily to su¢ cient statistics.8 Instead, we will characterize the bene(cid:133)ts qualitatively and compare our measurements of the costs to existing estimates of the bene(cid:133)ts. Finally, there is an emerging literature on the theoretical bene(cid:133)ts of liquidity requirements,basedonpreventingbankrunsor(cid:133)resales,including,forexample,Calomiris,Heider, and Hoerova (2015), Diamond and Kashyap (2016), Kara and Ozsoy (2019), Kashyap, Tsomocos, and Vardoulakis (2020), and Vives (2014). Quantitative, positive examinations of the e⁄ects of liquidity and capital requirements are presented by Corbae and D(cid:146)Erasmo (2021), who examine their e⁄ects on bank risk-taking, market structure, e¢ ciency, and stability in a model of industry dynamics, by De Nicolo, Gamba, and Lucchetta (2014), who take a micro-prudential perspective, and by Covas and Driscoll (2014), who introduce these requirements into a DSGE model. The rest of this paper is organized as follows. The next section presents the model and analyzes agents(cid:146)decision problems. Section 3 provides an initial, qualitative overview of the welfareimplicationsofbankregulation,followedbyapositiveanalysisofgeneralequilibrium in section 4. Section 5 presents su¢ cient statistics for the social costs of regulation. These su¢ cient statistics areused in section 6to measure thewelfarecosts of increasesin liquidity and capital requirements, as well as a hypothetical government-imposed switch to narrow 6This is also the key friction in Gorton and Winton (2017) and Van den Heuvel (2008), who also show that bank capital requirements can have a social cost because they reduce the ability of banks to create liquidity in equilibrium. In addition, in Elenev at al. (2021)(cid:146)s model, the specialness of bank debt as a safe asset is one among four frictions that lead to a failure of Modigliani-Miller at banks. 7DÆvila (2019) uses a su¢ cient statistics approach to examine welfare-maximizing bankruptcy exemptions. Similarly, DÆvila and Goldstein (2021) use this approach to study optimal deposit insurance. 8Moreover,the parametersgoverning theirsize are especially hard to calibrate,asdiscussed in section 7. 6

banking. Section 7 revisits the welfare bene(cid:133)ts, and the (cid:133)nal section concludes. 2 The Model As mentioned, the model extends Van den Heuvel (2008), which adds two features to the standard growth model: (cid:133)rst, households have a need for liquidity and, second, certain institutions, labelled banks, are able to create (cid:133)nancial assets, bank deposits, which provide liquidityservices. Asanovelelementinthismodelrelativetoitsprecursor, bondsissuedby the government can also serve as liquid assets for households and businesses. In addition, government bonds can be used by banks to deal with liquidity risk and to satisfy liquidity regulation (cid:150)the main other new features in this paper. Since a central goal of the model is to provide a framework not just for illustrating, but for actually measuring the welfare cost of liquidity and capital requirements, it is important to model the preferences for liquidity in a way that is not too restrictive. As much as possible, the data should be allowed to provide the answer, not special modeling choices. To that end, I follow Sidrauski (1967) and a large literature in monetary economics in adopting the modeling device of putting liquidity services in the utility function. The advantage of this approach is its (cid:135)exibility. Crucially, all main results will be derived without making any assumptions on the functional form of the utility function, beyond the standard assumptions that it is increasing and concave, thus allowing the data to speak. Of course, this approach does not further our understanding of why households like liquid assets, but this is simply not the topic of this paper. That said, it is important to know that the Sidrauski modeling device is functionally equivalent to a range of more specialized, micro-founded models of liquidity demand, such as the Baumol-Tobin transaction technology or cash-in-advance, as shown by Feenstra (1986). In that equivalence, the utility function with money (or deposits) as an argument is simply a derived utility function. Because we will not impose any restrictions on that derived utility function, all results will hold for any of those more primitive models. The economy consists of households, banks, (non(cid:133)nancial) (cid:133)rms, and a government. Households own both the banks and the non(cid:133)nancial (cid:133)rms. These (cid:133)rms combine capital and labor to produce the single good. The rest of this section describes and analyzes these agents(cid:146)decision problems. 7

2.1 Households There is a continuum of identical households with mass one. Households are in(cid:133)nitely lived dynasties and value consumption and liquidity services. Households can obtain these liquidity services by allocating some of their wealth to bank deposits, an asset created by banks for this purpose. In addition, households also derive a convenience value from holding government bonds, which stems from their liquidity and safety. One can think of the household sector in this model as also encompassing certain non-bank (cid:133)nancial (cid:133)rms, suchasmoneymarketfunds, bondfunds, orpensionfunds, whichoftenmanagehouseholds(cid:146) holdings of government bonds on their behalf. Besides holding bank deposits, denoted d , or government bonds, b , households can t t storetheirwealthbyholdingequity,e . Theysupplya(cid:133)xedquantityoflabor,normalizedto t one,forawage,W . Taxesarelump-sumandequaltoT . Thereisnoaggregateuncertainty, t t so the representative household(cid:146)s problem is one of perfect foresight: max 1 (cid:12)tu(c ;d ;b ) t t t f ct;dt;bt;et g1t=0 t=0 X s.t. d +b +e +c = W 1+RDd +RBb +REe T t+1 t+1 t+1 t t t t t t t t t (cid:0) and subject to a no-Ponzi-game condition and initial wealth constraint for d +b +e . c 0 0 0 t is consumption in period t, (cid:12) is the subjective discount factor and RD, RB and RE are the t t t returns on bank deposits, government bonds, and (bank or (cid:133)rm) equity, respectively. The returns and the wage are determined competitively, so the household takes these as given. There is no distinction between bank and (cid:133)rm equity, since, in the absence of risk, they are perfect substitutes for the household and will thus yield the same return. The utility function is assumed to be concave, at least once continuously di⁄erentiable on R 3 ++ , increasing in all arguments, and strictly so in consumption: u c (c;d;b) (cid:17) @u(c;d;b)=@c > 0, u (c;d;b) @u(c;d;b)=@d 0 and u (c;d;b) @u(c;d;b)=@b 0. d b (cid:17) (cid:21) (cid:17) (cid:21) The (cid:133)rst-order conditions to the household(cid:146)s problem are easily simpli(cid:133)ed to RE = ((cid:12)u (c ;d ;b )=u (c ;d ;b )) 1 (1) t c t t t c t 1 t 1 t 1 (cid:0) (cid:0) (cid:0) (cid:0) RE RD = u (c ;d ;b )=u (c ;d ;b ) (2) t t d t t t c t t t (cid:0) RE RB = u (c ;d ;b )=u (c ;d ;b ) (3) t t b t t t c t t t (cid:0) Equation (1), which determines the return on equity, is the standard intertemporal Euler equation for the consumption-saving choice, with one di⁄erence: the marginal utility of consumption may depend on deposits and bond holdings. Because there is no aggregate 8

risk, the return on equity is essentially a risk-free rate on an asset that does not provide any liquidity bene(cid:133)ts.9 Equation (2) captures the convenience yield on bank deposits. If u > 0, the interest rate on bank deposits is below the equity return re(cid:135)ecting the liquidity d services provided by deposits. Equation (3) relates the spread between equity and bonds to the liquidity services of bonds in a similar fashion. 2.2 Banks Thereisacontinuumofbanks,whichmakeloanstonon(cid:133)nancial(cid:133)rms,mayholdgovernment bonds, and (cid:133)nance these assets by accepting deposits from households and issuing equity. The ability of banks to create liquidity through deposit contracts is their de(cid:133)ning feature. Banks last until they fail or choose to exit.10 Banks(cid:146)technology exhibits constant returns to scale and there is free entry into banking, so banks operate in an environment of perfect competition. The mass of banks is normalized to one. The balance sheet, and the notation, for the representative bank during period t is: Assets Liabilities L Loans D Deposits t t B Bonds E Bank Equity t t Thebankcanmakesafeorriskyloanstonon(cid:133)nancial(cid:133)rms. Risklessloansyieldagross rate of return RL, for sure, at the end of the period t. RL is determined competitively in t t equilibrium, so each bank takes it as given. Risky loans will be discussed below. Similarly, the bank takes as given the return on the (riskless) government bonds, RB, and the interest t rate on (insured) deposits, RD. t For quantitative realism the model allows for resource costs associated with servicing deposits and/or making loans. Speci(cid:133)cally, a bank incurs a noninterest cost g(D;L) to service those (cid:133)nancial contracts. g is assumed to be nonnegative, twice continuously di⁄erentiable, (weakly) increasing, convex and homogenous of degree 1, i.e. it exhibits constant returns to scale. Note that costless intermediation is included as a special case (g 0), as (cid:17) is a linear cost function. 9Any bank- or (cid:133)rm-speci(cid:133)c idiosyncratic risk in equity returns (described below) is perfectly diversi(cid:133)ed by households. 10Becausetherearenoadjustmentcosts,noranyagencyproblemsbetweenbanksandtheotheroptimizing agents (households and (cid:133)rms), each bank(cid:146)s decision problem can be separated into a series of independent static decision problems. As explained below, a bank can fail due to loan defaults or liquidity stress, if it engages in excessive risk taking. Exiting takes the form of operating with scale set to zero. 9

Regulation Banks are subject to regulation, as well as supervision, by the government. First,banksfaceacapitalrequirement,whichrequiresthemtohaveaminimumamount ofequityasafractionofrisk-weightedassets. Inthecontextofthissimplemodel,thecapital requirement states that equity needs be at least a fraction (cid:13) of loans for a bank to be able to operate: E (cid:13)L t t (cid:21) For the moment, the capital requirement is merely assumed. It will later be shown how it can be socially desirable to have such a requirement, as it mitigates the moral hazard problemthatarisesduetothepresenceofdepositinsurance, discussedfurtherbelow. There is no rationale in the model for requiring equity against the bank(cid:146)s holdings of government bonds (which are assumed to be riskless). Accordingly, I have assumed that government bonds have a zero risk weight. Second, banks must satisfy a liquidity requirement by holding a minimum level government bonds, set equal to a fraction (cid:21) of deposits: B (cid:21)D t t (cid:21) Again, for the moment this regulation is merely assumed, but later it will be shown how it can be socially desirable in the presence of liquidity risk and the externalities associated with deposit insurance. 2.2.1 Assumptions Pertaining to the Bene(cid:133)ts of Regulation Theremainingassumptionsregardingbanks,detailedinthissubsection,areonlyrelevantto the bene(cid:133)ts of the capital and liquidity requirements, and for characterizing the conditions under which (cid:133)nancial crises occur in the model, but not for the welfare costs of regulation. These assumptions add features (cid:150)deposit insurance, credit risk, and liquidity risk (cid:150)that shape the moral hazard problem of excessive risk taking. It is worth noting, however, that the equilibrium analysis in section 4 will primarily focus on the case that regulation is su¢ ciently stringent (cid:150)according to conditions that will be derived (cid:150)so that banks are deterred from engaging in excessive risk taking, a deterrence that is socially optimal under plausible conditions, as argued in section 3. In addition, the formulas for the gross welfare costs in section 5, and their measurement in section 6, are identical with or without the following assumptions. Deposit insurance A government-run deposit insurance fund ensures that no depositor su⁄ers a loss in the event of a bank failure. That is, all deposits are fully insured. The 10

rationale for the deposit insurance is left unmodeled. However, it has been argued that deposit insurance improves the ability of banks to create liquidity.11 Depositinsurancecreatesamoralhazardproblem: thebankhasanincentivetoengage in excessive risk taking. As this is the justi(cid:133)cation for capital and liquidity regulation, two bank risk choices are introduced, through credit and liquidity risk. Loans with credit risk By directing a fraction of its lending to (cid:133)rms with a risky technology, described below, the bank can create a loan portfolio with riskiness (cid:27) that t pays o⁄ RL + (cid:27) " , where " is an idiosyncratic shock with negative mean, denoted (cid:24) t t t t (cid:0) ((cid:24) 0). Further, " has a cumulative distribution function, F , that has bounded support " (cid:21) [";"], with < " < 0 < (cid:22)" < and Pr[" > 0] > 0. " is i:i:d: across banks and time (cid:0)1 1 periods. These properties of " derive from the risky technology, described below. Thenegativemeanoftheshockimpliesthattheexpectedreturnoftheloanportfoliois decreasing in its risk. It is in this sense that risk-taking is excessive: absent a moral hazard problem due to deposit insurance, the bank would always prefer (cid:27) = 0. While the bank t chooses (cid:27) , bank supervision imposes an upper bound: (cid:27) (cid:27)(cid:22). This will be explained more t t (cid:20) fully in the discussion of the government. Liquidity risk of deposits Deposits come with liquidity risk for the bank. This feature is introduced not just for realism, but also to provide a rationale for liquidity regulation. Consistent with the design principles of Basel III(cid:146)s liquidity coverage ratio (LCR) requirement, it is assumed that, with a small probability, an unusually high fraction of depositors decide to withdraw early, before the bank has received the income from the loans it has made. The occurrence of this event, termed (cid:147)liquidity stress,(cid:148)can be thought of as the realization of bank-speci(cid:133)c liquidity risk.12 The bank can cover these early withdrawals by drawing down its stock of liquid securities, i.e. its holdings of government debt. In contrast, loans are fully illiquid and no secondary market exists for loans. As a consequence, if the bank does not have su¢ cient government bonds to cover the intra-period withdrawals, the bank defaults and goes into bankruptcy protection. Shareholders get zero in this case, while depositors are made whole by the deposit insurance scheme. The resolution through the deposit insurance fund is dis- 11Diamond and Dybvig (1983) provide a model of liquidity provision by banks, in which socially undesirable, panic-based bank runs can occur, and in which deposit insurance can prevent these runs. 12The model is silent about whether this liquidity stress is panic-based or based on a fundamental (preference) shock. 11

cussed in more detail below. The assumption of complete illiquidity of loans is admittedly an extreme one. The key idea, however, is that loans are less liquid, especially in times of stress, and that this can make it desirable for banks to hold more liquid securities in anticipation of stressed out(cid:135)ows. The question then becomes whether the private incentives to hold liquid assets are as strong as the social bene(cid:133)ts. Formally, let (cid:17) be a random variable that takes on the value one when liquidity stress materializes, and zero otherwise, and let p be the probability that a bank su⁄ers liquidity stress. Thus, (cid:17) = 1withprobabilitypand(cid:17) = 0withprobability1 p. Itisassumedthat(cid:17) (cid:0) and"aremutuallyindependent,and,like",(cid:17)isi:i:d:acrossbanksandtimeperiods. Denote the fraction of depositors who decide to withdraw early when (cid:17) = 1 by w. Thus, a bank fails due to liquidity stress if B < wD and (cid:17) = 1. It is assumed that early withdrawers use their funds to make payments to other households, who then deposit the funds into the banking system. To economize on notation and avoid having to keep track of intraperiod balance sheet changes, I adopt the simplifying assumption that those banks that experienced the liquidity out(cid:135)ows are also shortly thereafter recipients of liquidity in(cid:135)ows of the same magnitude (regardless of whether they survived the acute liquidity stress or are in FDIC resolution). Although this is clearly not the most realistic assumption, it simpli(cid:133)es the analysis and more realistic assumptions would lengthen the exposition without yielding additional insights. As mentioned, the assumptions regarding the deposit insurance, excessive risk taking throughlending,liquiditystressandbanksupervisiongiverisetothebene(cid:133)tsofregulations, but do not matter for their welfare costs, including measurements thereof. 2.2.2 The Bank(cid:146)s Decision Problem The objective of the bank is to maximize shareholder value, net of the initial equity investment:13 (cid:25)B = max E (1 1 (cid:17)) (RL+(cid:27)")L+RBB RDD g(D;L) + =RE E (cid:27);L;B;D;E (cid:0) f B<wD g f (cid:0) (cid:0) g (cid:0) h i s.t. L+B = E +D; E (cid:13)L; B (cid:21)D; and (cid:27) [0;(cid:27)(cid:22)] (4) (cid:21) (cid:21) 2 The notation x + stands for max(x;0) and 1 is an indicator variable taking the B<wD f g f g value1ifB < wD andzerootherwise, re(cid:135)ectingthefactthatthebankwillfacebankruptcy 13Each bank is potentially long-lived. However, because there are no adjustment costs, nor any agency problems between banks and the other optimizing agents (households and (cid:133)rms), its decision problem can beseparatedintoaseriesofindependentstaticdecisionproblemswithoutlossofgenerality. Inwhatfollows, time subscripts will be used only where necessary to avoid confusion. 12

due to liquidity stress if both B < wD and (cid:17) = 1. The constraints are, respectively, the balance sheet identity, the capital requirement, the liquidity requirement, and the supervisory bound on (cid:27). The term (RL + (cid:27)")L + RBB RDD g(D;L) is the bank(cid:146)s net cash (cid:135)ow at the (cid:0) (cid:0) end of the period, provided there was no failure due to liquidity stress. It consists of interest income from loans and bonds, minus any possible charge-o⁄s on the loans, minus the interest owed to depositors, and minus the resource cost of intermediation. If the net cash-(cid:135)ow is positive, shareholders are paid this full amount in dividends. If the net cash (cid:135)ow is negative, the bank fails and the deposit insurance fund must cover the di⁄erence in order to indemnify depositors, as limited liability of shareholders rules out negative dividends. Shareholders receive zero in this event or if the bank has already failed due to liquidity stress, so dividends equal the expression inside the square brackets. E is the initial investmentoftheshareholders. Atthebeginningofperiodtshareholdersdiscountthevalue of end-of-period dividends by the opportunity cost of holding this particular bank(cid:146)s equity. This opportunity cost is RE, the market return on equity. If (cid:27) > 0 or if B < wD, dividends are risky, but this risk is perfectly diversi(cid:133)able, so shareholders do not price it.14 2.2.3 Analysis of the Bank(cid:146)s Problem The analysis will start with the credit risk choice of the bank. Next, we will turn to its other balance sheet choices for the case that the liquidity requirement exceeds the level of stressed withdrawals ((cid:21) w), forcing the bank to self-insure against the liquidity stress. (cid:21) And, (cid:133)nally, we will examine the bank(cid:146)s liquidity risk choice and other choices when (cid:21) < w. Credit risk choice First, consider the choice of loan risk, (cid:27), conditional on L, B, D and E. For convenience, de(cid:133)ne r RL +RB(B=L) RD(D=L) g(D=L;1) > 0, a measure (cid:17) (cid:0) (cid:0) of the bank(cid:146)s return on assets without excessive risk taking.15 In this notation, expected dividends are E[ (r+(cid:27)")L +] if B wD, or (1 p)E[ (r+(cid:27)")L +] if B < wD (using f g (cid:21) (cid:0) f g the independence of (cid:17) and "). Due to the max operator, the expected value is a convex function of (cid:27).16 For low values of (cid:27), expected dividends are decreasing in (cid:27), re(cid:135)ecting the negative mean of the shock " (cid:150)this is the cost of excessive risk-taking. But at higher levels of (cid:27), there is not enough equity to absorb the loss in the event of a large negative 14Hence, the treatment of RE as nonstochastic in the household problem is also still correct, since, even if banks are risky, households would not leave any such risk undiversi(cid:133)ed. 15Recall that g is linear homogenous. 16See Appendix A.1 for proof of convexity. 13

realization of ". In that event, the excess loss is covered by the deposit insurance fund. Increasing risk further at this point can increase expected dividends, as it raises the payo⁄ to shareholders in the good states (" > 0) without lowering it in (some of the) bad states (cid:150)this is the bene(cid:133)t of excessive risk-taking to shareholders. Put di⁄erently, the value of the put option associated with the deposit insurance fund rises with (cid:27).17 Because of the convexity of expected dividends, only (cid:27) = 0 and (cid:27) = (cid:27)(cid:22) need to need to be considered as candidates for the optimal choice of risk. Comparing expected dividends for these two values, and imposing further optimality conditions, yields the following result: Proposition 1 (Credit risk choice) A su¢ cient condition for no excessive credit risk taking ((cid:27) = 0) is given by: (cid:30) (cid:27)(cid:22) (cid:13)RE (5) " (cid:20) This condition is also necessary if the capital requirement binds and B wD. If (cid:27) = 0, (cid:21) 6 then (cid:27) = (cid:27)(cid:22). Proof: See Appendix A. Here, (cid:30) is a (cid:147)value-at-risk(cid:148)statistic derived from the distribution of ". It is implicitly " de(cid:133)ned by (cid:30) (cid:0) " ("+(cid:30) )dF (") (cid:24) (6) " " (cid:17) (cid:0) " Z TheassumptionsmaderegardingthedistributionfunctionF implythat(cid:30) exists,isunique " " and satis(cid:133)es 0 < (cid:30) " (see the appendix). (cid:30) " for small values of (cid:24) .18 Note that " " (cid:20) (cid:0) (cid:25) (cid:0) condition (5) depends only on variables that the bank takes as given. Intuitively, a su¢ ciently high capital requirement ((cid:13)) can deter excessive risk taking by ensuring that the bank internalizes enough of the losses that may arise as a result of such risk taking. Excessive risk is also less appealing if supervision is strong (captured by a low (cid:27)(cid:22)). Conversely, risk taking is more attractive if the bank has little (cid:145)skin-in-the-game(cid:146)(a low (cid:13)), if supervision is weak, if the distribution of " has a long and fat left tail (high (cid:30) ), or if " the cost of excessive risk-taking is small (low (cid:24), which also implies a higher value of (cid:30) ).19 " 17As an example, suppose " equals either (cid:24)+a or (cid:24) a, with equal probability, and with a > (cid:24) (so (cid:0) (cid:0) (cid:0) that Pr[">0]>0 as assumed). Then E f (r+(cid:27)")L g + = 0:5(r (r +(cid:0)(cid:27) (cid:27) (a (cid:24))L (cid:24)) i ) f L (cid:27) (cid:20)if r (cid:27) =(a r + =( (cid:24) a ) +(cid:24)) (cid:0) (cid:21) 18For the illustrative distribution in th(cid:2)e previous foo(cid:3)tnot(cid:8)e, (cid:30) " =a (cid:0) (cid:24). As a further example based on a continuous distribution, suppose " is uniformly distributed on the interval [ (cid:24) a; (cid:24)+a] with a > (cid:24) (so (cid:0) (cid:0) (cid:0) that (cid:22)">0 as required). Then it is straightforward to show that (cid:30) = pa p(cid:24) 2 (0;a). " (cid:0) 2 19Inmoredetail,(cid:30) " isthevalueatrisksuchthatlossesinexcessofth(cid:0)atvalue(" (cid:1) < (cid:0) (cid:30) " )areinexpectation just equal to the mean of ", (cid:24). When (cid:30) (cid:27) = (cid:13)RE, the bank becomes just insolvent when " = (cid:30) . In (cid:0) " (cid:0) " 14

An interesting corollary is that the liquidity requirement has no impact on the bank(cid:146)s incentives to make excessively risky loans.20 There are two reasons for this invariance result. First, this type of excessive risk taking occurs through lending, creating credit risk, notliquidityrisk. Theresultmaystillseemsurprising,sincetheliquidityregulationrequires the bank to hold more safe assets (government bonds), which ought to reduce its overall credit risk and the scope for excessive risk taking. However, this is where the second reason comes in: There are no assumptions in the model that arti(cid:133)cially limit the level of the bank(cid:146)s total assets. True, a higher liquidity requirement requires the bank to hold more bonds, but the bank does not have to reduce its loan portfolio as a result (cid:150)it can simply raise more deposits and invest the proceeds in bonds, leaving the scope for excessive risk taking through lending unchanged. Theremainderofthissectionwillfocusonthecasethatthecapitalrequirementsatis(cid:133)es condition (5); that is, (cid:13) (cid:30) (cid:27)(cid:22)=RE. In that case, the bank opts not to take on excessive " (cid:21) credit risk. The next section will return to the alternative case, with excessive credit risk, and will explain why that case is socially undesirable due to the costs and distortions associated with bank failures. Solution to the bank(cid:146)s problem under a high liquidity requirement ((cid:21) w) This (cid:21) part considers the case of a high liquidity requirement, de(cid:133)ned as (cid:21) w. In that case, the (cid:21) regulation forces the bank to self-insure against liquidity stress, so that the bank cannot engage in excessive risk taking through liquidity risk. With capital regulation stringent according to condition (5), the bank additionally forgoes excessive risk in lending ((cid:27) = 0), as explained. The bank(cid:146)s maximization problem in (4) then simpli(cid:133)es to: (cid:25)B = max E RLL+RBB RDD g(D;L) =RE E (7) L;B;D;E (cid:0) (cid:0) (cid:0) (cid:2) (cid:3) s.t. L+B = E +D; E (cid:13)L; B (cid:21)D (cid:21) (cid:21) It is straightforward to solve this problem (see Appendix A.2). To summarize the results, it is convenient to (cid:133)rst de(cid:133)ne the all-in cost of (cid:133)nancing a unit of loans with deposits, taking into account the liquidity requirement (but setting aside the transaction costs g(D;L)): (cid:21) R~D((cid:21)) RD + (RD RB) (8) (cid:17) 1 (cid:21) (cid:0) (cid:0) that situation, the expected bene(cid:133)ts of risk shifting for the bank (the expected shortfall) are equal to the expected costs (the reduction in NPV due to the negative mean of "). 20In fact, condition (5) is identical to the analogous condition in Van den Heuvel (2008, equation 10), exceptthatthere(cid:30) =1,re(cid:135)ectinganormalizingassumptionadoptedonthedistributionof"inthatpaper. " 15

This re(cid:135)ects the fact that a fraction (cid:21) of the deposits must be invested in bonds, rather than loans, so to (cid:133)nance one unit of loans with deposits, 1=(1 (cid:21)) deposits must be raised, (cid:0) of which (cid:21)=(1 (cid:21)) are put in bonds. If the return on bonds is less than the interest paid (cid:0) to depositors, then the liquidity requirement e⁄ectively increases the cost of (cid:133)nancing loans with deposits. To build intuition, I (cid:133)rst characterize the results for the special case of zero resource costs of intermediation (g 0):21 (cid:17) Proposition 2 (Solution for costless intermediation) For the special case of costless intermediation (g 0) and with regulation satisfying (cid:21) w and (cid:13) (cid:30) (cid:27)(cid:22)=RE (so (cid:27) = 0), " (cid:17) (cid:21) (cid:21) the existence of a (cid:133)nite solution to the bank(cid:146)s problem requires RB RD RL RE and, (cid:20) (cid:20) (cid:20) hence, R~D((cid:21)) RD. The solution satis(cid:133)es the zero-pro(cid:133)t condition: (cid:21) RL = (cid:13)RE +(1 (cid:13))R~D((cid:21)) (9) (cid:0) resulting in (cid:25)B = 0. Finally, The liquidity requirement binds (so B = (cid:21)D) if and only if RB < RD or, equivalently, (cid:15) if and only if R~D((cid:21)) > RD. The capital requirement binds (so E = (cid:13)L) if and only if RL < RE or, equivalently, (cid:15) if and only if R~D((cid:21)) < RE. Proof: See Appendix A.2. Equation (9) has the interpretation of a zero-pro(cid:133)t condition. For a bank with a binding capital requirement, one unit of lending is (cid:133)nanced by (cid:13) in equity and (1 (cid:13)) (cid:0) in deposits. Thus, competition will equalize the rate of return to lending to the similarly weighted average of the required rates of return of equity and deposits. Condition (9) also takes into account that deposit (cid:133)nance of loans e⁄ectively costs R~D((cid:21)) due to the liquidity requirement(cid:146)sprescriptionthatsomeofthedepositsraisedbeinvestedingovernmentbonds. The all-in cost of (cid:133)nancing loans with deposits, R~D((cid:21)), exceeds RD when the return on bonds is less than the interest paid to depositors and it is under that condition, RB < RD, that the liquidity requirement binds; otherwise, if bonds and deposits yield the same, R~D((cid:21)) = RD. (RB > RD is ruled out, as it is incompatible with a (cid:133)nite solution: it would yield in(cid:133)nite pro(cid:133)ts, since the bank can always raise deposits and invest the proceeds in governmentbonds, whichdonot requirecapital; this starkimplicationis relaxedin thecase of costly (cid:133)nancial intermediation.) 21This proposition is really a corollary to the more general proposition that follows it. 16

The capital requirement binds if equity (cid:133)nance is more expensive than deposit (cid:133)nance, taking into account the impact of the liquidity requirement on the all-in cost of deposit (cid:133)nance. In that situation, the rate on loans will be strictly in between RE and R~D((cid:21)). In contrast, the capital requirement is slack when RE = R~D((cid:21)) and then RL = RE = R~D((cid:21)), so (9) holds trivially. Regardless of whether the two regulatory constraints are slack or binding (cid:150)all four cases are possible (cid:150)economic pro(cid:133)ts are zero due to the constant returns to scale and perfect competition. Shareholders simply get the competitive return, RE. The following proposition summarizes the results for the more general case with nonnegative intermediation costs. Proposition 3 (Solution for general intermediation cost) Assume regulation satis- (cid:133)es (cid:21) w and (cid:13) (cid:30) (cid:27)(cid:22)=RE, so (cid:27) = 0. A (cid:133)nite solution to the bank(cid:146)s problem requires " (cid:21) (cid:21) RB RD +g (D;L) RL g (D;L) RE (10) D L (cid:20) (cid:20) (cid:0) (cid:20) The liquidity requirement binds if and only if the (cid:133)rst inequality is strict. The capital requirement binds if and only if the last inequality is strict or, equivalently, if and only if R~D((cid:21))+ 1 g (D;L) < RE. The solution satis(cid:133)es the zero-pro(cid:133)t condition: 1 (cid:21) D (cid:0) 1 RL g (D;L) = (cid:13)RE +(1 (cid:13)) R~D((cid:21))+ g (D;L) (11) L D (cid:0) (cid:0) 1 (cid:21) (cid:26) (cid:0) (cid:27) resulting in (cid:25)B = 0. Four cases are possible: 1. If RB= RE, then both regulatory requirements are slack, and all relations in (10) hold with equality. 2. If RB< RE, RB< RD+g ((cid:26);1) and RL g ((cid:26);1) RE , then the liquidity requirement D L (cid:0) (cid:21) binds, so B = (cid:21)D, the capital requirement is slack, and RB< RD+g (D;L) < RL g (D;L) = RE= R~D ((cid:21))+ 1 g (D;L) D (cid:0) L 1 (cid:21) D (cid:0) 3. If RB< RE, RB RD+g ((cid:26);1) and RL g ((cid:26);1) < RE , then the liquidity requirement is D L (cid:21) (cid:0) slack, the capital requirement binds, so E = (cid:13)L, and RB= RD+g (D;L) < RL g (D;L) = (cid:13)RE+(1 (cid:13))RB< RE D L (cid:0) (cid:0) 4. IfRB< RE,RB< RD+g ((cid:26);1)andRL g ((cid:26);1) < RE ,thenbothregulatoryrequirements D L (cid:0) bind, so B = (cid:21)D and E = (cid:13)L, all inequalities in (10) are strict, and RL= (cid:13)RE+(1 (cid:13))R~D ((cid:21))+g((cid:26);1) (12) (cid:0) 17

Proof: See Appendix A.2. Here,(cid:26)isde(cid:133)nedasthevalueoftheratioD=Lwhenbothregulatoryconstraintsarebinding (so B = (cid:21)D and E = (cid:13)L): 1 (cid:13) (cid:26) (cid:0) (cid:17) 1 (cid:21) (cid:0) Proposition 2 closely mirrors proposition 1, except that the interest rates are now adjusted for the marginal resource cost of intermediation. Speci(cid:133)cally, the cost of deposit (cid:133)nance now includes not only the interest rate paid to depositors, RD, but also the cost of servicing an additional unit of deposits, g (D;L). Similarly, the bank deducts the marginal D cost of screening and servicing loans, g (D;L), from the lending rate that it receives from L borrowers. With these adjustments, the conditions for a binding liquidity or for a binding capital requirement remain the same, and competition still equalizes the lending rate to the weighted average of the required return on equity and the all-in cost of (cid:133)nancing loans with deposits, with the weights determined by the capital requirement. Because g exhibits constant returns to scale, the marginal zero-pro(cid:133)t condition in (11) continues to imply zero total economic pro(cid:133)ts ((cid:25)B = 0). Also re(cid:135)ecting constant returns to scale, only ratios (not levels)ofthebalancesheetcomponentsaredeterminedbytheaboveoptimalityconditions.22 The four cases enumerate when each of the regulatory requirements is binding or slack as a function of variables that the bank takes as given only (that is, independent of D and L). Liquidity risk choice and solution under a low liquidity requirement ((cid:21) < w) As mentioned, analysis of the bank(cid:146)s general problem in (4) di⁄ers when liquidity regulation is not as stringent (cid:150)that is, when (cid:21) < w (cid:150)since the bank must decide whether or not to self-insure against liquidity problems. That said, many of the details of the analysis are similar to the case (cid:21) w and so are relegated to Appendix A.3. I focus here on describing (cid:21) and explaining the results. The basic intuition is as follows. If holding government bonds is costly (in that RB < RD +g ) and the probability of liquidity stress is low, then the bank will be tempted to D forego the option to self-insure and to set B at its minimum level, (cid:21)D, which is less than stressed withdrawals, wD, and entails the risk of failure due to liquidity stress. This is especially likely when the required liquidity ratio (cid:21) is far below w. It is also especially likely if the bank has little equity, so that most of the losses from such an event are borne by the government. Thus, a capital requirement also turns out to improve incentives to self-insure 22Thepartialderivativesofgarehomogenousofdegreezero,sog (D;L)=g (D=L;1),etc. Asdiscussed D D below, equilibrium conditions pin down aggregate levels. 18

against liquidity risk, in addition to improving incentives with regard to the credit risk pro(cid:133)le of the bank. Formally, the following proposition describes the conditions under which the bank opts to hold liquid assets at a level commensurate with its liquidity risk pro(cid:133)le. Proposition 4 (Liquidity risk choice & solution with a low requirement) Suppose (cid:21) < w and (cid:13) (cid:30) (cid:27)(cid:22)=RE, so (cid:27) = 0. Let " (cid:21) w (cid:21) p (cid:16) (1 (cid:13)) (RD RB) (cid:13) RE +h (cid:17) (cid:0) 1 w (cid:0) 1 (cid:21) (cid:0) (cid:0) 1 p (cid:18) (cid:0) (cid:0) (cid:19) (cid:18) (cid:0) (cid:19) If(cid:16) 0thenthebankself-insuresagainstliquiditystressbysettingB wD andproposition (cid:20) (cid:21) 7 applies; the bank acts as if (cid:21) = w. If (cid:16) > 0, then the bank sets B = (cid:21)D, is at risk of failure due to liquidity stress, and proposition 8 applies. Proof: See Appendix A.3. The variable h is de(cid:133)ned in equation (34) and collects terms related to di⁄erences in intermediation costs between the two business models (that is, B wD versus B = (cid:21)D). (cid:21) For the special case of costless intermediation (g 0), h = 0. Propositions 7 and 8 can be (cid:17) found in the appendix and closely mirror proposition 2, with the following exceptions: If (cid:16) > 0,thebankfailsifthereisanepisodeofliquiditystress. Moreover,becauseshareholders get zero in that event, they require that the realized return on equity conditional on such stress not occurring is higher by a factor 1=(1 p). If (cid:16) 0, the bank(cid:146)s behavior is identical (cid:0) (cid:20) to a bank whose liquidity requirement equals w. As expected, an imprudent liquidity risk pro(cid:133)le is especially tempting if the spread RD RB is high, or if the level of stressed withdrawals w is high relative to the liquidity (cid:0) requirement (cid:21) (cid:150)these factors relate to the cost of self-insurance against liquidity stress. The temptation is smaller if the odds of liquidity stress ( p ) are high or if the capital 1 p (cid:0) requirement (cid:13) is high (cid:150)these factors relate to the expected (private) bene(cid:133)ts of insuring against distress from liquidity problems. A high capital requirement can help incentivize a prudent liquidity risk pro(cid:133)le by ensuring that the bank internalizes more of the potential losses from liquidity stress. In addition, even a liquidity requirement that is somewhat below w still has a similar positive incentivee⁄ect. Bothe⁄ectsarehelpfulif(realistically)theexactvalueofw isnotknownto the regulator. For example, if the regulator estimates that w is, say, 20 percent, but could be as high as, say, 30 percent for some banks, a 20 percent liquidity requirement could be su¢ cient if the capital requirement is high enough, avoiding the economic ine¢ ciency of a uniform 30 percent liquidity requirement. 19

2.3 Firms Non(cid:133)nancial (cid:133)rms cannot create liquidity though deposits. They can, however, produce output of the good using capital and labor as inputs. Capital (K ) is purchased at the t beginning of the period and can be (cid:133)nanced by issuing equity to households (EF) and by t borrowing from banks (L ), so K = EF +L .23 t t t t Firms can employ a riskless or a risky production technology. The riskless technology is standard. Output in period t is F(K ;H ), where H is hours of labor input and F() is a t t t well-behaved production function exhibiting constant returns to scale. A fraction (cid:14) of the capital stock depreciates during the period. Firms last for one period24 and each period, there is a continuum of (cid:133)rms with mass normalized to one, so each (cid:133)rm takes prices as given. The (cid:133)rm maximizes shareholder value net of initial equity investment, subject to the constraint that equity cannot be negative: (cid:25)F = max F(K;H)+(1 (cid:14))K WH RL(K EF) =RE EF K;H;EF 0 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:21) (cid:0) (cid:1) Here loans have been substituted out using the balance sheet identity. The (cid:133)rst-order conditions for the choices of labor and capital are standard: (H) F (K;H) = W (13) H (K) F (K;H)+1 (cid:14) = RL (14) K (cid:0) (EF) RL=RE = 1 (cid:22); (cid:22) 0; (cid:22)EF = 0 (15) (cid:0) (cid:21) A (cid:133)nite solution requires RE RL. If RE > RL, then EF = 0, so K = L. In other (cid:21) words, if bank loans are cheaper than equity (cid:133)nance, the (cid:133)rm chooses to use only bank loans to (cid:133)nance its capital. If RE = RL, the (cid:133)rm(cid:146)s (cid:133)nancial structure is not determined by individual optimality. These optimality conditions, together with the constant returns to scale assumption, imply that economic pro(cid:133)ts, (cid:25)F, equal zero. Instead of this riskless technology, (cid:133)rms can also choose to use a risky technology, in which case output is F(K;H)+(cid:27) "K, where " is the same negative mean, idiosyncratic RF shock as de(cid:133)ned in the subsection on loans with credit risk (and (cid:27) (cid:27)(cid:22)). The optimal RF (cid:21) loan contract with such a (cid:133)rm is the type of risky loan described above, which provides a 23In the model, non(cid:133)nancial (cid:133)rms do not want to hold any government bonds as RB RL RE in (cid:20) (cid:20) equilibrium, so explicit consideration of the possibility would not change any of the results. 24The absence of adjustment costs and agency problems implies that this is without loss of generality. One can think of ongoing (cid:133)rms as repurchasing their capital stock each period. 20

rationale for capital regulation.25 As mentioned, the analysis will mostly focus on the case that (5) holds, so that banks do not engage in excessive risk taking. No risky (cid:133)rms then exist in equilibrium. 2.4 Government The government runs (cid:133)scal policy, manages the deposit insurance fund, sets capital and liquidity requirements, and conducts bank supervision. The purpose of bank supervision is not only to enforce regulations, but also to monitor excessive risk taking by banks, (cid:27). Supervisors can to some degree detect such behavior and stop any bank that is (cid:145)caught(cid:146) attemptingtotakeonexcessiveriskinordertoprotectthedepositinsurancefund. Itseems reasonable to assume that a small amount of risk taking is harder to detect than a large amount. The largest level of risk-taking that is still just undetectable is (cid:27)(cid:22). The assumption of imperfect observability of excessive risk taking is important. If regulators could perfectly observe each bank(cid:146)s riskiness, they could simply adjust each bank(cid:146)s deposit insurance premium so as to make the bank pay for the expected loss to the deposit insurance fund, thus eliminating any moral hazard. They could also achieve this by adjusting each bank(cid:146)s capital requirement in response to its true risk. But such perfect observability is simply not realistic, whence the moral hazard problem.26 The government(cid:146)s (cid:133)scal policy is to maintain a constant level of government debt, B(cid:22). T is tax revenue spent on bank supervision. The model allows for a resource cost arising from the resolution of failed banks by the government. 0 denotes the deadweight Liq (cid:21) resolutioncostperunitofloansinbanksthatfailearlyduetoliquiditystress,while 0 Sol (cid:21) denotes the resolution cost per unit of loans in banks that fail due to insolvency. Lump-sum taxes are T = (RB 1)B(cid:22) +T +p1 ( (r (cid:27) (cid:24)))L (16) t t (cid:0) f Bt<wDt g Liq (cid:0) t (cid:0) t t rt=(cid:27)t (cid:0) +(1 p1 ) ( (r +(cid:27) "))L dF (") (cid:0) f Bt<wDt g Sol (cid:0) t t t " Z(cid:0)1 25See Van den Heuvel (2008), Appendix B, for details. 26The supervisory bound on (cid:27) can be viewed as a more granular risk-based capital requirement or a riskbased deposit insurance premium, but one based on observable risk. Under that interpretation, regulators deter detectable excessive risk taking by imposing a su¢ ciently high capital requirement, or a su¢ ciently highdepositinsurancepremiumonthatriskwhendetected.Theprecisevalueofthisrequirementorpremium when (cid:27) > (cid:27)(cid:22) is irrelevant, as it is never implemented in equilibrium. Not inconsistent with this, the model assumes that the bank actually pays a deposit insurance premium equal to zero; this is the actuarially fair deposit insurance premium when (5) holds (cid:150)the case I will focus on. 21

The terms on the right-hand side are respectively the (net) interest on the government debt, supervision spending, the cost of resolving banks that fail due to liquidity stress, net of gains/losses from the operation of these banks in resolution by the deposit insurance fund, andthelosstodepositinsurancefundduetobankfailuresassociatedwithloanlosses, including deadweight resolution costs. If (5) holds and (cid:21) w (or (cid:16) 0; see prop. 4), then (cid:21) (cid:20) taxes are simply: T = (RB 1)B(cid:22) +T. t t (cid:0) 3 Financial Stability Policy This section provides an initial, qualitative overview of the welfare implications of bank regulation and characterizes the best division of labor between capital and liquidity requirements. Optimal regulatory policy involves macroprudential trade-o⁄s. On the cost side, regulation reduces the ability of banks to create liquidity and impacts investment, as will be shown explicitly in later sections. On the bene(cid:133)t side, the capital requirement can deter excessive risk taking by banks, whether from lending to excessively risky borrowers or from holding inadequate bu⁄ers of liquid assets. The liquidity requirement only addresses the latter threat to (cid:133)nancial stability, but not the former. When it happens, banks(cid:146)excessive risk taking causes a high rate of bank failures, resulting in a situation that resembles a (cid:133)nancial crisis. Speci(cid:133)cally, in the model, the bank failure rate when banks take excessive risk is p (for liquidity risk), F ( r =(cid:27)(cid:22)) (for " t (cid:0) credit risk), or p + (1 p)F ( r =(cid:27)(cid:22)) (if both are present). Bank failures entail negative " t (cid:0) (cid:0) externalities: Losses are transferred onto the deposit insurance fund and ultimately to taxpayers. Moreover, this is not a mere transfer from taxpayers to banks, as bank failures and (cid:145)bailouts(cid:146)come with additional deadweight costs and negative spillovers, and their prospect can create distortions ex-ante. In the model, these costs are very simple: the ex-post resolution costs, and , and the ex-ante direct cost of lending to ine¢ cient, Liq Sol excessively risky (cid:133)rms, (cid:24). Of course, in reality such costs are vastly more complex. If these costs (that is, , , and (cid:24) in the model) are su¢ ciently high, then it will Liq Sol be socially optimal to deter banks(cid:146)excessive risk taking, even as the capital and liquidity regulations also entail welfare costs due to reduced liquidity creation by banks, as will be shownexplicitly. Motivatedbyestimatesofthecostsof(cid:133)nancialcrises,Iwilladopttheview thatavoiding(cid:133)nancialcrises (cid:150)and thus excessive risktaking (cid:150)is infact sociallydesirable.27 27Preventing (cid:133)nancial crises in the model requires fully deterring excessive risk taking, since banks(cid:146)risk choices have dichotomous solutions (cid:150)that is, either (cid:27) =0 or (cid:27) =(cid:27)(cid:22), and, if (cid:21)<w, liquid assets are either 22

Figure 1 illustrates the welfare implications of di⁄erent regulatory choices under this view. It shows the level of welfare as a function of the capital requirement for two levels of the liquidity requirement: (cid:21) = 0 and (cid:21) = w. The (cid:133)gure relies on results derived later in this paper regarding the welfare costs of the two requirements, on estimates of the magnitude of theirbene(cid:133)tsintermsofreducingtheexpectedcostsof(cid:133)nancialcrises,obtainedfromBCBS (2010), and on assumptions regarding the values of certain parameters (see Appendix B for details). The parameters in question are di¢ cult to know or estimate with any precision, so the (cid:133)gure should be regarded as illustrative. In particular, the numbers on the axes should not be taken seriously. Thesolidblacklinerepresentswelfarewithoutliquidityregulation. Forlowlevelsofthe capital requirement, welfare is low, re(cid:135)ecting the costs of widespread bank failures, which put the economy in a crisis-like state. These failures occur without adequate regulation as banks take on excessive credit and liquidity risk. Once the capital requirement is increased to its threshold level for no excessive credit risk taking ((cid:13) = (cid:30) (cid:27)(cid:22)=RE (cid:150)see proposition 1), " banksareincentedtorefrainfromsuchriskandthereisanupwardjumpinwelfare,re(cid:135)ecting a reduced risk of bank failures. Moving further to the right, a second upward jump occurs at a higher level of the capital requirement, at which banks have enough (cid:145)skin-in-the game(cid:146) (according to the condition in proposition 4) so that they self-insure against liquidity stress (B wD), further reducing the rate of bank failures and improving welfare.28 Outside the (cid:21) jumps, the relation between the capital requirement and welfare is negative, as indicated by the negative slope of the line segments. This re(cid:135)ects the gross welfare cost of the capital requirement due to reduced liquidity creation by banks, an e⁄ect that is characterized more precisely and quanti(cid:133)ed in sections 5 and 6. Welfare with liquidity regulation is depicted by the solid blue line. Speci(cid:133)cally, the liquidity requirement is set at the rate of deposit withdrawals in the event of liquidity stress ((cid:21) = w). As a result, there are no bank failures due to liquidity stress, and welfare is strictly higher for most levels of the capital requirement, as the gain from the reduction in bank failures exceeds any cost of reduced net liquidity creation by banks which now have to satisfy the liquidity requirement. Only for levels of the capital requirement that are high enough to incentivize prudent liquidity management with (cid:21) = 0 is welfare equal with and minimal((cid:21)D)orsu¢ cienttoforestallallliquiditystress(wD)(cid:150) andsinceallbanksmakethesamechoices. 28Thenetincreaseinwelfarere(cid:135)ectsthegainfromthereductioninbankfailuresminusthecostofreduced net liquidity creation by banks, which start to hold more government bonds. The (cid:133)gure is constructed such that the net increase is positive, consistent with the evidence showing high costs of (cid:133)nancial crises and the view that preventing such crises is socially desirable. 23

Figure 1: Welfare implications of (cid:133)nancial stability policies without liquidity regulation. As shown in the chart, the strictly highest level of welfare is achieved with liquidity regulation and with the capital requirement set at its (cid:133)rst threshold level, which deters excessive credit risk taking. This combination prevents bank failures from both forms of excessiverisktaking(cid:150)liquidityandcredit(cid:150)atthelowestcost. Althoughliquidityregulation isnotnecessarytopreventallexcessiverisktaking,usingonlycapitalregulationisine¢ cient becauseitrequiresahighercapitalrequirement(cid:150)whichiscostly(cid:150)anditresultsinB wD (cid:21) inanycase,sothatthegrosswelfarecostofincreasedgovernmentbondholdingsbybanksis the same as with (cid:21) = w. The liquidity requirement thus addresses the problem of excessive liquidity risk more directly and therefore more e¢ ciently. In sum, the socially optimal policy is to use both tools and set (cid:13) = (cid:30) (cid:27)(cid:22)=RE and (cid:21) = w. " This represents a simple division of labor: it is optimal to use the liquidity requirement to deal with liquidity risk and let the capital requirement deal with credit risk.29 29From proposition 4, it can be seen that slightly lower levels of (cid:21) (levels that will keep (cid:16) 0 given (cid:20) 24

The discrete changes in risk taking and welfare at threshold levels of the requirements (cid:150)the jumps in the chart (cid:150)are a stark implication of the model. They occur because banksarehomogenousandtheirriskchoiceshavedichotomoussolutions, witheitherzeroor maximumrisk.30 Althoughanalyticallyconvenient, thisimplicationisunlikelytogeneralize to environments with heterogenous banks or regulatory uncertainty. Toillustratethis,supposethattheregulatorisuncertainaboutthevaluesofthethresholds. This seems plausible since the underlying parameters, like the (cid:147)value-at-risk(cid:148)associated with excessively risk loans ((cid:30) ) and the probability and rate of stressed withdrawals (p " andw),aredi¢ culttoknowwithgreatprecision. Thedashedlinesin(cid:133)gure1showexpected welfare when there is uncertainty about the threshold levels of the capital requirement.31 As can be seen, the relation between the requirements and welfare is much smoother in the presence of regulatory uncertainty. Two further conclusions emerge in this setting. First, the welfare-maximizing level of the capital requirement is higher with uncertainty and, although not shown explicitly, it is also increasing in the degree of uncertainty. The higher level is needed as a precaution to ensure that the probability of excessive risk taking and associated bank failures is kept acceptably low. Exactly what (cid:145)acceptably low(cid:146)means depends on the cost of crises and the gross welfare cost of raising the capital requirement. If there is little cost of tightening regulation, then the probability of a crisis should be brought to (nearly) zero by setting a very high capital requirement. This leads to the second conclusion: with regulatory uncertainty, the optimal capital requirement depends negatively on the marginal welfare cost of the capital requirement. Similar conclusions hold for the liquidity requirement if the regulator is uncertain about the level of withdrawals in liquidity stress (w). Although not modelled, one might also expect similarimplicationsifbankswereheterogenousin,say,theirrisk-takingopportunities,which would likely result in a smoother relationship between the aggregate rate of bank failures and regulation. The next section will present a positive analysis of general equilibrium. Motivated by (cid:13) =(cid:30) (cid:27)(cid:22)=RE) may be able achieve the same level of welfare as with (cid:21)=w. In addition, to the extent that " the quality of supervision can be improved (perhaps by devoting more resources to bank supervision), that could improve welfare by reducing (cid:27)(cid:22). 30That is, either (cid:27) =0 or (cid:27) =(cid:27)(cid:22), and, if (cid:21)<w, liquid assets are either minimal ((cid:21)D) or high enough to forestall all liquidity stress (at least wD). 31Theuncertaintyiscapturedbythresholdsthatarenormallydistributed withstandarddeviationsequal to 0.02 around means set at 0.08 and 0.13. 25

the arguments above, it will focus on the case that regulation is successful in preventing excessive risk taking and (cid:133)nancial crises. The remainder of the paper will then characterize and quantify the gross welfare costs of the two requirements and, (cid:133)nally, revisit the welfare bene(cid:133)ts. 4 General Equilibrium Given a government policy (cid:21), (cid:13), and T, an equilibrium is de(cid:133)ned as a path of consumption, capital, employment, and (cid:133)nancial quantities and returns, for t = 0;1;2;:::; such that households, banks and (cid:133)rms all solve their maximization problems, taxes are set according to (16), and all markets clear: e = E +EF; d = D ; L = K EF; B +b = B(cid:22); H = 1 (17) t t t t t t t t t t t (cid:0) and, for the goods market, F(K ;1) (cid:24)(cid:27) L +(1 (cid:14))K = c +K +g(D ;L )+p1 L (18) t (cid:0) t t (cid:0) t t t+1 t t f Bt<wDt g Liq t +(1 p1 )F ( r =(cid:27) ) L +T (cid:0) f Bt<wDt g " (cid:0) t t Sol t Forthereasonsexplainedintheprevioussection,Iwillfocusonthecasethatregulatory policy deters excessive risk taking by banks: (cid:21) w and (cid:13) (cid:30) (cid:27)(cid:22)=RE. The government " (cid:21) (cid:21) can achieve this by setting (cid:21) and (cid:13) su¢ ciently high. In that case, there are no bank failures, as 1 = 0, (cid:27) = 0, and F ( r =(cid:27) ) = 0. By combining this with the market f Bt<wDt g t " (cid:0) t t clearing conditions, equations (1), (2), (3), (8), (13), (14) and (15), and proposition 3, the resulting equilibrium allocation can be characterized as a dynamic system in (K ;c ). This t t system is shown in full in Appendix C, which also provides a more technical discussion of its characteristics than what follows. Here, I highlight some key features of the equilibrium. First, the bank capital requirement typically binds inequilibriumduetotheconvenience yield on deposits, which makes them a cheaper source of funds for banks than equity (see (2)). For example, without noninterest costs of banking (g = 0), the capital requirement binds whenever the convenience yield on deposits exceeds a fraction (cid:21) of the convenience yield on government bonds.32 32With positive noninterest costs, the capital requirement binds under the same condition provided the convenienceyieldondepositsistakennetoftheirmarginalnoninterestcosts;thatis,thecapitalrequirement binds if u (c ;d ;b ) g (d ;L )u (c ;d ;b ) > (cid:21)u (c ;d ;b ). Moreover, as shown in Appendix C, this d t t t D t t c t t t b t t t (cid:0) condition always holds in equilibrium if g (d ;L )>0. L t t 26

Second, the liquidity requirement may or may not bind, depending on the convenience yield of government bonds relative to bank deposits. Speci(cid:133)cally, it binds when, at the margin, the convenience yield of Treasuries exceeds the convenience yield of bank deposits, net of the marginal noninterest costs of servicing those deposits; that is, if u (c ;d ;b ) > u (c ;d ;b ) g (d ;L )u (c ;d ;b ) b t t t d t t t D t t c t t t (cid:0) Third, investment can be a⁄ected by the capital requirement as well as the liquidity requirement, if binding. In equilibrium, the marginal product of capital is equated with the banks(cid:146)lending rate, which can be lower than the cost of equity and depend on these requirements. Two features of the model are key to understanding how and when this can happen. First, as noted, households(cid:146)liquidity preference implies that the pecuniary return on deposits is lower than the return on equity. Second, competitive banks will pass on the cheapdeposit(cid:133)nanceintheformofalowerlendingrate,butthispass-throughismoderated by regulation. Take the case that the capital and liquidity requirements both bind. Then the cheap deposit(cid:133)nancelowersthelendingrateby(1 (cid:13))(RE R~D((cid:21))),asafraction(cid:13) ofloansisstill (cid:0) (cid:0) (cid:133)nanced with bank equity and as the liquidity regulation raises the all-in cost of (cid:133)nancing loans with deposits by R~D((cid:21)) RD = (cid:21) (RD RB). Using the households(cid:146)(cid:133)rst-order (cid:0) 1 (cid:21) (cid:0) (cid:0) conditions for deposits and bonds and taking into account noninterest costs yields a net reduction in the lending rate, relative to the return on equity, that is equal to 1 (cid:13) u (c ;d ;b ) u (c ;d ;b ) RE RL = (cid:1) (cid:0) d t t t g (d ;L ) (cid:21) b t t t g (d ;L ) (19) t (cid:0) t K;t (cid:17) 1 (cid:21) u (c ;d ;b ) (cid:0) D t t (cid:0) u (c ;d ;b ) (cid:0) L t t c t t t c t t t (cid:0) (cid:18) (cid:19) Thisistheequilibriumanaloguetothebank(cid:146)szeropro(cid:133)tcondition(11). Thus,themarginal product of capital is given by u F K (K t ;1)+1 (cid:0) (cid:14) = R t L = (cid:12) (cid:0) 1 u c;t (cid:0) 1 (cid:0) (cid:1) K;t (20) c;t (see (1) and (14)). If the spread (cid:1) is positive, so that RL < RE, (cid:133)rms will rely exclusively K onthecheaperbankloansto(cid:133)nanceinvestmentandL = K insucha(cid:145)pure bank (cid:133)nance(cid:146) equilibrium.33 In that equilibrium, because banks pass on the low cost of deposits to (cid:133)rms, thesteadystatecapitalstockishigherthanthelevelimpliedbythestandardgrowthmodel(cid:146)s modi(cid:133)ed golden rule. 33Appendix C shows that the condition (cid:1) > 0 is a necessary and su¢ cient condition for a pure bank K (cid:133)nance equilibrium (L=K), regardless of the bindingness of the regulatory requirements. That said, in a pure bank (cid:133)nance equilibrium the capital requirement always binds, because RL <RE (see proposition 3). 27

Moreover, as a consequence, the steady state levels of the capital stock and income per capitaarenotinvarianttochangesintheliquidityrequirementorinthecapitalrequirement, as these requirements in(cid:135)uence the spread RE RL (see (19)). With respect to the capital (cid:0) requirement, this non-invariance result is similar to the one obtained in Van den Heuvel (2008)andexploredmorefullyinBegenau(2020)andVandenHeuvel(2006). Withrespect to the liquidity requirement, the non-invariance result is, as far as I know, novel within the context of this type of model.34 As a fourth key feature, the equilibrium can also be characterized by (cid:145)mixed (cid:133)nance,(cid:146) where (cid:133)rms (cid:133)nance investment with a combination of bank and non-bank funding (so L < K). Inthemodel,non-bankfundingtakestheformof(cid:133)rmequity,butthiscanbeinterpreted more broadly as representing any funds raised on capital markets or through non-bank (cid:133)nancialintermediaries,includingshadowbanks. Technically,themixed(cid:133)nanceequilibrium occurs if (cid:1) < 0 when evaluated at L = K. In such an equilibrium, (cid:133)rms use both bank K loans and equity, in such proportion that, in equilibrium, their costs are exactly equal: RL = RE, and the relative size of the banking sector is then determined endogenously by that condition (or, equivalently, by (cid:1) = 0).35 K Fifth, very stringent regulation can lead to disintermediation or a shift to shadow banking. Intuitively, the mixed (cid:133)nance equilibrium prevails when the resource cost of bank intermediation, g, is high relative to the liquidity services of deposits, when the capital requirement is high, or because of the combination of a high liquidity requirement, (cid:21), and a high liquidity premium (low yield) on government bonds; see (19). Thus, a very high capital or liquidity requirement can cause migration toward non-bank (cid:133)nance, whether through capital markets or through non-bank intermediaries, such as shadow banks. For the liquid- 34The result contrasts starkly with the well-known superneutrality result of the Sidrauski (1967) model. In that model, liquidity preference and monetary policy do not in(cid:135)uence the steady state capital stock. A key di⁄erence is that the supply of liquid assets (money) is exogenous in the Sidrauski model. 35In contrast to the pure bank (cid:133)nance case, with RL = RE in a mixed (cid:133)nance equilibrium, the steady statelevelofthecapitalstock satis(cid:133)esthemodi(cid:133)ed golden rule(MPK =(cid:12)(cid:0) 1,asthereturn on equity does not command a convenience yield). Accordingly, the long-run level of capital is independent of liquidity preference or any banking or regulatory parameters, although these elements do in(cid:135)uence the composition of the (cid:133)nancial sector, as noted below. Given that (cid:133)rms in the real world do not exclusively use bank loans, it may seem that the mixed (cid:133)nance equilibriumismorerealistic,andthatthedependenceofeconomicactivityinthelongrunonregulationisa meretheoreticalpossibility. However,thatwouldbetakingthemodeltooliterallyinmyview,as,inreality, bank and non-bank funding are not perfectsubstitutes forall(cid:133)rms, asthey are in the modelby simplifying assumption. Inreality,some(cid:133)rmsarebank-dependent,andeven(cid:133)rmsthatcanaccesscapitalmarketsoften rely on backup lines of credit from banks to facilitate that access. 28

ity requirement, this is more likely to happen if the convenience yield on government bonds is high; that is, when the supply of government bonds or close substitute high-quality liquid assets is low relative to the demand for such assets. 5 The Welfare Costs of Regulation To quantify the welfare cost of the liquidity and capital requirements, we will use a social planner(cid:146)s problem that is constrained to respect the regulations. This planner(cid:146)s problem is designed to replicate the decentralized equilibrium, rather than to solve for the (cid:133)rst-best. After showing that the planner(cid:146)s allocation is indeed identical to the decentralized equilibrium, this equivalence will then be exploited to analytically derive two simple formulas that will serve as su¢ cient statistics for the welfare costs of the requirements. De(cid:133)ne the following constrained social planner(cid:146)s problem: V ((cid:18)) = max 1 (cid:12)tu(c ;d ;b ) (21) 0 t t t f ct;dt;bt;Bt;Lt;Kt+1 g1t=0 t=0 X s.t. K = F(K ;1)+(1 (cid:14))K c g(d ;L ) T; t+1 t t t t t (cid:0) (cid:0) (cid:0) (cid:0) B (cid:21)d ; (1 (cid:13))L +B d ; B +b = B(cid:22); K L t t t t t t t t t (cid:21) (cid:0) (cid:21) (cid:21) where (cid:18) = ((cid:21);(cid:13);K ). The constraints correspond to the social resource constraint (for 0 (cid:27) = 0and(cid:21) w), theliquidityrequirement, thecapitalrequirement, bondmarketclearing, (cid:21) and the nonnegativity constraint on (cid:133)rm equity, in that order. Appendix C shows that the allocation of this planner is identical to the decentralized equilibrium when regulation satis(cid:133)es (cid:21) w and condition (5) for all t 0, so that there is no excessive risk taking. (cid:21) (cid:21) Under these conditions, therefore, the constrained social planner(cid:146)s problem replicates the decentralized equilibrium and welfare in that equilibrium is equal to V ((cid:18)). 0 With that, it is straightforward to derive expressions for the marginal welfare costs of the two regulations by di⁄erentiating V ((cid:18)) with respect to (cid:21) and (cid:13), using the envelope 0 theorem. Combining the resulting expressions with the planner(cid:146)s (cid:133)rst-order conditions, exploiting the equivalence of the planner(cid:146)s allocation with the decentralized equilibrium(cid:146)s, and using the households(cid:146)optimality conditions for the choices of deposits and bonds, (2) and (3), yields the main results (cid:150)two su¢ cient statistics for the marginal welfare costs. These are presented in the next two propositions, starting with the liquidity requirement: 29

Proposition 5 (Gross welfare cost of the liquidity requirement) Assumethattheeconomy is in steady state in the current period, that (cid:21) w and that (5) holds. Consider (cid:21) permanently increasing the liquidity requirement (cid:21) by (cid:1)(cid:21). A (cid:133)rst-order approximation to the resulting welfare loss, expressed as the welfare-equivalent permanent relative loss in consumption, is (cid:23) (cid:1)(cid:21), where LIQ d (cid:23) = RD +g (d;L) RB (1 (cid:21)) 1 (22) LIQ D (cid:0) c (cid:0) (cid:0) (cid:0) (cid:1) Proof: See Appendix C. The above formula is empirically implementable. Remarkably, it does not rely on any assumptions about the functional form of preferences, beyond the standard assumptions of monotonicity, di⁄erentiability and concavity. Instead, the formula relies on asset yields to reveal the strength of the household(cid:146)s preference for liquidity. In addition, the measurements presented below will also avoid making any functional form assumptions on the cost function g. As is common for (cid:147)su¢ cient statistics(cid:148)formulas, there are multiple combinations of primitive parameters and functional forms that are consistent with the inputs to the formulas, and all such combinations have the same welfare implications (Chetty, 2009). The result shows that there is a positive (gross) welfare cost associated with bank liquidity regulation only to the extent that the interest rate on deposits, plus the marginal cost of servicing deposits, exceeds the interest rate on government bonds. The logic is simple: from the perspective of the other agents, the liquidity requirement e⁄ectively forces banks to transform some government bonds into deposits, both instruments prized for their liquidity. Thus, imposing a liquidity requirement entails a social cost only to the extent that the liquidity services of deposits are, at the margin and net of the noninterest cost of creating these services, valued less than those of Treasuries; only then is there a costly net reduction in liquidity available to investors. The deposit-Treasury spread, adjusted for the noninterest cost of deposits, reveals whether this is true or not. The formula takes into account gains and losses associated with the move to a new steady state and is valid whether the equilibrium is characterized by pure bank (cid:133)nance or by mixed bank and nonbank (cid:133)nance and even if the liquidity requirement does not bind (in which case RD +g (d;L) = RB, so (cid:23) = 0 as expected). The regulation entails a D LIQ gross social cost whenever the requirement binds and is costless otherwise. Moreover, the formula is valid even if we discard all of the assumptions adopted to generate the bene(cid:133)ts of regulation (section 2.2.1).36 36In that case, w and (cid:30) should be taken to equal zero in propositions 5 and 6. (cid:15) 30

The next key proposition presents a su¢ cient statistic for the marginal gross welfare cost of the capital requirement: Proposition 6 (Gross welfare cost of the capital requirement) Assumethattheeconomy is in steady state in the current period, that (cid:21) w and that (5) holds. Consider per- (cid:21) manently increasing the capital requirement (cid:13) by (cid:1)(cid:13). A (cid:133)rst-order approximation to the resulting welfare loss, expressed as the welfare-equivalent permanent relative loss in consumption, is (cid:23) (cid:1)(cid:13), where CAP L (cid:23) = RE R~D((cid:21)) (1 (cid:21)) 1g (d;L) (23) CAP (cid:0) D c (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) Proof: See Appendix C. Recall that R~D((cid:21)) RD + (cid:21) (RD RB). Again, the above formula is empirically imple- (cid:17) 1 (cid:21) (cid:0) (cid:0) mentable, does not rely on any assumptions about the functional form of preferences, and takes into account gains and losses associated with the move to a new steady state. It is valid whether the equilibrium is characterized by pure bank (cid:133)nance or by mixed (cid:133)nance, even if the capital requirement does not bind (in which case RE = R~D((cid:21))+(1 (cid:21)) 1g , (cid:0) D (cid:0) so (cid:23) = 0 as expected), and even if all assumptions adopted to generate the bene(cid:133)ts of CAP regulation (section 2.2.1) are discarded. An increase in the capital requirement beyond the threshold necessary for (cid:133)nancial stability lowers welfare by constraining the ability of banks to issue deposit-type liabilities, which are valued by households for their liquidity. The spread between the risk-adjusted37 return on bank equity and the pecuniary return on deposits, RE RD, reveals the strength (cid:0) of households(cid:146)preferences for the liquidity services of deposits. However, the production of these services also entails noninterest costs, g (d;L), and requires banks to hold more D government bonds (cid:150)which are also prized for their liquidity(cid:150)to satisfy the liquidity requirement. To account for this, the formula deducts the marginal noninterest cost of deposits38 andfactorsintheimpactoftheliquidityrequirement, (cid:21), byusingtheall-incostof(cid:133)nancing loans with deposits, R~D((cid:21)), instead of RD. Only if a positive spread remains after these adjustment, is a scarcity of deposits due the capital requirement revealed and only then is there a welfare e⁄ect at the margin. 37Recall that bank equity is aggregate-risk-free in the model, so there are no Modigliani-Miller (cid:145)o⁄sets.(cid:146) However,whenthemodelisconfrontedwiththedatainthenextsection,thiswillbeakeyareaofconcern. 38Thefactor1=(1 (cid:21))multiplyingg (d;L)re(cid:135)ectsthefactthatthebankmustraise1=(1 (cid:21))indeposits D (cid:0) (cid:0) to (cid:133)nance one unit of lending, while satisfying the liquidity requirement. 31

This result generalizes the one obtained in Van den Heuvel (2008), which does not feature liquidity regulation. Proposition 1 in the latter paper is nested by setting (cid:21) = 0 in (23):39 (cid:23) = (L=c) RE RD g (d;L) (24) CAP (cid:21)=0 D j (cid:0) (cid:0) Relatedly, the impact of the liquidity requir(cid:0)ement on the welfare c(cid:1)ost of the capital requirement can be seen more explicitly by rewriting (23) using (22) and (8): (cid:23) = (cid:23) (L=d)(cid:21)(cid:23) (25) CAP CAP (cid:21)=0 LIQ j (cid:0) Thus, for given observables (RE, RD, etc.) imposing a liquidity requirement lowers the welfare cost of the capital requirement if the liquidity requirement binds. Of course, these observables are generally not completely invariant to changes in the liquidity requirement. 6 Measurement of the Welfare Costs The goal of this section is to measure the gross welfare costs of bank liquidity requirements and capital requirements by combining the formulas derived in the previous section with data. To that end, I use annual aggregate balance sheet and income statement data for all FDIC-insured commercial banks in the United States. These data are obtained from the FDIC(cid:146)s Historical Statistics on Banking (HSOB) and are based on regulatory (cid:133)lings ((cid:145)call reports(cid:146)). I employ data from the period 1986 to 2019.40 A key challenge to the empirical application of the formulas in (22) and (23) is the measurement of the marginal net noninterest cost of servicing deposits, g . This includes D the cost of ATMs, some of the cost of maintaining a network of branches, etc. Fees on deposits should be netted out. The call reports contain data on noninterest expense and revenue, and the di⁄erence, net noninterest cost, is nontrivial, averaging 1.3 percent of total assets on an annual basis over the 1986-2019 period. However, there is little information in the data permitting a breakdown by activity (e.g. servicing deposits, screening loan applications, collecting payments, etc.). Fortunately, however, the model suggests a way to infer the marginal net noninterest cost of deposits when banks voluntarily hold Treasuries on their balance sheet. Speci(cid:133)cally, 39The statement of proposition 1 in Van den Heuvel (2008) also uses the fact that, in that model, L = d=(1 (cid:13)) if RE RD g (d;L)=0. D (cid:0) (cid:0) (cid:0) 6 40Regulation Q, which placed restrictions on banks(cid:146)deposit rates, was fully phased out on January 1, 1986. 32

Figure 2: U.S. Treasuries and excess reserves held by U.S. depository institutions proposition 3 shows that whenever the liquidity requirement is not binding, then RD +g (d;L) = RB (26) D The interpretation is banks will only hold more Treasuries than required if investing in Treasuries and (cid:133)nancing them with deposit-type liabilities is not a money-losing activity, taking into account noninterest costs. Thus, by (cid:133)nding a historical period when banks held Treasuries well in excess of any regulatory requirements, we can infer g from that period(cid:146)s D Treasury-deposit spread. Figure2showsU.S.TreasuriesandexcessreservesheldbyU.S.depositoryinstitutions, expressed as a share of total assets, from 1986 to 2019.41 As can be seen from the chart, between1986and2000, banksinvestedasigni(cid:133)cantpartoftheirbalancesheetinTreasuries 41Inasense,excessreservescanbethoughtofasholdingTreasuriesthroughtheFederalReserve(cid:146)sbalance sheet. Before 2008, excess reserves were not renumerated by the Fed and were a negligible share of banks(cid:146) total assets. In 2008, the Fed started paying interest on excess reserves and embarked on large scale asset purchases, developments that boosted excess reserves in the following years. See Ennis and Wolman (2015) for an empirical analysis of banks(cid:146)excess reserves in this period. 33

(more than 1 percent of total assets). This asset allocation was voluntary, as there was no Basel-style liquidity requirement applicable during this period (so (cid:21) = 0 in the sense of the model).42 Reserve requirements were in place, but these could only be satis(cid:133)ed by holding reserves at the Fed, not by holding Treasuries. Thus, I use data from the 1986-2000 period to infer g using equation (26).43 D ForRB, Iusethe3-monthTreasurybillrateonthesecondarymarket. Theaveragenet interest rate on deposits, RD 1, is calculated as the HSOB(cid:146)s Interest on Total Deposits di- (cid:0) videdbyTotalDeposits.44 Fortheperiod1986-2000, theresultingaverageTreasury-deposit spread, RB RD, equals 1:22 percent. Accordingly, I set the marginal net noninterest cost (cid:0) associated with deposits at g = 0:0122 per annum. D As a robustness check, a second estimate for the marginal net noninterest cost of depositscanbeobtainedfromHanson,Schleifer,Stein,andVishny(2015). Theyuseahedonic regressionapproachandestimatetheaveragecostofservicingdepositsat1:30percent. Netting out the noninterest income from service charges on deposit accounts (0:49%) yields a marginal net noninterest cost of 0:81 percent, a little below the (cid:133)rst estimate. It is useful to compare these estimates to an upper bound that can be obtained by attributing all net noninterest cost to servicing deposits and none to lending. Maintaining the assumption of constant returns to scale of g, this upper bound equals g (D;L) = D g(D;L)=D.45 The latter ratio is equal to 0:0216 per annum, on average for the same time period. Consistent with the model, the upper bound exceeds the spread-based estimate and suggests that about half of total net noninterest cost can be attributed to deposits (slightly more than half for the spread-based estimate and a bit less than half for the Hanson et al.-based estimate). This implication strikes me as plausible. To map the data into the remaining variables, I largely follow Van den Heuvel (2008). For deposits, D, the HSOB(cid:146)s Total Deposits is used. For consumption, c, personal consumption expenditures from the NIPA is used. For loans, I use Total Assets net of U.S. Treasuries and excess reserves. To quantify the welfare costs, I calculate long run averages of the ratios and the spreads in the formulas in propositions 5 and 6, starting with the 42Onecouldviewthisperiodasonewhereliquidityregulationwasnotneededbecausehigh-qualityliquid assets were abundant so banks voluntarily held them in su¢ cient amounts. 43Startingaround2014,banksalsoheldnotableamountofTreasuries. However,asdiscussedinmoredetail below, holdings in this period are a⁄ected by the anticipation and implementation of liquidity regulation. 44All data are nominal. While the model is real, using nominal data consistently is correct, because the formulas for the welfare costs contain only ratios of quantities and spreads of returns. 45Constant returns to scale imply g(D;L)=g (D;L)D+g (D;L)L g (D;L)D: D L D (cid:21) 34

welfare cost of liquidity requirements. 6.1 Liquidity Regulation I use data from two distinct periods to the gauge the long-run economic costs of liquidity regulation. The (cid:133)rst period is 2001-2007, a time without liquidity regulation and when the introduction of such regulation likely would have been binding. As can be seen in (cid:133)gure 2, in each of these years, banks(cid:146)holding of Treasuries plus excess reserves were less than 1 percent of total assets, indicating that even a modest liquidity requirement would have necessitated changes to banks(cid:146)balance sheets and making this period a good candidate to gauge its potential welfare cost.46 The second period, 2016-2019, covers the time after the implementation of the LCR, a key Basel III liquidity requirement.47 Starting with the pre-Basel III period, over 2001-2007, the average nominal yields on Treasuries and deposits are, respectively, 2.80% and 2.04%, so the average spread is 76 basis points, less than the marginal noninterest cost of servicing deposits, which we have already estimated at 122 basis points. The mean deposit to consumption ratio is 0.67, and, as already noted, there was no liquidity requirement in place ((cid:21) = 0). Applying (22), the (cid:133)rst-order approximation to the gross welfare cost of introducing a liquidity requirement set at (cid:21) > 0 is: new d (cid:23) (cid:21) = RD +g (d;L) RB (1 (cid:21)) 1(cid:21) LIQ new D (cid:0) new c (cid:0) (cid:0) = 0:6(cid:0)7 (0:0122 0:0076)(cid:1) 1 (cid:21) = 0:0031 (cid:21) new new (cid:2) (cid:0) (cid:2) (cid:2) (cid:2) Tointerpretthisnumber,considerthecostofa10percentliquidityrequirement,alevelthat is roughly comparable to Basel III(cid:146)s LCR requirement.48 Its gross welfare cost is equivalent to a permanent loss in consumption of (cid:23) 0:1 = 0:0031 0:1 100% = 0:031% LIQ (cid:2) (cid:2) (cid:2) 46The 2001-2007 period might be viewed as time where banks counted on the lender of last resort and the interbank market, but also as a time when there was an overreliance on such backstops and excessive liquidity risk taking. Under that view, liquidity regulation was in fact necessary in these years, with the global (cid:133)nancial crisis that followed these years in part the result of its absence. 47Theyearsbetweenthetwoperiodscontainedtheintensephaseoftheglobal(cid:133)nancialcrisis,threerounds of large scale asset purchases (QE) by the Fed, and several announcements that provided increasing clarity about the Basel III liquidity rules, making those years less arguably attractive for measuring their steady state welfare costs. 48The LCR requirement depends on more detailed balance sheet information than can be captured in a macroeconomicmodel. However,internationally,10%appearsareasonableratiotocapturetheLCR.Inthe U.S., the ratio appears somewhat higher, as elaborated below. 35

or about $4.5 billion per year (using 2019 consumption). While perhaps not trivial, this is a relatively small number compared to estimates of the welfare cost of in(cid:135)ation (see e.g. Lucas (2000)), or compared to many existing estimates of the cost of capital requirements (see e.g. BCBS (2010)). Even modest (cid:133)nancial stability bene(cid:133)ts would easily justify this cost.49 Using the alternative estimate for the net noninterest cost of servicing deposits that is based on Hanson, Schleifer, Stein, and Vishny (2015) (g = 0:81 percent) results in D an even lower measurement of the marginal welfare cost: (cid:23) = 0:0003. Using this, the LIQ gross welfare cost of a 10 percent liquidity requirement is equivalent to a permanent loss in consumption of only 0:003 percent, a tiny e⁄ect. Our second set of measurements covers the period following the implementation of the LCR, 2016-2019.50 Admittedly, this time span includes fewer years, raising the risk that short-term (cid:135)uctuations in spreads have an outsize in(cid:135)uence on the results. With that said, during this period, the average nominal yields on Treasuries and deposits are, respectively, 1.31%and0.50%,sotheaveragespreadis81basispoints,actuallyquitesimilartothe2000s. The mean deposit to consumption ratio is 0.94, a somewhat higher value. To obtain a value for (cid:21), I use the period(cid:146)s average ratio of depository institutions(cid:146)holdings of Treasuries plus excess reserves to their total deposits. This results in (cid:21) = 0:17.51 Combining these measurements with the (cid:133)rst estimate of the net noninterest cost of servicing deposits (1:22%), yields a gross marginal welfare cost of the liquidity requirement 49Itisinterestingtonotethatthecostisnon-negative,aspredictedbythemodel. Thereisnothinginthe empiricalmethodologythatguaranteedanon-negativenumber,sothismightbeviewedasasmallempirical validation of the model. 50The liquidity coverage ratio and the modi(cid:133)ed liquidity coverage ratio, a similar but less stringent requirementforsmallerbanks,werephasedinat90percentoftheir(cid:133)nalvaluesbeginningonJan. 1,2016and at100percentbeginningonJan.1,2017. Startingthemeasurementperiodin2017makeslittledi⁄erenceto the results. Similarly, extending the sample to 2020, a year a⁄ected by the onset of the pandemic, has little impact. 51Because, as mentioned, the actual the LCR requirement depends on more detailed balance sheet information than can be captured in a tractable model, I measure its implied value for (cid:21) under the assumption thattheLCRwasbinding,oratleastdynamicallybinding. Inpractice,banksreportedlyheldabu⁄erstock of liquid assets above minimum LCR requirements (at roughly 10-20% of the requirement). Adjusting for thatwouldslightlyreducethemeasuredwelfarecost. Itisalsoworthnotingthatthelevelofexcessreserves in the measurement period may have been elevated due to the lack of a complete unwind of the large scale assetpurchasesbytheFedthatoccurredbetween2008and2014. Focusingontheyearleasta⁄ectedbythis, 2019, would result in (cid:21)=0:15. Again, using that value barely changes the measured welfare cost, reducing it just slightly. 36

inthepost-implementationperiodequalto(cid:23) = 0:0046. Again, tointerpretthisnumber, LIQ we can consider the gross welfare cost of a 10 percentage point increase in the liquidity requirement. To a (cid:133)rst-order approximation, this cost is equivalent to a permanent loss in consumption of 0:046%, or about $6.6 billion per year. While it this is about 50 percent higher than its pre-implementation value, it may still be considered a small welfare cost. Using the alternative estimate for the net noninterest cost of deposits (0:81%) again results in a lower marginal welfare cost: (cid:23) = 0:0001, or a tiny 0:001 percent of consumption for LIQ a 10 percentage point increase in the liquidity requirement. As a caveat, these estimates are, as noted, (cid:133)rst-order approximations and may be less accurate for large changes in liquidity regulation. In particular, it is possible that further or larger increases in the liquidity requirement would entail more than proportionally larger welfare costs, as the stock of Treasuries that remains available to the non-bank public progressively shrinks. 6.2 Capital Regulation To measure the welfare cost of capital requirements, we need an estimate of the required return on bank equity. Whereas the model abstracts from aggregate risk, a risk-adjusted measure is in fact called for. A risk adjustment captures the degree to which equity(cid:146)s required return adjusts in response to changes in leverage, such as those brought about by changes to capital requirements. In particular, for given asset risk, a decline in leverage should make bank shares less risky, and in theory this should lower the required return that shareholders demand. Indeed, under the idealized conditions underlying Modigliani and Miller(cid:146)s propositions, the strength of this e⁄ect is just such that the weighted average cost of funds does not depend on its leverage at all, so that the (cid:133)rm can change its leverage at zero cost.52 In reality, there are several reasons why the Modigliani-Miller theorem does not hold (cid:150)agency problems, taxes, bankruptcy costs, etc. (cid:150)and it is especially unlikely to hold for banks in light of the special nature of their debt. Indeed, in the model presented, the liquidity of bank debt is simultaneously the reason that banks exist and the chief reason the Modigliani-Miller theorem fails to hold for them.53 On this point, empirical analysis by Baker and Wurgler (2015) (cid:133)nds that, while better-capitalized banks have lower risk as 52For this reason, this risk-adjustment is sometimes referred to as a (cid:145)Modigliani-Miller o⁄set.(cid:146) 53The only other source of its failure is the moral hazard problem, which manifests itself only when the capital requirement is too low, and thus leverage is too high, according to the threshold in proposition 1. 37

expected, lower-risk banks tend to have higher stock returns on a risk-adjusted or even raw basis, so that an increase in capital ratios would result in a (possibly sharply) higher weightedaveragecostofcapital, anoutcomethatwouldbequalitativelyconsistentwiththe model. Nonetheless, even if the Modigliani-Miller theorem does not hold exactly in reality (as well as in the model), it is still possible that the expected return on equity adjusts to changes in bank leverage, and the empirical approach should take this into account. Thus, whereas the model abstracts from aggregate risk, a risk-adjusted measure of the required return on equity is needed from the data. Following Van den Heuvel (2008), I use the average return on subordinated bank debt as a proxy for the risk-adjusted return on equity. The reason for this choice is that (a) subordinated debt counts towards regulatory equity capital, albeit within certain limits, and (b) defaults on this type of debt have historically been rare, so the debt is not very risky, certainly compared to common equity. This proxy avoids the di¢ culties inherent in measuring the (ex ante) risk premium on common equity,54 and how that premium adjusts to changes in leverage. Concretely, (RE (cid:0) 1) is measured by Interest on Subordinated Notes divided by Subordinated Notes.55 The limits on the use of subordinated debt for regulatory purposes as well as it tax treatment imply that this is a conservative measure for the risk-adjusted required return on bank equity. First, subordinated debt can count only towards tier 2 capital, so it only helps to satisfy the risk-based total capital ratio requirement, not the risk-based tier 1, common equity tier 1, or leverage ratio requirements. Second, until the recent adoption of Basel III, the amount of subordinated debt in tier 2 was limited to 50 percent of the bank(cid:146)s tier 1 capital. So if the tier 1 capital ratio was close to binding, subordinated debt could count for at most approximately 25 percent of total capital. Third, relative to common equity, interest on subordinated debt receives favorable corporate income tax treatment. Due to factors, it is possible that for many banks the required return on subordinated debt is lower than the pre-tax, risk-adjusted required return on common equity. Again, we will use two measurement periods to quantify the welfare cost, one pre- and one post-implementation of Basel III. The pre-Basel III period is set to 1993-2010. The 54Forexample,thehistoricalaverageexcessreturnonbankequitywouldimplyahighpremium,butdoes this equal the ex ante expected premium? In addition, depending on what interest rate is used to measure the excess return on equity, one would run the risk of contaminating the measured risk premium with a liquidity premium, which one would de(cid:133)nitely want to avoid in the present context. 55PartoftheHSOB(cid:146)sSubordinatedNotesdoesnotqualifyasregulatorycapital. However,cross-checking with the call reports (item RCFD5610) indicates that the di⁄erence is minimal after 1992. Also, some subordinated bank debt is callable. Flannery and Sorescu (1996) (cid:133)nd that the average call option value for callable bank sub-debt is 0.19%, so the point is minor for the present purpose. 38

start date is motivated by the fact that the (cid:133)rst Basel Accord and the FDICIA legislation enacting it were not fully implemented until January 1, 1993, and prior to Basel the use of subordinated debt for regulatory purposes was rather limited. 2010 is chosen as end date, because the Basel III package was published in December 2010. That said, extending this sample period by a few years, or letting it start earlier, has little impact on the results. The post-Basel III-implementation period is set at 2016-2019, the same time span as for liquidity regulation. For 1993-2010, the average nominal interest rates on subordinated debt and deposits are, respectively, 5.45% and 2.43%, so the average cost-revealing spread is 302 basis points. Themean loanstoconsumptionratio is0.95(usingtotal assetsminus Treasuries andexcess reserves for loans). As explained, the net noninterest cost of servicing deposits is set at 122 basis points and (cid:21) = 0 for this period. Combining these measurements with the analytical result in (24) yields a marginal gross welfare cost of the capital requirement equal to (cid:23) = (L=c) RE RD g (d;L) CAP (cid:21)=0 D j (cid:0) (cid:0) = 0:95 (cid:0)(0:0302 0:0122) = 0(cid:1):0171 (cid:2) (cid:0) Thus, the gross welfare cost of an increase in the capital requirement by 10 percentage points is equivalent to a permanent loss in consumption of about (cid:23) 0:1 100% = 0:17% CAP (cid:2) (cid:2) This is substantially higher than obtained for a 10 percent liquidity requirement, a point I discuss in more detail below. Further, the cost is similar to the welfare cost of a permanent increase in in(cid:135)ation by a few percentage points, as measured by Lucas (2000). Of course, it should really be compared to the (cid:133)nancial stability bene(cid:133)ts of such an increase in the capital requirement, captured in proposition 1. Section 3 has already discussed this at a conceptual level, and section 7 below will provide further analysis. Using the alternative, Hanson et al. based estimate for the net noninterest cost of deposits (g = 0:81 percent) yields a modestly larger measurement of the gross marginal D welfarecost: (cid:23) = 0:0210,resultinginagrosswelfarecostof0:21percentofconsumption CAP for an increase in the capital requirement by 10 percentage points. Turning to the more recent period, it worth noting that the U.S. rule implementing Basel III(cid:146)s new capital regime was already (cid:133)nalized and published in 2013, but its phase-in was an especially gradual one, starting on Jan. 1, 2014 and ending on Jan. 1, 2019. That said, in part re(cid:135)ecting anticipation e⁄ects and in part the banks(cid:146)desire to pass the stress tests, most of the post-crisis capital buildup, which was substantial, had occurred by the 39

end of 2015.56 In light of this, and since a sample period starting in 2019 would seem too short, IemploythesameBaselIIIsampleperiodasforliquidityregulation, 2016-2019, even though technically this time span includes some of the phase in. Reassuringly, shifting the start of the measurement period up or down by one or two years does not have a big impact on the results. For 2016-2019, the average nominal interest rates on subordinated debt and deposits are, respectively, 3.86% and 0.50%, yielding an average spread of 336 basis points. With liquidity regulation in place in this period, adjustments are needed to use the all-in cost of (cid:133)nancing loans with deposits (see (8)) as well as for the noninterest costs of servicing deposits, (cid:133)rst set at 1:22%. With (cid:21) = 0:17 now, these adjustments reduce the spread to 206 basis points. The mean loans to consumption ratio is now 1.07. Combining these measurementswiththeanalyticalresult(23)inproposition6yieldsamarginalgrosswelfare cost of the capital requirement equal to (cid:23) = 0:022. Thus, for the Basel III period, CAP as a (cid:133)rst-order approximation, the gross welfare cost of a further increase in the capital requirement by 10 percentage points is equivalent to a permanent loss in consumption of 0:22 percent, a bit more than before Basel III(cid:146)s implementation. Using the alternative, Hanson et al.-based estimate for the net noninterest cost of deposits (g = 0:81%) again yields a modestly larger measurement of the gross marginal D welfare cost: (cid:23) = 0:027, or a gross welfare cost of 0:27 percent of consumption for a 10 CAP percentage points increase in the capital requirement. 6.3 Comparative Assessment For comparison, table 1 recaps the measurements of the gross welfare costs of both regulations. It presents the permanent consumption loss, in percent, that is to a (cid:133)rst-order approximation welfare-equivalent to a 10 percentage point increase in each requirement, for each of the two measurement periods and taking the average across the two estimates of the net noninterest costs of servicing deposits. The numbers below, in parentheses, represent the full range depending on these cost estimates. 56For example, for the largest U.S. banks (global systemically important bank holding companies) the ratio of common equity tier 1 capital to risk-weighted assets increased from 6.3% at the end of the great recession (2009Q2) to 12.0% at year-end 2015, rising only marginally further to 12.2% by the start of 2019. SeetheFederalReserve(cid:146)sFinancialStabilityReport,https://www.federalreserve.gov/publications/(cid:133)nancialstability-report.htm 40

Table 1. Gross Welfare Costs of the Liquidity and Capital Requirements Measurement period Welfare cost of: Pre-Basel III Basel III 10% liquidity requirement 0.017 0.023 (0.003-0.031) (0.001-0.046) 10% capital requirement 0.191 0.245 (0.171-0.210) (0.219-0.272) Note: Entries are (cid:133)rst-order approximations of the permanent consumption loss, expressed in percent, that is welfare-equivalent to a 10 percentage point increase in each requirement, holding (cid:133)xed the incidence of (cid:133)nancial crises, and taking the average across estimates of the net noninterest costs of servicing deposits (0.81%and1.22%). Numbersinparenthesesindicatethefullrangeofmeasuredwelfarecosts,withtheupper end associated with high (low) net noninterest costs for the liquidity (capital) requirement. (cid:145)Pre-Basel III(cid:146) means 2001-2007 for liquidity and 1993-2010 for capital. (cid:145)Basel III(cid:146)means 2016-2019. The main comparative takeaway from table 1 is that the welfare cost of an increase in the capital requirement is roughly ten times as large as the cost of a similarly sized increase in the liquidity requirement. The divergence by an order of magnitude appears to hold regardless of the measurement period. This key result re(cid:135)ects an insight obtained from the model: capital requirements reduce the supply of safe, liquid assets available to the public by much more than liquidity requirements do. Even with the general equilibrium feedbacks that change the size of the banking sector,57 capital requirements e⁄ectively reduce the supply of bank deposits, replacing them to an important degree with bank equity, an instrument that does not provide liquidityservices. Incontrast, liquidityrequirementse⁄ectivelytransformsomegovernment bonds held by the public into bank deposits (which fund the banks(cid:146)holdings of government bonds). These are both liquid instruments that command a convenience yield, so the net reductioninliquidityservicesavailabletothenon-bankpublicismuchsmaller. Inthedata, this is manifested by a smaller spread between Treasuries and bank deposits than between equity and deposits, and that is why the welfare cost estimates di⁄er as much as they do. 57Indeed,the(potentiallylarge)changesinthesizeofthebankingsector,thecapitalstock,andconsumption (cid:150)taking into account changes in both the transition and the steady state (cid:150)have a combined e⁄ect on welfare that is second-order. Mathematically, this is a manifestation of the envelope theorem. 41

While capital requirements entail substantially higher welfare costs according to these measurements, it is important to recall that they also have broader (cid:133)nancial stability bene(cid:133)ts than liquidity requirements in the model. As explained, liquidity regulation only addresses liquidity risk, while capital regulation can ameliorate both credit and liquidity risk-taking. The two requirements are thus not perfect substitutes. In fact, one might say that you get what you pay for. A secondary (cid:133)nding that is apparent from the table is that the gross marginal welfare costs of both tools are higher now than in the pre-Basel III period, by 37 and 29 percent, respectively, for the liquidity and the capital requirement (based on the period-speci(cid:133)c averages). SinceBaselIIIraisedbothrequirements,this(cid:133)ndingisconsistentwithincreasing marginal costs, i.e. convex welfare costs. Fromanaccountingperspective,theincreaseinmeasuredcostsre(cid:135)ectsbothparametric factors and observables. Speci(cid:133)cally, for the liquidity requirement, the rise in marginal welfare cost can be attributed to the direct e⁄ect of a higher (cid:21) (due to the factor 1=(1 (cid:21)) (cid:0) in(22))andanobservedincreaseintheratioofdepositstoconsumption,asbanksexpanded their balance sheets, primarily through larger holdings of liquid assets. There was relatively little movement in the relevant spread. For the capital requirement, the rise in marginal costisaccountedforbyincreasesintheloanstoconsumptionratioandinthecost-revealing equity-deposit spread, which outweighed the adjustments for a higher (cid:21); see (23).58 6.4 Narrow Banking A narrow bank is a depository institution that holds only safe, liquid assets. Proponents of narrowbankingarguethatsuch(cid:133)rmswouldbe(virtually)immunetobankrunsandfailures, thus eliminating the economic harm caused by failures of deposit-taking (cid:133)rms. Loans to (cid:133)rms and households would instead be made by non-deposit-taking (cid:133)nancial (cid:133)rms, such as (cid:133)nance companies, or would be replaced by market-based (cid:133)nance, such as bonds or commercial paper. In our model, narrow banking is already permitted: a bank can become a narrow bank by maintaining a balance sheet with only government bonds and deposits. Moreover, other banks could opt to become a (cid:133)nance company by making only equity-(cid:133)nanced loans, and 58AsshowninBegenau(2020)andVandenHeuvel(2006),thegeneralequilibriume⁄ectofanincreasein thecapitalrequirementonbanklendingcanbepositive. Whileperhapssurprising,thisoutcomeoccurswhen the interest-elasticity ofthe demand for deposits is low. In that case, an increase in the capitalrequirement leads to an increase in the deposit spread (RE RD) that is su¢ ciently large to reduce banks(cid:146)lending rate (cid:0) (RL), despite the costlier funding mix. 42

(cid:133)rms in the model can borrow directly from households, akin to market-based (cid:133)nance. For banks, neither business model would violate regulatory constraints.59 However, the data suggest that these strategies are not always pro(cid:133)table. The narrow bank is not pro(cid:133)table whenevertheliquidityrequirementbinds(thatis,wheneverRD+g > RB)or,equivalently, D whenever there is a positive welfare cost of liquidity regulation, which we have found to be the case for at least some periods.60 The (cid:133)nance company is not pro(cid:133)table whenever the capital requirement binds (that is, whenever RL g < RE) or, equivalently, whenever L (cid:0) there is a positive welfare cost of the capital requirement - again, we have found this to be true in the data. Whatthenwouldbethegrosswelfarecostofrequiring deposit-takinginstitutionstobe narrow banks? In the model, this can be achieved by imposing 100% liquidity and capital requirements.61 Wecangaugethecostofsuchapolicyinthesamewayaswehavemeasured the costs of smaller policy changes above. It must be stated at the outset, however, that doing so seems likely to come with large approximation error: our measurements rely on (cid:133)rst-order approximations, which may not perform well for such a big policy change. If we nonetheless proceed with that caveat in mind, relative to current regulations, the gross welfare cost of requiring deposit-taking institutions to be narrow banks is (very) roughly equal to (cid:23) (1 0:17) + (cid:23) (1 0:12) = 2:4% of consumption.62 Not LIQ CAP (cid:2) (cid:0) (cid:2) (cid:0) surprisingly, a move to narrow banking would be considerably costlier than the smaller policy changes contemplated in table 1. Intuitively, narrow banking is costly because it ignores the reality that, despite the demandable nature of many deposits, most depositors do not withdraw their funds all the time, allowing banks to use a portion of their deposits to fund more illiquid loans. 59This re(cid:135)ects the absence of a leverage ratio restriction from the model. A leverage ratio rule would require the narrow bank to maintain some equity against its government bonds. 60Hanson, Schleifer, Stein, and Vishny (2015) reach a similar conclusion by comparing the total cost of deposits to the return on T-bills (or, in our notation, RD+g to RB). D 61Technically,suchapolicy((cid:21)=1and(cid:13) =1)wouldnotimposetheseparationofnarrowbanksand(cid:133)nance companies. This is without loss of generality if g =0 or if g is separable in its arguments. However, if g is notzeronorseparable,thenitsassumedconvexityandlinearhomogeneityimplythattherearecostsavings from combining the narrow bank and (cid:133)nancing company in a single (cid:133)rm (g(D;L) g(D;0)+g(0;L)). In (cid:20) thatcase,thecalculationsthatfollowwillmisssomeofthecostsassociatedwithamovetonarrowbanking. 62The baseline level of the capital requirement is set at 0:12, which is the current ratio of common equity tier 1 capital to risk-weighted assets of the largest U.S. bank holding companies, and the baseline liquidity requirement is set at its current level, 0:17 (see section 6.1 for details). The calculation is based on the averagewelfarecostsfortheBaselIIIperiod,asreportedintable1for10p.pincreasesineachrequirement. Using the averages from the pre-Basel III measurement period results in an estimated welfare cost of 1:8%. 43

As mentioned, this number should not be view as a precise estimate. In particular, it may underestimate the cost of narrow banking because the convenience yield on Treasuries wouldalmostsurelyriseduetotheextrademandfortheseassetsfromnarrowbanks. Onthe other hand, it could overestimate the cost because the marginal welfare cost of raising the capitalrequirementisdecreasingintheproductoftheliquidityrequirementanditsmarginal welfare cost; see (25). Finally, whatever the precise gross cost of a move to narrow banking, incurring it would only be justi(cid:133)able within the model if maximum liquidity and capital requirements were necessary to prevent excessive risk taking (according to the thresholds in propositions 1 and 4). 7 The Welfare Bene(cid:133)ts of Regulation While the previous section provided a quanti(cid:133)cation of the welfare costs of regulation, the discussion of the welfare bene(cid:133)ts has so far been qualitative in nature. What would it take to quantify the bene(cid:133)ts of regulation and thereby the optimal levels of the requirements through the lens of the model? Those bene(cid:133)ts arise from the social costs of excessive risk taking and bank failures and from the ability of the two regulations to prevent these costs. In the model, the former depend on the ex-ante cost of lending to excessively risky (cid:133)rms, (cid:24), and the ex-post resolution costs, and . The latter are captured by the thresholds Sol Liq for the capital and liquidity requirements that are derived in propositions 1 and 4, which in turn depend on parameters such as the (cid:147)value-at-risk(cid:148)of excessively risk loans ((cid:30) ), the " quality of bank supervision (1=(cid:27)(cid:22)), and the rate and probability of stressed withdrawals (w and p), as well as equilibrium objects. Providing a compelling calibration or estimation of these parameters would be quite challenging. Moreover, in contrast to the welfare costs, measuring the bene(cid:133)ts does not appear amenable to a su¢ cient statistics approach that would be straightforward to implement empirically. One reason is that revealed preference arguments are unlikely to be very informative regarding the social costs from bank failures or the ability of regulation to prevent such failures, precisely because of their externalities. All that said, regulators do need to make quantitative choices about capital and liquidity requirements. In this respect, it is important to note that much of the quantitative analysis done by regulators for the calibration of such requirements relates to their bene- (cid:133)ts. For example, for capital requirements, regulators examine the historical distribution of credit losses for di⁄erent types of bank loans (akin to (cid:30) in the model). For the liquidity " regulation, the calibration of the LCR rule is importantly based on stressed withdrawal 44

rates of di⁄erent types of bank liabilities (w in the model). This paper has little to add to those very detailed analyses. Instead, it hopes to provide a convenient way to measure the social costs (cid:150)something that is often challenging for regulators. In that vein, it is useful to compare the measurements of the welfare costs in this paper to existing estimates of bene(cid:133)ts. For example, a study by the Basel Committee intended to inform the overall calibration of the Basel III reforms (BCBS 2010) estimates those bene(cid:133)ts as a reduction in the probability of a (cid:133)nancial crisis due to stricter regulation times the loss in output conditional on a crisis. It (cid:133)nds that the gross bene(cid:133)ts of increasing capital requirements from 7 to 15 percent are 0:8% to 2:64% of GDP, depending on whether the outpute⁄ectsofbankingcrisesareconsideredtobetemporaryorpermanent(seeAppendix B for details). Section 6 of this paper measures the gross welfare cost of such an increase as 0:019 (0:15 0:07) 100% = 0:15% of consumption.63 Comparing these numbers (cid:2) (cid:0) (cid:2) suggests that the increase in capital requirements is clearly welfare improving, regardless of whether the output e⁄ects of crises are temporary or permanent (provided, of course, that the estimates are su¢ ciently accurate). Similarly, BCBS (2010) estimates the gross bene(cid:133)ts of an LCR-style liquidity requirement at about 0:23% to 0:76% of GDP, again depending on the persistence of the output e⁄ects. These numbers are conditional on a capital requirement of 7 percent, and the estimates drop if the capital requirement is higher, as more capital already reduces the frequency of crises. For example, at a 10 percent capital requirement, the estimated bene- (cid:133)ts of liquidity regulation are 0:03% to 0:12% of GDP, and they drop further to 0:01% to 0:03% at a 12 percent capital requirement.64 For comparison, in the model, a 10% liquidity requirement (cid:150)a simpli(cid:133)ed but reasonable approximation to the LCR requirement65 (cid:150)en- 63The calculation is based on the average for the pre-Basel III period (reported in table 1 for a 10 p.p. increase in the requirement). This measurement period is most consistent with the timing of the BCBS (2010) study. Based on the more recent measurement period, the welfare cost of an increase in capital requirements by 8 percentage points amounts to 0:20% of consumption. 64To provide some perspective on these levels of the capital requirement, Basel III raised the minimum requirements(includingthecapitalconservationbu⁄er)to7%ofrisk-weightedassetsforcommonequitytier 1 capital (the highest quality capital); to 8.5% for tier 1 capital; and to 10.5% for total capital. However, those numbers do not include the surcharges applicable to the largest banks (1-3.5% in the U.S.), the countercyclicalcapitalbu⁄er(currentlysetat0%),noranyadditionalcapitalthatlargebanksmayneed to pass the stress tests (or, since 2020, satisfy the stress capital bu⁄er). Partly re(cid:135)ecting all these factors, and partlyre(cid:135)ectingvoluntarybu⁄ersaboveregulatoryrequirements,commonequitytier1capitalofU.S.bank holding companies is currently about 12 percent of their risk-weighted assets. 65As mentioned, the LCR requirement depends on more detailed balance sheet information than can be captured in a macro-style model, but, internationally, 10% appears a reasonable ratio to capture the LCR. 45

tails a gross welfare cost of about 0:02% of consumption, or slightly less as a percentage of GDP. Taken together, these estimates suggest that the liquidity requirement is also welfare improving, although it is a closer call at higher levels of the capital requirement. 8 Conclusion This paper has presented a framework for measuring the welfare costs of bank liquidity and capital requirements. While such requirements have important (cid:133)nancial stability bene(cid:133)ts, they also entail social costs because they reduce banks(cid:146)ability to create net liquidity in equilibrium. The cost of the capital requirement scales with the convenience yield on bank deposits, net of the non-interest costs of servicing those deposits. The cost of the liquidity requirement scales with the di⁄erence between the convenience yields on Treasuries and bank deposits (again, net o⁄ their non-interest costs). Using U.S. data, the welfare cost of a 10 percent liquidity requirement is found to be equivalent to a permanent loss in consumption of about 0:02% (cid:150)a modest impact. According to conservative estimates, the cost of a 10 percentage point increase in capital requirements is roughly ten times as large. At the same time, the (cid:133)nancial stability bene(cid:133)ts of capital requirements were found to be broader than those of liquidity requirements. In particular, liquidity requirements are not a substitute for capital requirements when it comes to credit risk, and optimal policy relies on both tools to safeguard (cid:133)nancial stability. Finally, comparing our measurements of the welfare costs of the increases in capital and liquidity requirements associated with Basel III to existing estimates of their bene(cid:133)ts, we found that the Basel III reforms produced net welfare gains. References Baker,Malcolm,andJe⁄reyWurgler,2015.(cid:147)DoStrictCapitalRequirementsRaisetheCostofCapital? Bank Regulation, Capital Structure, and the Low Risk Anomaly,(cid:148)American Economic Review, v. 105 n. 5, pp 315-320, May. Barr, Michael S., 2022. (cid:147)Making the Financial System Safer and Fairer,(cid:148)Speech at the at The Brookings Institution, Washington, D.C., September 7. In the U.S., as noted, the ratio appears somewhat higher, re(cid:135)ecting in part the balance sheet compositions oflarge U.S.banks and the details ofthe U.S.implementation ofthe LCR,factors not explicitly considered in BCBS (2010). 46

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Appendix A. Analysis of the Bank(cid:146)s Problem A.1. Credit Risk Choice, Part 1 Since loans are nonnegative, expected dividends equal E[ (r+(cid:27)") +]L if B wD or f g (cid:21) (1 p)E[ (r+(cid:27)")L +] if B < wD. Next, (cid:0) f g E (r+(cid:27)") + = (r+(cid:27)E") E r+(cid:27)" (cid:0) = (r (cid:27)(cid:24))+(cid:27)j( r=(cid:27)) f g (cid:0) f g (cid:0) (cid:0) (cid:2) (cid:3) (cid:2) (cid:3) where j is the following function, derived from the distribution function of " : x j(x) (x ")dF (") " (cid:17) (cid:0) " Z Note that j(x) is continuous and increasing in x, equals zero when x " (< 0) and strictly (cid:20) exceeds (cid:24) 0 when x = 0 (by the de(cid:133)nition of (cid:24) and the assumption (cid:22)" > 0). As shown in (cid:21) Van den Heuvel (2008), appendix D.1, (cid:27)j( r=(cid:27)) is a convex function of (cid:27). Hence, expected (cid:0) dividends are convex in (cid:27). Therefore, either (cid:27) = 0 or (cid:27) = (cid:27)(cid:22) is optimal. Comparing these two choices, (cid:27) = 0 if and only if j( r=(cid:27)(cid:22)) (cid:24) (cid:0) (cid:20) The de(cid:133)nition of (cid:30) in (6) can be rewritten in this notation, as follows: " j( (cid:30) ) (cid:24) " (cid:0) (cid:17) From the above-mentioned properties of j, it follows that (cid:30) exists, is unique and satis(cid:133)es " " (cid:30) < 0, so 0 < (cid:30) ". Using this notation, we have66 " " (cid:20) (cid:0) (cid:20) (cid:0) Lemma 1 (cid:27) = 0 if and only if (cid:30) (cid:27)(cid:22) r. Otherwise, (cid:27) = (cid:27)(cid:22). " (cid:20) The condition in proposition 1 imposes additional optimality conditions (a zero-pro(cid:133)t condition, really) on lemma 1. These additional optimality conditions are derived in Appendices A.2 and A.3, which also conclude the proof of proposition 1. A.2. The Bank(cid:146)s Problem When (cid:21) w and (cid:27) = 0 (Proof of Propositions 2 (cid:21) and 3) Under the condition in lemma 1, (cid:27) = 0. After scaling the resulting problem in (7) by RE and using the balance sheet identity to substitute out B, the Lagrangian and (cid:133)rst-order 66When (cid:30) (cid:27)(cid:22) =r, the bank is indi⁄erent. For convenience, it is assumed that (cid:27)=0 in that case. " 50

conditions (FOCs) are:67 = RLL+RB(E+D L) RDD g(D;L) REE+(cid:3)[E+(1 (cid:21))D L]+(cid:31)[E (cid:13)L] L (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (L) RL = RB +g +(cid:3)+(cid:13)(cid:31) L (E) RE = RB +(cid:3)+(cid:31) (D) RD +g = RB +(1 (cid:21))(cid:3) D (cid:0) The complementary slackness conditions are: (cid:31)[E (cid:13)L] = 0; (cid:31) 0; (cid:3)[E+(1 (cid:21))D L] = (cid:0) (cid:21) (cid:0) (cid:0) 0; (cid:3) 0. Note that (cid:21) RB +(cid:3) = RD +g +(cid:21)(cid:3) = RL g (cid:13)(cid:31) = RE (cid:31) (27) D L (cid:0) (cid:0) (cid:0) Since the Kuhn-Tucker multipliers must be nonnegative, a (cid:133)nite solution requires the ranking of returns shown in (10) in proposition 3, i.e.: RB RD +g RL g RE D L (cid:20) (cid:20) (cid:0) (cid:20) From FOC (D), (cid:3) = 1 (RD +g RB). Hence, the liquidity requirement binds if 1 (cid:21) D (cid:0) (cid:0) and only if RD +g > RB. D In addition, from (27), (cid:31) = 1 (RE (RL g )). Hence, the capital requirement binds 1 (cid:13) (cid:0) (cid:0) L (cid:0) if and only if RE > RL g . L (cid:0) Furthermore, (27) implies that RL g (cid:13)(cid:31) = (cid:13) RE (cid:31) +(1 (cid:13)) RD+g +(cid:21)(cid:3) . L D (cid:0) (cid:0) f (cid:0) g (cid:0) f g Rearranging and using the expression for (cid:3) as well as the de(cid:133)nition of R~D((cid:21)) (see (8)) yields the (marginal) zero-pro(cid:133)t condition: 1 RL g (D;L) = (cid:13)RE +(1 (cid:13)) R~D((cid:21))+ g (D;L) (28) L D (cid:0) (cid:0) f 1 (cid:21) g (cid:0) which is (11) in proposition 3. Moreover, this implies that RL g RE = (1 (cid:13))[ R~D((cid:21))+ 1 g RE], yielding (cid:0) L (cid:0) (cid:0) f 1 (cid:21) D g(cid:0) (cid:0) the equivalent, alternative condition for a (non)binding capital requirement: 1 RL g (D;L) < (=) RE R~D((cid:21))+ g (D;L) < (=) RE L D (cid:0) () 1 (cid:21) (cid:0) 67For brevity, the arguments of g(D;L) and its partial derivatives are often suppressed where this does not lead to confusion. 51

Zero pro(cid:133)ts follow from RE(cid:25)B = RLL+RBB RDD g(D;L) REE (cid:0) (cid:0) (cid:0) = RLL+RB(E +D L) RDD (Dg (D;L)+Lg (D;L)) REE D L (cid:0) (cid:0) (cid:0) (cid:0) = [RL g (D;L) RB]L [RD +g (D;L) RB]D [RE RB]E L D (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) = [(cid:3)+(cid:13)(cid:31)]L [(1 (cid:21))(cid:3)]D [(cid:3)+(cid:31)]E (cid:0) (cid:0) (cid:0) = (cid:31)(E (cid:13)L) (cid:3)[E +(1 (cid:21))D L] = 0 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) where the steps follow from Euler(cid:146)s theorem, the (cid:133)rst-order conditions and the complementary slackness conditions, in that order. Credit risk choice, part 2 Recall that r RL+RB(B=L) RD(D=L) g(D=L;1) and (cid:17) (cid:0) (cid:0) note that (cid:25)B = rL=RE E. Zero pro(cid:133)ts ((cid:25)B = 0) imply that r = (E=L)RE (cid:13)RE, so the (cid:0) (cid:21) critical value of (cid:27)(cid:22) for (cid:27) = 0 in lemma 1, r=(cid:30) , is at least (cid:13)RE=(cid:30) , and is equal to that value " " if the capital requirement binds. Hence, (cid:30) (cid:27)(cid:22) (cid:13)RE is a su¢ cient condition for (cid:27) = 0, and " (cid:20) this condition is also necessary if the capital requirement binds. This concludes the proof of proposition 1 for the case (cid:21) w. (cid:21) This also concludes the proof of the (cid:133)rst half of proposition 3 (up to (cid:145)four cases are possible(cid:146)). Proposition 2 in the main text follows immediately as a corollary by setting g = 0 in the (cid:133)rst half of proposition 3. Proof of second half of proposition 3 What follows is a proof of the second half of proposition 3 (after (cid:145)four cases are possible(cid:146)), which characterizes the solution further by showing when each of the two regulatory constraints is binding or slack as a function of objects that the bank takes as given only.68 Case1. Nonbindingconstraints((cid:31) = 0and(cid:3) = 0) From(27),RB = RD+g = RL D (cid:0) g = RE. Note that this case requires RD RB = RE RL: With the partial derivatives L (cid:20) (cid:20) of g homogenous of degree 0, the ratio D=L is determined by g (D=L;1) = RB RD and D (cid:0) by g (D=L;1) = RL RB. A solution requires that the con(cid:133)guration of returns is such L (cid:0) that both equations imply the same value for D=L. 68The issue is that the conditions derived so far (cid:150)that is, RD +g (D;L) > RB for a biding liquidity D requirement and RE > RL g (D;L) for a binding capital requirement (cid:150)still depend on two decision L (cid:0) variablesofthebank,D and L,so thatthecharacterization ofthesolution isnotcomplete. Thisissue does not arise when g=0, the case summarized in proposition 1 52

Case 2. Only liquidity requirement binds ((cid:31) = 0 and (cid:3) > 0) From the FOCs, RL g (D;L) = RE. From this and (28) it follows that L (cid:0) 1 RL g (D;L) = RE = R~D((cid:21))+ g (D;L) L D (cid:0) 1 (cid:21) (cid:0) Again, we have two equations that each pin down the ratio D=L. This case requires that RB < RD +g (D;L) < RL g (D;L) = RE and in particular RB < RE. Recall that (cid:26) is D L (cid:0) de(cid:133)ned as the value of D=L when both regulatory constraints are binding: (cid:26) = 1 (cid:13). Due 1(cid:0)(cid:21) (cid:0) to the nonbinding capital requirement here, D=L (cid:26). Hence, as g is convex, g (D;L) = D (cid:20) g (D=L;1) g ((cid:26);1) and g (D;L) = g (D=L;1) g ((cid:26);1). Thus, RB < RD +g ((cid:26);1) D D L L L D (cid:20) (cid:21) and RL RE +g ((cid:26);1). L (cid:21) Case 3. Only capital requirement binds ((cid:31) > 0 and (cid:3) = 0) From the FOCs, RB = RD +g (D;L) and D RL g (D;L) = (cid:13)RE +(1 (cid:13))(RD +g (D;L)) = (cid:13)RE +(1 (cid:13))RB L D (cid:0) (cid:0) (cid:0) ThiscaserequiresthatRB = RD+g (D;L) < RL g (D;L) < RE. Duetothenonbinding D L (cid:0) liquidity requirement, D=L (cid:26), so g (D;L) g ((cid:26);1) and g (D;L) g ((cid:26);1). Hence, D D L L (cid:21) (cid:21) (cid:20) RB RD +g ((cid:26);1) and RL < RE +g ((cid:26);1). Also, RD +g ((cid:26);1) < RE. D L D (cid:21) Case 4. Both requirements bind ((cid:31) > 0 and (cid:3) > 0) In this case, D=L = (cid:26), so (28) implies RL g ((cid:26);1) = (cid:13)RE +(1 (cid:13)) R~D((cid:21))+ 1 g ((cid:26);1) . Using Euler(cid:146)s theorem and (cid:0) L (cid:0) f 1 (cid:21) D g (cid:26) = (1 (cid:13))=(1 (cid:21)), this can also be written as RL = (cid:0) (cid:13)RE+(1 (cid:13))R~D((cid:21))+g((cid:26);1), which is (cid:0) (cid:0) (cid:0) (12). g((cid:26);1) is the total noninterest cost of making one unit of loans and servicing (cid:26) units of deposits. With (cid:31) > 0 and (cid:3) > 0, the inequalities in the ranking of returns (10) are all strict, and, with D=L = (cid:26), RB < RD +g ((cid:26);1) < RL g ((cid:26);1) < RE. D L (cid:0) This concludes the proof of proposition 3.69 A.3. The Bank(cid:146)s Problem When (cid:21) < w (Proof of Proposition 4 and conclusion of proof of proposition 1) Recall that if (cid:21) w, then 1 = 0, so that problem (4) simpli(cid:133)es to: B<wD (cid:21) f g (cid:25)B (cid:21) w = max E (RL+(cid:27)")L+RBB RDD g(D;L) + =RE E j (cid:21) (cid:27);L;B;D;E f (cid:0) (cid:0) g (cid:0) s.t. B(cid:2) (cid:21)D and (29) (cid:3) (cid:21) 69ThecaseRB <RE,RB RD+g ((cid:26);1)andRL g ((cid:26);1) RE ismissingfrom proposition3because D L (cid:21) (cid:0) (cid:21) it is incompatible with a (cid:133)nite solution. It is straightforward to show that feasible choices B = (cid:21)D and E =(cid:13)L result in strictly positive pro(cid:133)ts and an optimal scale that is in(cid:133)nite in this case. 53

where (29) collects the non-liquidity constraints to the bank(cid:146)s problem: L+B = E +D; E (cid:13)L; (cid:27) [0;(cid:27)(cid:22)] (29) (cid:21) 2 To analyze the case (cid:21) < w, we make use of the mathematical fact that if S = A B, then [ max f(x) = max max f(x);max f(x) , provided max f(x) and max f(x) x S x A x B x A x B 2 f 2 2 g 2 2 both exist. Thus, for (cid:21) < w, the general problem in (4) can be equivalently described as: (cid:25)B = max (cid:25)B ;(cid:25)B (cid:21)<w B<wD B wD j f j j (cid:21) g where (cid:25)B B wD = max E (RL+(cid:27)")L+RBB RDD g(D;L) + =RE E j (cid:21) (cid:27);L;B;D;E f (cid:0) (cid:0) g (cid:0) s.t. B(cid:2) wD and (29) (cid:3) (cid:21) (cid:25)B B<wD = max (1 p)E (RL+(cid:27)")L+RBB RDD g(D;L) + =RE E j (cid:27);L;B;D;E (cid:0) f (cid:0) (cid:0) g (cid:0) s.t. B [(cid:21)D;(cid:2)wD) and (29) (cid:3) 2 The statement of (cid:25)B also uses the independence of " and (cid:17). The strict inequality B<wD j constraint B < wD could lead to nonexistence of a solution to this problem. However, we willshowshortlythatthisissuedoesnotarise. Itturnsouttobemathematicallyconvenient to de(cid:133)ne a slightly modi(cid:133)ed problem (which di⁄ers only in the liquidity constraint): (cid:25)B liq:risk = max (1 p)E (RL+(cid:27)")L+RBB RDD g(D;L) + =RE E j (cid:27);L;B;D;E (cid:0) f (cid:0) (cid:0) g (cid:0) s.t. B (cid:21)D (cid:2)and (29) (cid:3) (cid:21) (Thesubscriptliq:risk standsforliquidity risk.) Notethat(cid:25)B (cid:25)B because liq:risk B<wD j (cid:21) j the set the of feasible choices is larger. However, if (cid:25)B > (cid:25)B , it must be liq:risk B<wD j j because the optimal choice involves B wD and for any such choice (cid:25)B (cid:25)B liq:risk B wD (cid:21) j (cid:20) j (cid:21) (as p > 0). Hence, (cid:25)B = max (cid:25)B ;(cid:25)B (cid:21)<w liq:risk B wD j f j j (cid:21) g Note that these latter two problems are isomorphic to (cid:25)B : (cid:21) w j (cid:21) (cid:25)B is identical to (cid:25)B if (cid:21) is set equal to w in the latter B wD (cid:21) w j (cid:21) j (cid:21) (cid:25)B is identical to (cid:25)B if RE replaced by RE=(1 p) in the latter liq:risk (cid:21) w j j (cid:21) (cid:0) 54

Analysis of (cid:25)B with (cid:27) = 0 (cid:21)<w j Assume the condition (cid:30) (cid:27)(cid:22) r is satis(cid:133)ed (see lemma 1). Then the pro(cid:133)t-maximization " (cid:20) problems simplify as follows: (cid:25)B = max RLL+RBB RDD g(D;L) =RE E B wD j (cid:21) L;B;D;E (cid:0) (cid:0) (cid:0) s.t. (cid:2)L+B = E +D; B wD; E (cid:3) (cid:13)L (cid:21) (cid:21) (cid:25)B = max (1 p) RLL+RBB RDD g(D;L) =RE E liq:risk j L;B;D;E (cid:0) (cid:0) (cid:0) (cid:0) s.t. L+B =(cid:2)E +D; B (cid:21)D; E (cid:13)L (cid:3) (cid:21) (cid:21) Recall that (cid:25)B = max (cid:25)B ;(cid:25)B . I will (cid:133)rst analyze (cid:25)B , then (cid:21)<w liq:risk B wD B wD j f j j (cid:21) g j (cid:21) (cid:25)B . Due to the isomorphisms between these two problems on the one hand and liq:risk j (cid:25)B on the other hand, the solutions follow almost immediately from proposition 2. (cid:21) w j (cid:21) Solution to (cid:25)B with (cid:27) = 0 Recall that the problem of maximizing (cid:25)B is B wD B wD j (cid:21) j (cid:21) the same as the problem of maximizing (cid:25)B if (cid:21) is replaced by w in the latter. Thus, (cid:21) w j (cid:21) de(cid:133)ning (cid:26) 1 (cid:13), adapting the notation R~D(w) = RD+ w (RD RB), and referring to w (cid:17) 1(cid:0)w 1 w (cid:0) (cid:0) (cid:0) the B wD constraint as the (cid:145)liquidity constraint,(cid:146)we have (cid:21) Proposition 7 (Solution to problem (cid:25)B with (cid:27) = 0.) A (cid:133)nite solution requires B wD j (cid:21) RB RD +g (D;L) RL g (D;L) RE (30) D L (cid:20) (cid:20) (cid:0) (cid:20) The liquidity constraint (B wD) binds if and only if the (cid:133)rst inequality is strict. The (cid:21) capital requirement binds if and only if the last inequality is strict or, equivalently, if and only if R~D(w)+ 1 g (D;L) < RE. The solution satis(cid:133)es the zero-pro(cid:133)t condition: 1 w D (cid:0) 1 RL g (D;L) = (cid:13)RE +(1 (cid:13)) R~D(w)+ g (D;L) (31) L D (cid:0) (cid:0) f 1 w g (cid:0) resulting in (cid:25)B = 0. Four cases are possible, which are as described in proposition 2, with (cid:26) in place of (cid:26), w in place of (cid:21), and (cid:145)liquidity constraint(cid:146)in place of (cid:145)liquidity requirement(cid:146). w Solution to (cid:25)B with (cid:27) = 0 Recall that the problem of maximizing (cid:25)B is liq:risk liq:risk j j the same as the problem of maximizing (cid:25)B if RE is replaced by RE=(1 p) in the (cid:21) w j (cid:21) (cid:0) latter. Hence, 55

Lemma 2 (Solution to problem (cid:25)B with (cid:27) = 0) A (cid:133)nite solution requires liq:risk j RB RD +g (D;L) RL g (D;L) RE=(1 p) (32) D L (cid:20) (cid:20) (cid:0) (cid:20) (cid:0) The liquidity requirement binds if and only if the (cid:133)rst inequality is strict. The capital requirement binds if and only if the last inequality is strict or, equivalently, if and only if R~D((cid:21))+ 1 g (D;L) < RE=(1 p). The solution satis(cid:133)es the zero-pro(cid:133)t condition: 1 (cid:21) D (cid:0) (cid:0) 1 RL g (D;L) = (cid:13)RE=(1 p)+(1 (cid:13)) R~D((cid:21))+ g (D;L) (33) L D (cid:0) (cid:0) (cid:0) f 1 (cid:21) g (cid:0) resulting in (cid:25)B = 0. Four cases are possible, which are as described in proposition 2, with RE=(1 p) in place of RE. (cid:0) Inthiscase,expectedeconomicpro(cid:133)tsarezero,butrealizedpro(cid:133)tsarestate-contingent. Economic pro(cid:133)ts conditional on ((cid:17) = 0) are pE=(1 p) so shareholders earn a rate of return (cid:0) RE(1+ p ) = RE=(1 p) in that event. Economic pro(cid:133)ts conditional on ((cid:17) = 1) are E 1 p (cid:0) (cid:0) (cid:0) as shareholders lose all their investment in that event. Solution to (cid:25)B with (cid:27) = 0 Recall that (cid:21)<w j (cid:25)B = max (cid:25)B ;(cid:25)B (cid:21)<w B wD liq:risk j f j (cid:21) j g Let (cid:14) be an optimal choice for the ratio D=L associated with problem (cid:25)B and de(cid:133)ne (cid:3)w B wD j (cid:21) (cid:14) analogously for problem (cid:25)B . The zero-pro(cid:133)t condition (31) for problem (cid:25)B (cid:3)lr liq:risk B wD j j (cid:21) provides an expression for the breakeven lending rate for this problem that is consistent with optimal choice: 1 RL breakeven = (cid:13)RE +(1 (cid:13)) R~D(w)+ g ((cid:14) ;1) +g ((cid:14) ;1) j B (cid:21) wD (cid:0) f 1 w D (cid:3)w g L (cid:3)w (cid:0) (Recallthatthepartialderivativesofgarehomogenousofdegree0.) Similarly,for(cid:25)B , liq:risk j 1 RL breakeven = (cid:13)RE=(1 p)+(1 (cid:13)) R~D((cid:21))+ g ((cid:14) ;1) +g ((cid:14) ;1) j liq:risk (cid:0) (cid:0) f 1 (cid:21) D (cid:3)lr g L (cid:3)lr (cid:0) To have a (cid:133)nite solution to (cid:25)B , it must be the case that (cid:21)<w j RL = min RL breakeven;RL breakeven B wD liq:risk f j (cid:21) j g The reason is as follows: (i) if RL < min RL breakeven;RL breakeven , no bank would operate f jB wD jliq:risk g (cid:21) with a strictly positive scale (lest it earns strictly negative pro(cid:133)ts), a situation that is ruled out by equilibrium conditions; (ii) if RL > min RL breakeven;RL breakeven , then the f jB wD jliq:risk g (cid:21) 56

business model with the lowest breakeven rate would yield in(cid:133)nite pro(cid:133)ts by operating at in(cid:133)nite scale, which is incompatible with a (cid:133)nite solution. (In all this, recall that g is linear homogenous and all constraints are linear, so each of the problems is linear homogenous in (L;B;D;E).) Moreover, the business model with the lowest break-even lending rate will be operated in equilibrium. That is, provided a (cid:133)nite solution exists, (cid:25)B = (cid:25)B (cid:21)<w B wD j j (cid:21) if RL breakeven RL breakeven and (cid:25)B = (cid:25)B otherwise. Now, jB wD (cid:20) jliq:risk j (cid:21)<w j liq:risk (cid:21) 1 w (cid:21) RL breakeven RL breakeven = (cid:13)RE 1 +(1 (cid:13)) (RD RB)+h j B (cid:21) wD (cid:0) j liq:risk (cid:0) 1 p (cid:0) 1 w (cid:0) 1 (cid:21) (cid:0) (cid:18) (cid:0) (cid:19) (cid:18) (cid:0) (cid:0) (cid:19) wherehcollectstermsrelatedtodi⁄erencesinintermediationcostsbetweenthetwobusiness models:70 h (cid:26) g ((cid:14) ;1)+g ((cid:14) ;1) (cid:26)g ((cid:14) ;1) g ((cid:14) ;1) (34) w D (cid:3)w L (cid:3)w D (cid:3)lr L (cid:3)lr (cid:17) (cid:0) (cid:0) Hence, we have: Lemma 3 Suppose (cid:21) < w and (cid:30) (cid:27)(cid:22) r, so (cid:27) = 0. Let " (cid:20) w (cid:21) p (cid:16) (1 (cid:13)) (RD RB) (cid:13) RE +h (cid:17) (cid:0) 1 w (cid:0) 1 (cid:21) (cid:0) (cid:0) 1 p (cid:18) (cid:0) (cid:0) (cid:19) (cid:18) (cid:0) (cid:19) If (cid:16) 0 and a (cid:133)nite solution to (cid:25)B exists, then (cid:25)B = (cid:25)B and proposition B wD (cid:21)<w B wD (cid:20) j (cid:21) j j (cid:21) 7 applies; the bank self-insures against liquidity stress. If (cid:16) > 0 and a (cid:133)nite solution to (cid:25)B exists, then (cid:25)B = (cid:25)B and proposition 8 applies; the bank is at risk of liq:risk (cid:21)<w liq:risk j j j failure due to liquidity stress. Proposition 8 (shown immediately below) simply imposes (cid:16) > 0 on the solution to (cid:25)B inlemma2. Itisstraightforwardtoshowthat(cid:16) > 0impliesRB < RD+g (D;L), liq:risk D j so the liquidity requirement binds whenever (cid:16) > 0, simplifying lemma 2 as follows: Proposition 8 (Solution to problem (cid:25)B with (cid:27) = 0 and (cid:16) > 0) A (cid:133)nite soluliq:risk j tion requires RB RD +g (D;L) RL g (D;L) RE=(1 p) D L (cid:20) (cid:20) (cid:0) (cid:20) (cid:0) With (cid:16) > 0, the (cid:133)rst inequality is strict, so the liquidity requirement always binds and B = (cid:21)D. The capital requirement binds if and only if the last inequality is strict or, equivalently, if and only if R~D((cid:21))+ 1 g (D;L) < RE=(1 p). The solution satis(cid:133)es the 1 (cid:21) D (cid:0) (cid:0) zero-pro(cid:133)t condition: RL g (D;L) = (cid:13)RE=(1 p)+(1 (cid:13)) R~D((cid:21))+(1 (cid:21)) 1g (D;L) L (cid:0) D (cid:0) (cid:0) (cid:0) f (cid:0) g 70If the capital requirement and the liquidity constraint are both binding for each problem, then h = g((cid:26) ;1) g((cid:26);1) (using Euler(cid:146)s theorem). As (cid:21)<w, (cid:26)<(cid:26) , so h 0 in this case. w (cid:0) w (cid:21) 57

resulting in (cid:25)B = 0. Two cases are possible: 1. If RL g ((cid:26);1) RE=(1 p), then the capital requirement is slack, and L (cid:0) (cid:21) (cid:0) RL g (D;L) = RE=(1 p) = R~D ((cid:21))+(1 (cid:21)) 1g (D;L) (cid:0) L (cid:0) (cid:0) (cid:0) D 2. If RL g ((cid:26);1) < RE=(1 p), then the capital requirement binds, so E = (cid:13)L, and L (cid:0) (cid:0) RL= (cid:13)RE=(1 p)+(1 (cid:13))R~D ((cid:21))+g((cid:26);1) (cid:0) (cid:0) Risk choices with (cid:21) < w Credit risk choice, part 3 From lemma 1, (cid:27) = 0 if and only if (cid:30) (cid:27)(cid:22) r ( RL + " (cid:20) (cid:17) RB(B=L) RD(D=L) g(D=L;1)), and recall that, in that case, (cid:133)nite solutions satisfy (cid:0) (cid:0) (cid:25)B = rL=RE E = 0 and (cid:25)B = (1 p)rL=RE E = 0 (see prop. 7 and 8). B wD liq:risk j (cid:21) (cid:0) j (cid:0) (cid:0) Hence: If (cid:16) 0, so that (cid:25)B = (cid:25)B = 0, then r = RE(E=L) RE(cid:13), so (cid:30) (cid:27)(cid:22) (cid:13)RE (cid:21)<w B wD " (cid:15) (cid:20) j j (cid:21) (cid:21) (cid:20) is a su¢ cient condition for no excessive credit risk taking (and is necessary if the capital requirement binds). If (cid:16) > 0, so that (cid:25)B = (cid:25)B = 0, then r = REE=((1 p)L) RE(cid:13)=(1 p), (cid:21)<w liq:risk (cid:15) j j (cid:0) (cid:21) (cid:0) so (cid:30) (cid:27)(cid:22)(1 p) (cid:13)RE is a su¢ cient condition for no excessive credit risk taking (and " (cid:0) (cid:20) is necessary if the capital requirement binds). Interestingly, a (slightly) lower level of capital requirement is su¢ cient to deter excessive risk taking if the optimal choice involves liquidity risk, taking RE and (cid:27)(cid:22) as given. More importantly, (cid:30) (cid:27)(cid:22) (cid:13)RE is always su¢ cient, even if (cid:21) < w. Having dealt with the case " (cid:20) (cid:21) w in Appendix A.2, this concludes the proof of proposition 1. (cid:21) Liquidity risk choice Combining proposition 1 with lemma 3 yields proposition 4 in the main text. QED. Appendix B. Notes to Figure 1 Figure 1 is based on the following measurements and assumptions. These are not used elsewhere in the paper, except that section 7 discusses the BCBS (2010) estimates further (and the welfare costs measurements are obtained from section 6, as noted). 58

Object Value Basis/Explanation Gross marginal welfare cost of liquidity requirement 0:0031 Measured in section 6 ((cid:23) ) LIQ capital requirement 0:0171 Measured in section 6 ((cid:23) ) CAP Macroeconomic cost of liquidity risk taking 0:23% BCBS (2010), bene(cid:133)t of liquidity req. credit risk taking 0:57% BCBS (2010), bene(cid:133)t of capital req. Stressed withdrawals (w) 0:1 Assumption Capital requirement threshold for no excessive credit risk 0:08 Assumption; equals (cid:30) (cid:27)(cid:22)=RE (proposition 1) " no excessive liquidity risk 0:13 Assumption; (cid:16) = 0 with (cid:21) = 0 (prop. 4) Except for w, all objects depend on multiple parameters (and, in some cases, the functional forms of u(:) and g(:)). For example, the (cid:133)rst capital requirement threshold depends on the (cid:147)value-at-risk(cid:148)of excessively risky loans ((cid:30) ) and the quality of bank supervision (indexed " by (cid:27)(cid:22)); the macroeconomic cost of credit risk taking depends on resolution costs ( ), Sol the distribution of credit risk (F ), and other parameters. As the values of many of these " parameters are di¢ cult to know or estimate, the (cid:133)gure instead relies on measurements of, or assumptions regarding, the values of the 7 objects listed above. A large number of combinations of underlying parameter values can be consistent with these choices. The avoidance of the macroeconomic costs associated with excessive risk provides the bene(cid:133)ts to regulation in the model. The (cid:133)gure uses existing estimates of the bene(cid:133)ts of capital and liquidity requirements from BCBS (2010), which estimates those bene(cid:133)ts as a reduction in the probability of a (cid:133)nancial crisis due to stricter regulation times the loss in output conditional on a crisis. The numbers shown above are expressed as a percent of GDP and are obtained from Table 8 of the BCBS report under (cid:145)no permanent output losses from crises(cid:146).71 The discussion in section 7 also uses these estimates, as well as those reported under (cid:145)moderate permanent e⁄ect(cid:146)in the same table. The slope of the line segments in the (cid:133)gure equals the negative of the gross marginal 71This yields the smallest estimate of bene(cid:133)ts; estimates that include some permanent output losses are substantially larger, and using those would make the jumps in the charts so large that it would be hard to discern the negative slope outside the jumps. For the liquidity requirement, gross bene(cid:133)ts are calculated from the entries in Table 8 as net bene(cid:133)ts plus expected costs at the baseline capital requirement (7%) withtheliquidityrequirementmetminusthatsamenumberwithouttheliquidityrequirementmet. Forthe capital requirement, gross bene(cid:133)ts of deterring excessive credit risk (i.e. the (cid:133)rst threshold) are calculated asnetbene(cid:133)tsplusexpectedcostsatthehighestcapitalrequirement(15%)minusthatsamenumberatthe baseline, net of the bene(cid:133)ts attributed to the liquidity requirement. 59

welfarecostofthecapitalrequirementasmeasuredinsection6(usingg = 0:0122). Welfare D is expressed in consumption equivalents and is normalized to 100 for (cid:13) = 0:08 and (cid:21) = w: Appendix C. Equilibrium and Planner(cid:146)s Problem Competitive equilibrium As explained, we focus on the case that regulation satis(cid:133)es (cid:21) w and (cid:13) (cid:30) (cid:27)(cid:22)=RE, so " (cid:21) (cid:21) that 1 = 0, (cid:27) = 0, and F ( r =(cid:27) ) = 0. Inserting this into (18), and combining f Bt<wDt g t " (cid:0) t t with (1), (2), (3), (8), (13), (14) (15), (17), and proposition 3, the resulting equilibrium allocation can be characterized in terms of a dynamic system in (K ;c ) with RE, L , d , t t t t t and b as auxiliary variables: t K = F(K ;1)+(1 (cid:14))K c g(d ;L ) T (35) t+1 t t t t t (cid:0) (cid:0) (cid:0) (cid:0) RE = ((cid:12)u (c ;d ;b )=u (c ;d ;b )) 1 (36) t c t t t c t 1 t 1 t 1 (cid:0) (cid:0) (cid:0) (cid:0) F (K ;1)+1 (cid:14) = RL = RE (cid:1) (c;d ;b ;L ) (37) K t t t K t t t (cid:0) (cid:0) with the spread (cid:1) = RE RL de(cid:133)ned by:72 K (cid:0) u (c ;d ;b ) u (c ;d ;b ) d t t t b t t t (cid:1) (c ;d ;b ;L ) (cid:26) g (d ;L ) (cid:21) g (d ;L ) (38) K t t t t D t t L t t (cid:17) u (c ;d ;b ) (cid:0) (cid:0) u (c ;d ;b ) (cid:0) c t t t c t t t (cid:18) (cid:19) and where L , d , and b are jointly determined by the following equilibrium versions of the t t t (cid:133)rms(cid:146)and banks(cid:146)complementary slackness conditions: If (cid:1) (c ;d ;b ;K ) > 0, then L = K ; else (cid:1) (c ;d ;b ;L ) = 0 (39) K t t t t t t K t t t t If (cid:1) (c ;d ;b ;L ) > g (d ;L ), then d = (1 (cid:13))L +B(cid:22) b ; (40) K t t t t L t t t t t (cid:0) (cid:0) (cid:0) else g (d ;L ) = (cid:1) (c ;d ;b ;L ) = 0 and (d B(cid:22) +b )=(1 (cid:13)) L K L t t K t t t t t(cid:0) t (cid:0) (cid:20) t(cid:20) t If (cid:1) (c ;d ;b ;L ) > 0, then B(cid:22) b = (cid:21)d ; else (cid:1) (c ;d ;b ;L ) = 0 (41) B t t t t t t B t t t t (cid:0) with the wedge (cid:1) = RD +g RB given by: B D (cid:0) u (c ;d ;b ) u (c ;d ;b ) b t t t d t t t (cid:1) (c ;d ;b ;L ) +g (d ;L ) (42) B t t t t D t t (cid:17) u (c ;d ;b ) (cid:0) u (c ;d ;b ) c t t t c t t t 72The expression for (cid:1) follows from (11) and (8): RE RL = (1 (cid:13))(RE R~D((cid:21))) (cid:26)g (D;L) K D (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) g (D;L)=(cid:26)(RE RD) (cid:21)(cid:26)(RE RB) (cid:26)g (D;L) g (D;L). L D L (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 60

The (cid:133)rst and second equations restate, respectively, the social resource constraint with (cid:27) = 0 and (cid:21) w and the household(cid:146)s intertemporal optimality condition (1), which de- (cid:21) termines the required return on (riskless) equity. The marginal product of capital is equal to the lending rate, which can be below the equity return, as acknowledged by (37). The spread between the return on equity and then lending rate, (cid:1) , is given by (38), which is K the equilibrium version of the bank(cid:146)s zero pro(cid:133)t condition (11). A pure bank (cid:133)nance equilibrium obtains when RL < RE, which implies L = K (see (15)). In equilibrium, this requires (cid:1) (c ;d ;b ;K ) > 0, as stated in (39). Such an K t t t t equilibrium is always characterized by a binding capital requirement, which is implied by RL < RE (see proposition 3). A pure bank (cid:133)nance with binding capital and liquidity requirementsisanequilibriumif(cid:1) (c;(cid:26)K;B(cid:22) (cid:21)(cid:26)K;K) > 0and(cid:1) (c;(cid:26)K;B(cid:22) (cid:21)(cid:26)K;K) > K B (cid:0) (cid:0) 0. A pure bank (cid:133)nance equilibrium with a nonbinding liquidity requirement (and binding capital requirement) occurs if instead (cid:1) (c;(cid:26)K;B(cid:22) (cid:21)(cid:26)K;K) 0. In that case, bond B (cid:0) (cid:20) holdingsbybanksandbyhouseholdsaredeterminedbytheequilibriumcondition(cid:1) (c;(1 B (cid:0) (cid:13))K +B(cid:22) b;b;K) = 0 (see (41)). (cid:0) When (cid:1) (c;d;b;K) < 0, the equilibrium is characterized by mixed (cid:133)nance (L < K), K with RE = RL and the relative size of the banking sector determined endogenously by (cid:1) (c ;d ;b ;L ) = 0(see(39)). Underthiscondition,thecapitalandliquidityrequirements K t t t t can each be slack or binding, according to conditions (40) and (41), respectively.73 Thus, all four cases listed in proposition 3 are theoretically possible as part of a mixed (cid:133)nance equilibrium. Equivalence of the planner(cid:146)s problem and the competitive equilibrium The Lagrangian and (cid:133)rst-order conditions to the constrained planner(cid:146)s problem in (21) are: = max 1 (cid:12)t u(c ;d ;b )+! sp [F(K ;1)+(1 (cid:14))K c K g(d ;L ) T] t t t t t t t t+1 t t L f ct;dt;bt;Lt;Kt+1 g1t=0 t=0 f (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) X +(cid:3) sp [B(cid:22) b (cid:21)d ]+(cid:31) sp [(1 (cid:13))L +B(cid:22) b d ]+(cid:22) sp [K L ] t t t t t t t t t t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) g 73An equilibrium with a slack capital requirement requires that (cid:1) (c ;d ;b ;L )+g (d ;L )=0. Since K t t t t L t t (cid:1) 0 in equilibrium, this requires that g =0 as indicated by (40), which is ruled if at least some of the K L (cid:21) non-interest costs are due to lending. Note also that with g > 0, an equilibrium with banks requires that u (cid:21)u > 0. Otherwise, we would have RL > RE (see (38)), and the demand for banks loans would be d b (cid:0) in(cid:133)nitely negative (see (15)). 61

sp (c) u (c ;d ;b ) = ! c t t t t sp sp sp (d) u (c ;d ;b ) = ! g (d ;L )+(cid:3) (cid:21)+(cid:31) d t t t t D t t t t sp sp (b) u (c ;d ;b ) = (cid:3) +(cid:31) b t t t t t sp sp sp (L) (cid:31) (1 (cid:13)) = ! g (d ;L )+(cid:22) t t L t t t (cid:0) (K) ! sp [F (K ;1)+1 (cid:14)] = (cid:12) 1! sp (cid:22) sp t K t (cid:0) (cid:0) t 1(cid:0) t (cid:0) with (cid:3) sp 0, (cid:3) sp [B(cid:22) b (cid:21)d ] = 0, (cid:31) sp 0, (cid:31) sp [(1 (cid:13))L +B(cid:22) b d ] = 0, (cid:22) sp 0, and t t t t t t t t t t (cid:21) (cid:0) (cid:0) (cid:21) (cid:0) (cid:0) (cid:0) (cid:21) sp (cid:22) [K L ] = 0. t t t (cid:0) Subtract (cid:21) times the (cid:133)rst-order condition with respect to bonds (FOC (b)) from FOC sp sp (d) to obtain u (cid:21) u = (1 (cid:21))(cid:31) ! g (omitting arguments for brevity). Solving for d b t t D (cid:0) (cid:0) (cid:0) sp (cid:31) and inserting the result into FOC (L) and using FOC (c) yields: t sp (cid:22) u (c ;d ;b ) u (c ;d ;b ) t = (cid:26) d t t t (cid:21) b t t t g (d ;L ) g (d ;L ) (cid:1) (c ;d ;b ;L ) ! sp u (c ;d ;b ) (cid:0) u (c ;d ;b ) (cid:0) D t t (cid:0) L t t (cid:17) K t t t t t (cid:18) c t t t c t t t (cid:19) (Recall that (cid:26) = (1 (cid:13))=(1 (cid:21)).) Inserting this into FOC (K) and using FOC (c) yields: (cid:0) (cid:0) u (c ;d ;b ) F K (K t ;1)+1 (cid:14) = (cid:12) (cid:0) 1 c t (cid:0) 1 t (cid:0) 1 t (cid:0) 1 (cid:1) K (c t ;b t ;d t ;L t ) (cid:0) u (c ;d ;b ) (cid:0) c t t t This replicates equations (36), (37) and (38) in the characterization of the decentralized sp sp sp equilibrium. Furthermore, (39) follows from (cid:1) (c ;d ;b ;L ) = (cid:22) =! , ! = u > 0, K t t t t t t t c sp sp (cid:22) 0, and (cid:22) [K L ] = 0. t t t t (cid:21) (cid:0) sp sp sp sp sp sp Since (cid:22) +! g = (cid:31) (1 (cid:13)) (from FOC(L)), (cid:22) 0, ! > 0, g 0, (cid:31) 0 and t t L t t t L t (cid:0) (cid:21) (cid:21) (cid:21) (cid:31) sp [(1 (cid:13))L +B(cid:22) b d ] = 0, it follows that (cid:31) sp > 0 and d = (1 (cid:13))L +B(cid:22) b if t t t t t t t t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:22) sp > 0 or if g > 0 (or both); otherwise (cid:31) sp = 0 and d (1 (cid:13))L +B(cid:22) b , a result that t L t t t t (cid:20) (cid:0) (cid:0) is equivalent to (40) in the decentralized equilibrium. Taking the di⁄erence between FOC(b) and FOC(d) yields sp (cid:3) u (c ;d ;b ) u (c ;d ;b ) (1 (cid:21)) t = b t t t d t t t +g (d ;L ) (cid:1) (c ;d ;b ;L ) (43) (cid:0) ! sp u (c ;d ;b ) (cid:0) u (c ;d ;b ) D t t (cid:17) B t t t t t c t t t c t t t sp sp sp the expression in (42). (41) follows from (cid:1) (c ;d ;b ;L ) = (1 (cid:21))(cid:3) =! , ! = u > 0, B t t t t t t t c (cid:0) (cid:3) sp 0, and (cid:3) sp [B(cid:22) b (cid:21)d ] = 0: Finally, equation (35) in the characterization of the t t t t (cid:21) (cid:0) (cid:0) decentralized equilibrium is included as one of the constraints of the planner(cid:146)s problem. Collectingtheseresults,itisapparentthattheallocationsofK ,c ,b ,d andL implied t t t t t by the planner(cid:146)s problem are identical to those of the decentralized equilibrium summarized in equations (35)-(42). Hence, the constrained social planner(cid:146)s problem replicates the decentralized equilibrium if (cid:21) w and (5) holds for all t 0 in that equilibrium, so that (cid:21) (cid:21) (cid:27) = 0. Moreover, under those conditions, welfare equals V ((cid:18)), as de(cid:133)ned in (21). t 0 62

Proof of proposition 5 Call the current period 0: Using the envelope theorem, the marginal e⁄ect on welfare of raising the liquidity requirement (cid:21) is: @V ((cid:18)) 0 = 1 (cid:12)t(cid:3) sp d @(cid:21) (cid:0) t=0 t t X d = 1 (cid:12)t u (c ;d ;b ) u (c ;d ;b )+u (c ;d ;b )g (d ;L ) t b t t t d t t t c t t t D t t (cid:0) t=0 f (cid:0) g1 (cid:21) (cid:0) X (see(43)). Sincetheallocationsofc , d , b andL areidenticaltothoseofthedecentralized t t t t equilibrium, their equilibrium values can be used. Moreover, in that equilibrium, we have, by taking the di⁄erence between the household(cid:146)s (cid:133)rst-order conditions (2) and (3), u (c ;d ;b ) u (c ;d ;b ) = u (c ;d ;b )(RD RB) b t t t d t t t c t t t t t (cid:0) (cid:0) Thus, with the assumption that the economy is in steady state in period 0, @V ((cid:18)) u (c ;d ;b )(RD RB +g (d ;L ))d 0 = c 0 0 0 0 (cid:0) 0 D 0 0 0 @(cid:21) (cid:0) (1 (cid:12))(1 (cid:21)) (cid:0) (cid:0) As is standard, compare this to the welfare e⁄ect of a permanent change in consumption by a factor (1+(cid:23)), which equals, to a (cid:133)rst-order approximation, (cid:12)tu (c ;d ;b )c (cid:23), 1t=0 c t t t t or u (c ;d ;b )c (cid:23)=(1 (cid:12)) with a steady state reigning in period 0. Equating this to the c 0 0 0 0 P (cid:0) right-hand side of the previous equation yields proposition 5. QED. Proof of proposition 6 Call the current period 0: Using the envelope theorem, the marginal e⁄ect on welfare of raising (cid:13) is: @V ((cid:18)) 0 = 1 (cid:12)t(cid:31) sp L @(cid:13) (cid:0) t=0 t t X L = 1 (cid:12)t u (c ;d ;b ) (cid:21) u (c ;d ;b ) u (c ;d ;b )g (d ;L ) t d t t t b t t t c t t t D t t (cid:0) t=0 f (cid:0) (cid:0) g1 (cid:21) (cid:0) X where the second equality follows from the planner(cid:146)s (cid:133)rst-order conditions for bonds, deposits and consumption. Since the allocations of c , d ;b and L are identical to those t t t t of the decentralized equilibrium, their equilibrium values can be used. Moreover, in that equilibrium, we have, from the household(cid:146)s (cid:133)rst-order conditions (2) and (3), u (c ;d ;b ) (cid:21)u (c ;d ;b ) = u (c ;d ;b )(RE RD (cid:21)(RE RB)) d t t t b t t t c t t t t t t t (cid:0) (cid:0) (cid:0) (cid:0) = u (c ;d ;b )(1 (cid:21))(RE R~D((cid:21))) c t t t t t (cid:0) (cid:0) Hence, @V ((cid:18)) g (d ;L ) 0 = 1 (cid:12)tu (c ;d ;b )(RE R~D((cid:21)) D t t )L @(cid:13) (cid:0) t=0 c t t t t (cid:0) t (cid:0) 1 (cid:21) t (cid:0) X 63

With the assumption that the economy is in steady state in period 0, @V ((cid:18)) g (d ;L ) 0 = (1 (cid:12)) 1u (c ;d ;b )(RE R~D((cid:21)) D t t )L @(cid:13) (cid:0) (cid:0) (cid:0) c 0 0 0 0 (cid:0) 0 (cid:0) 1 (cid:21) 0 (cid:0) With a steady state in period 0, the welfare e⁄ect of a permanent change in consumption by a factor (1+(cid:23)) equals, to a (cid:133)rst-order approximation, u (c ;d ;b )c (cid:23)=(1 (cid:12)). Equating c 0 0 0 0 (cid:0) this to the right-hand side of the previous equation yields the proposition. QED. 64